ABSTRACTS...Special Session 87 Direct, Inverse and Control Problems for Di erential Equations 295...

11th AIMS International Conferenceon Dynamical Systems, Differential Equations and ApplicationsFriday, July 1 - Tuesday, July 5, 2016Hyatt Regency OrlandoOrlando, FL, USA

American Institute of Mathematical Sciences

AIMS

Orlando, FL 201

6

11th Co

nference

ABSTRACTSAIMS

Department of Mathematics & StatisticsUniversity of North Carolina Wilmington

The 11th AIMS Conference onDynamical Systems,Di↵erential Equations and Applications

July 1 – July 5, 2016Orlando, Florida, USA

ABSTRACTS

Organizers:

The American Institute of Mathematical SciencesThe University of North Carolina Wilmington

Committees

Scientific Committee

Shouchuan Hu (chair)Jerry Bona William Bray Alberto BressanGunduz Caginalp Danielle Hilhorst Peter LaxAlain Miranville Roger Temam Enrico ValdinociMarcelo Viana

Organizing Committee

Xin Lu (chair)Yaw Chang Wei Feng Michael FreezeJonathan Rowell Beth Casper

Global Organizing Committee

Shouchuan Hu (chair)Michael Barnsley Hakima Bessaih Jaeyoung ByronTomas Caraballo Alessandro Carlotto Marcelo Moreira CavalcantiGianluca Crippa Diogo Aguiar Gomes John R. GraefYi Li Xin Lu Gianni Dal MasoDorina I. Mitrea Sarka Necasova Mitsuharu OtaniM. A. Peletier Benedetto Piccoli Jie ShenJunping Shi Peter Takac Yuwen WangGail Wolkowicz

Table of Contents

Invited Plenary Lectures 1

Special Sessions

Special Session 01 Nonlinearity in Climate and the Geosciences, A Special Session HonoringPeter D. Lax

8

Special Session 02 Emergence and Dynamics of Patterns in Nonlinear Partial Di↵erential Equa-tions

12

Special Session 03 Nonlinear Evolution PDEs, Interfaces and Applications 19Special Session 04 Classical and Geophysical Fluid Dynamics 24Special Session 05 Spatial and Evolutionary Aspects in Ecology and Epidemiology 28Special Session 06 Numerical Approximation of Fractional and Integral Di↵erential Equations 33Special Session 08 New Trends in Calculus of Variations and Partial Di↵erential Equations 36Special Session 09 Stochastic Modeling in Fluid Dynamics: Theory and Approximation 39Special Session 10 Complex Systems and Nonlinear Dynamics 42Special Session 11 PDEs with Applications in Biology, Fluid Mechanics and Material Sciences 44Special Session 12 Propagation Phenomena in Reaction-Di↵usion Systems 47Special Session 13 Chemotactic Cross-di↵usion in Complex Frameworks 50Special Session 14 Nonlinear Evolution Equations and Related Topics 54Special Session 15 Special Session on Monotone Dynamical Systems and Applications 58Special Session 16 Dissipative Systems and Applications 62Special Session 17 Quantitative Geometric and Functional Inequalities and New Trends in Non-

linear PDEs68

Special Session 18 Tra�c Flow Models and Their Application in Tra�c Engineering 71Special Session 19 Modern Applications of Mathematical and Computational Sciences 74Special Session 20 Models for Treatment of Prostate Cancer 78Special Session 21 Bifurcations and Asymptotic Analysis of Solutions of Nonlinear Models 80Special Session 22 Dynamics and Games 85Special Session 23 Numerical Methods for Phase-field Models 87Special Session 24 SPDEs/SDEs and Stochastic Systems with Control/Optimization and Ap-

plications90

Special Session 25 Applied Analysis and Dynamics in Engineering and Sciences 93Special Session 26 Hamiltonian Systems and the Planetary Problem 95Special Session 27 Advances in the Mathematical Modeling of Failure Phenomena and Inter-

faces in Materials98

Special Session 28 Recent Developments Related to Conservation Laws and Hamilton-JacobiEquations

101

Special Session 29 Advances in Theory and Application of Reaction Di↵usion Models 105Special Session 30 High Order Numerical Methods for Partial Di↵erential Equations 109Special Session 31 Celestial Mechanics and Beyond 112Special Session 32 Global or/and Blowup Solutions for Nonlinear Evolution Equations and

Their Applications115

Special Session 33 Nonlinear Waves in Dispersive Equations 119Special Session 34 Di↵erential Equations and Applications to Biological Models 123Special Session 35 Control and Optimization Theory for Partial Di↵erential Equations 126Special Session 36 New Trends in Nonlinear Partial Di↵erential Equations and Applications 129Special Session 37 Recent Advances in Dynamical Systems with Applications to Ecology and

Epidemiology132

Special Session 38 Evolution Equations and Integrable Systems 136Special Session 40 Polymer Dynamics Models and Applications to Neurodegenerative Disease 140Special Session 41 Stochastic Partial Di↵erential Equations 143Special Session 42 Dynamics of Evolution Equations in Viscoelasticity and Thermoelasticity 147Special Session 43 Long Time Dynamics, Numerical Analysis and Control of Evolutionary Sys-

tems149

Special Session 44 Fractal Geometry, Dynamical Systems, and Their Applications 153Special Session 45 Nonlinear Waves and Singularities in Optical and Hydrodynamic Systems 159Special Session 47 Mathematical Contribution Towards the Understanding of the Dynamics of

the 2014 Ebola Epidemic in West Africa163

Special Session 48 Uncertainty Quantification in Dynamical Systems 165Special Session 49 Recent Advances of Di↵erential Equations with Applications in Life Sciences 167

Special Session 50 Transition Dynamics of Parabolic Type Equations 172Special Session 51 Advances in Population Dynamics and Epidemiology 174Special Session 52 Function Spaces and Inequalities 178Special Session 53 Interface Dynamics and Transport Phenomena 181Special Session 54 Nonlinear PDEs and Variational Methods 184Special Session 56 Junior Session on Nonlinear Hyperbolic Equations and Related Topics 186Special Session 57 Lie Symmetries, Conservation Laws and Other Approaches in Solving Non-

linear Di↵erential Equations189

Special Session 58 Qualitative Properties of Nonlinear Di↵erential Equations of Elliptic andParabolic Type

196

Special Session 59 Mathematical Models of Cell Motility and Cancer Progression in Microen-vironment: Design, Experiments, Mathematical Framework, and HypothesisTest

199

Special Session 60 Infinite-Dimensional Dynamical Systems from Di↵erential Equations underSingular Perturbations

204

Special Session 61 Recent Trends in Navier-Stokes Equations and Related Problems 208Special Session 62 Imaging Methods in Coupled Physics Models 212Special Session 63 Topological Methods for Nonlinear Boundary and Initial Value Problems 215Special Session 64 Dynamics of Evolutionary Equations in the Applied Sciences 219Special Session 65 On Singular Problems Related to Distance Functions and Very Weak Solu-

tions223

Special Session 66 Mathematical Oncology 225Special Session 67 Applications of Mathematical Modeling in Developmental and Cell Biology 229Special Session 68 Rate-Dependent and Rate-Independent Evolution Problems in Continuum

Mechanics: Analytical and Numerical Aspects233

Special Session 69 Dispersive E↵ects in Nonlinear PDEs 237Special Session 70 Vortex Dynamics and Geometry: Analysis, Computations and Applications 239Special Session 71 Elliptic Equations and Systems, and Concentration Phenomena 242Special Session 72 Optimal Control and its Applications 244Special Session 73 Mathematical Modeling and Computations 248Special Session 74 Infinite Dimensional Stochastic Systems and Applications 250Special Session 75 Recent Trends on PDEs Driven by Gaussian Processes with Applications 254Special Session 76 Advances in the Numerical Solution of Nonlinear Evolution Equations 257Special Session 77 Delay Di↵erential Equations with State-Dependent Delays and their Appli-

cations259

Special Session 78 Advances in Analysis of Mathematical Problems arising from Materials andBiological Science

263

Special Session 79 Nonlinear Di↵erential Dynamic systems in Fluid Dynamics 268Special Session 80 Inverse Limits in Dynamical Systems 270Special Session 81 Advances in Computer Assisted Proofs for Dynamical Systems and Di↵eren-

tial Equations274

Special Session 82 Numerical Simulations and Computations for Stochastic Dynamics 278Special Session 83 Dynamical Systems and their Applications 281Special Session 84 Recent Advances and Challenges in Coastal Dynamics 284Special Session 85 Di↵erential Equation Modeling and Analysis for Brain and other Complex

bio-systems286

Special Session 86 Pattern Formation and Recognition in Structured Information and BiologicalSystems with External Forcing

293

Special Session 87 Direct, Inverse and Control Problems for Di↵erential Equations 295Special Session 88 Data Assimilation and Nonlinear Filtering 297Special Session 89 Dynamics and Computation 300Special Session 91 Harmonic Analysis and Partial Di↵erential Equations 303Special Session 92 Variational, Topological and Set-Valued Methods for Nonlinear Problems 306Special Session 93 Nonlinear Dispersive Equations and Integrable Systems 309Special Session 94 Infinite Dimensional Dynamics in Analysis 313

Special Session 95 PDEs from Gauge Field Theories and Mathematical Physics 316Special Session 96 Complex Biological and Ecological Systems 319Special Session 97 Qualitative and Quantitative Techniques for Di↵erential Equations arising in

Economics, Finance and Natural Sciences323

Special Session 98 Inverse Problems and Imaging and Their Applications 331Special Session 99 Theory and Applications of Boundary-Domain Integral and Pseudodi↵eren-

tial Operators333

Special Session 100 Nonstandard Analysis, Quantizations and Singular Perturbations 336Special Session 101 Randomness Meets Life 337Special Session 102 Recent Developments of High-Order Numerical Methods 341Special Session 103 Mixing in Dynamical Systems: Theory, Modelling, and Applications, from

Micro- to Geophysical Scales344

Special Session 104 Nonlinear Elliptic Equations and Fractional Laplacian 350Special Session 105 Recent Advances in Computational PDEs and their Applications 353Special Session 106 Nonlinear Waves: Coherent Structures and Complex Dynamics 356Special Session 107 Analysis of Nonlinear Dispersive Wave Equations and Integrable Systems 358Special Session 108 New Developments in Porous Media 361Special Session 109 Applied and Integrable Nonlinear PDEs 365Special Session 110 Computational and Mathematical Methods for Complex Biological Systems 368Special Session 111 Geometric Methods in Mechanics and Di↵erential Equations 371Special Session 113 Inverse Problems, Variational Inequalities, and Applications 375Special Session 114 Uncertainty Quantification 377Special Session 117 Partial Di↵erential Equations from Fluid Dynamics 379Special Session 118 Mean Field Games and Applications 381Special Session 119 Geometric Functional Inequalities and Application to PDEs 385Special Session 120 Global Bifurcations and Complex Dynamics 388Special Session 121 Recent Advancements in Computational Methods Involving Implicit or Non-

Parametric Interfaces389

Special Session 122 Variational Convergence and Degeneracies in PDEs: Fractal Domains, Com-posite Media, Dynamical Boundary Conditions

391

Contributed SessionsContributed Session 1 ODEs and Applications 394Contributed Session 2 PDEs and Applications 399Contributed Session 3 Modeling, Math Biology and Math Finance 406Contributed Session 4 Control and Optimization 412Contributed Session 5 Scientific Computation and Numerical Algorithms 415Contributed Session 6 Bifurcation and Chaotic Dynamics 419

Poster Session 421

Student Paper Competition Session 426

List of Contributors 428

Invited Plenary Lectures

Suncica CanicUniversity of Houston, USAhttp://www.math.uh.edu/⇠canic/

Suncica Canic earned her Ph.D. in 1992 in the area of nonlinear hyperbolicconservation laws from the Department of Applied Mathematics and Statisticsat SUNY Stony Brook. Upon her move to the University of Houston in 1999,she began collaborating with several medical specialists at the Texas MedicalCenter in Houston on problems related to cardiovascular treatment and diag-nosis. She was honored for her research by the National Science Foundationas Distinguished MPS Lecturer in 2007, and received the US CongressionalRecognition for Top Women in Technology in 2006. Her research received

local and national media attention, and was featured in several publications by NSF, NIH, and AMS. Canicwas also invited to present a Congressional Briefing on Applied Mathematics, on Capitol Hill on December6th, 2011. She serves on the Board of Governors of the Institute for Mathematics and its Applications inMinneapolis, and was the Program Director of the SIAM Activity Group on Partial Di↵erential Equations.In 2014 she was elected Fellow of the Society for Industrial and Applied Mathematics. She is the only womanwho holds a prestigious Cullen Distinguished Professorship position at the University of Houston.

Fluid-Composite Structure Interaction and Blood Flow

AbstractFluid-structure interaction problems with composite structures arise in many applications. One exampleis the interaction between blood flow and arterial walls. Arterial walls are composed of several layers,each with di↵erent mechanical characteristics and thickness. No mathematical results exist so far thatanalyze existence of solutions to nonlinear, fluid-structure interaction problems in which the structureis composed of several layers. In this talk we summarize the main di�culties in studying this class ofproblems, and present a computational scheme based on which a proof of the existence of a weak solutionwas obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of thinfluid-structure interface with mass regularizes evolution of FSI solutions. Implications of our theoreticalresults on modeling the human cardiovascular system will be discussed.

This is a joint work with Boris Muha (University of Zagreb, Croatia), and with Martina Bukac (U of NotreDame, US). Numerical results with vascular stents were obtained with S. Deparis and D. Forti (EPFL,Switzerland).

Alessio FigalliThe University of Texas at Austin, USAhttp://www.ma.utexas.edu/users/figalli/

Figalli received his master degree in mathematics from the Scuola NormaleSuperiore di Pisa in 2006, and earned his doctorate in 2007 under the super-vision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa and CedricVillani at the Ecole Normale Superieure de Lyon. In 2007 he was appointedCharge de recherche at the French National Centre for Scientific Research,in 2008 he went to the Ecole polytechnique as Professeur Hadamard. He hasbeen a professor at University of Texas at Austin since 2009. Starting from2013 he holds the R. L. Moore Chair. Amongst his several recognitions, Fi-galli has won an EMS Prize in 2012, he has been awarded the Peccot-Vimont

Prize 2011 and Cours Peccot 2012 of the College de France, and has been appointed Nachdiplom Lecturerin 2014 at ETH Zurich. His main research interests include Partial Di↵erential Equations and Calculus ofVariations.

From Isoperimetry to Random Matrices

AbstractThe optimal transport problem consists in finding the cheapest way to transport a distribution of massfrom one place to another. Apart from its natural applications in economics, optimal transport mapsprovide “e�cient” changes of variables that have been used to investigate the stability of minimizers togeometric/functional inequalities. However, in some cases, optimal maps may not alway been the “right”choice and other changes of variables may be more suitable. For instance, this happens to be the case in thestudy of universality in random matrix theory. In this talk I’ll give an overview of these results.

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2 11th AIMS CONFERENCE – ABSTRACTS

Irene FonsecaCarnegie Mellon University, USAhttp://irenefonseca.weebly.com/

Irene Fonseca is an educator and researcher in applied mathematics and iscurrently the Director of Carnegie Mellon University’s Center for NonlinearAnalysis (CNA). In recognition for her contributions to the advancement ofresearch in her area of expertise, Irene Fonseca was bestowed a knighthoodin the Military Order of St. James (Grande Oficial da Ordem Militar de San-tiago da Espada) by the then-President of Portugal, Jorge Sampaio, in 1997.Irene Fonseca received the Mellon College of Science chair in Mathematics in2003 and was appointed to the rank of University Professor in 2014. In 2012,she was elected President of the Society for Industrial and Applied Mathemat-ics (SIAM), one of the largest organizations dedicated to mathematics andcomputational science in the world. Irene Fonseca was also named a Fellowof SIAM in 2009 and a Fellow of the American Mathematical Society in 2012.

Irene Fonseca’s recent work is focused on variational techniques as they apply to contemporary problemsin materials sciences and computer vision, including the mathematical study of ferroelectric and magneticmaterials, composites, thin structures, phase transitions, epitaxy and dislocations, and image segmentationand denoising in imaging science.

A Chromaticity-Brightness Model for Color Images Denoising in a Meyer’s “u + v” Framework

AbstractA variational model for imaging denoising aimed at restoring color images is proposed. The model combinesMeyer’s “u+v” decomposition with a chromaticity-brightness framework, and is expressed in terms of aminimization of energy integral functionals depending on a small parameter " > 0. The asymptotic behavioras " ! 0+ is characterized, and convergence of infima, almost minimizers, and energies are established. Inparticular, an integral representation of the lower semicontinuous envelope, with respect to the L1-norm, offunctionals with linear growth and defined for maps taking values on a compact manifold is provided.

Michael GhilEcole Normale Superieure, Paris, FranceUniversity of California Los Angeles, USAhttp://research.atmos.ucla.edu/tcd/GHIL/

Michael Ghil obtained his Ph.D. from New York University’s Courant Insti-tute of Mathematical Science with Peter D. Lax in 1975. He is a DistinguishedProfessor of Geosciences (emeritus) at the Ecole Normale Superieure, Paris,past Head of its Geosciences Department (2003–2009) and founder of its En-vironmental Research and Teaching Institute. He is also a DistinguishedResearch Professor at the University of California, Los Angeles, where he wasChair of the Department of Atmospheric Sciences (1988–1992) and Directorof the Institute of Geophysics and Planetary Physics (1992–2003). Ghil is afounder of theoretical climate dynamics, as presented in his Springer-Verlag

(1987) book with Steve Childress, as well as of advanced data assimilation methodology, as presented inthe Springer-Verlag (1981) book co-edited with Lennart Bengtsson and Erland Kallen. He has applied sys-tematically ideas and methods from dynamical systems theory to planetary-scale flows, atmospheric andoceanic. Ghil has used these methods to proceed from simple flows with high temporal regularity and spa-tial symmetry to the observed flows, with their complex behavior in space and time. His studies of climatevariability on many time scales have used a full hierarchy of models, from the simplest “toy” models all theway to atmospheric, oceanic and coupled general circulation models. Ghil has worked on Climate Dynamics,Dynamical and Complex Systems, Extreme Events, Numerical and Statistical Methods, and (most recently)Mathematical Economics. He is the author or editor of a dozen books and author or co-author of over300 research and review articles. Many of the latter can be found on the web site of his research groupat UCLA, http://www.atmos.ucla.edu/tcd/. His honors and awards include the L.F. Richardson Medal ofthe European Geosciences Union (EGU, 2004), the E.N. Lorenz Lecture of the American Geophysical Union(2005), a Plenary Lecture at the 7th International Congress on Industrial and Applied Mathematics (ICIAM2011), the Alfred Wegener Medal of the EGU (2012), and Membership in the Academia Europaea (1998).

INVITED PLENARY LECTURES 3

A Mathematical Theory of Climate Sensitivity or, A Tale of Deterministic and StochasticDynamical Systems

AbstractThe climate system is nonlinear, complex and variable on many scales of time and space. It is typicallystudied across a hierarchy of models from low-dimensional systems of ordinary di↵erential equations (ODEs)to infinite-dimensional systems of partial and functional di↵erential equations (PDEs and FDEs). Thetheory of di↵erentiable dynamical systems (DDS) has provided a road map for climbing this hierarchy andfor comparing theoretical results with observations. The climate system is also subject to time-dependentforcing, both natural and anthropogenic, e.g. volcanic eruptions and changing greenhouse gas concentrations.Hence increased attention has been paid recently to applications of the theory of non-autonomous and randomdynamical systems (NDS and RDS). This talk will review the road from the classical DDS applications tolow-dimensional ODE climate models to current e↵orts at applying NDS and RDS theory to non-autonomousFDE and stochastic PDE models. The debt of the lecturer and of his co-authors over the years to Peter D.Lax is immense, and a modest tribute will be paid to Peter’s contributions to pure and applied mathematics.

Martin HairerThe University of Warwick, UKhttp://www.hairer.org

Martin Hairer received his PhD in theoretical physics in 2001 from the Univer-sity of Geneva. He then moved to the University of Warwick where he becameAssociate Professor in 2004. In 2009, he was appointed at the Courant In-stitute in New York and then moved back to the University of Warwick asa full professor in 2010. His main areas of research are stochastic dynamics,stochastic analysis, and stochastic partial di↵erential equations. In recent

years, his emphasis was mainly on developing the theory of regularity structures which provides a robustframework in which to interpret large classes of stochastic PDEs whose mathematical meaning had so farbeen unclear. His work has been distinguished with a 2008 Whitehead prize, a 2013 Fermat prize and a2014 Fields medal.

Evolution on Random Loops

AbstractA “rubber band” constrained to remain on a manifold evolves by trying to shorten its length, eventuallysettling on some minimal closed geodesic, or collapsing entirely. It is natural to try to consider a noisyversion of such a model where each segment of the band gets pulled in random directions. Trying to buildsuch a model turns out to be surprisingly di�cult and generates a number of nice geometric insights, as wellas some beautiful algebraic and analytical objects. We will survey some of the main results obtained on theway to this construction.


Anatole KatokPenn State University, USAhttp://www.personal.psu.edu/axk29/

Entropy in Dynamical Systems: Complexity, Flexibility and Rigidity

Several interrelated concepts of entropy as well as closely related notions of Lyapunov characteristic ex-ponents play a central role in the modern theory of dynamical systems.Those notions give quantitativeexpression of the measure of exponential complexity present in a deterministic system. After a general re-view of those concepts and principal relations between them I will discuss results and open problems relatedto two complimentary phenomena. of flexibility and rigidity. The general paradigm of flexibility can berather vaguely formulated as follows:

Under properly understood general restrictions within a fixed class of smooth dynamical systems quanti-tative dynamical invariants take arbitrary values.

Precise calculations are possible only in very few cases, primarily of algebraic nature such as homogeneousor a�ne systems. Most known constructions are perturbative and hence at best would allow to cover a smallneighborhood of the values allowed by the model, or more often, not even that, since those models are often“extremal”. So establishing flexibility calls for non-perturbative or large perturbation constructions in largefamilies to cover possible values of invariants.On the other hand, there is the rigidity paradigm that is better developed. It has several aspects and inthe case of classical systems with discrete and continuos time one of them is related to these quantitativecharacteristics os exponential complexity:

Particular values of entropies or Lyapunov exponents or relations between those determine algebraic orsimilar models within a broad class of systems.

Rigidity becomes more common and even prevalent when one passes from classical systems to systems withmulti-dimensional time.

Wei-Ming NiUniversity of Minnesota, USAEast China Normal University, China

Wei-Ming Ni received his B.S. from National Taiwan University and Ph.D.from Courant Institute, New York University. He is currently on the facultyof University of Minnesota, and the Director of the Center for PDE at EastChina Normal University in Shanghai. His research interests are mainly inelliptic and/or parabolic equations/systems, including: Symmetry propertiesof solutions, solutions on entire space and their stability properties, peak

solutions (spiky Turing patterns), and interactions between spatial and/or temporal heterogeneities anddi↵usion.

From Logistic Equation to Lotka-Volterra Competition-Di↵usion System

AbstractIn this lecture I shall report some of the recent progress on the logistic equation and 2 ⇥ 2 Lotka-Volterracompetition-di↵usion system when spatial heterogeneity and/or temporal periodicity are present.

INVITED PLENARY LECTURES 5

Stan OsherUniversity of California Los Angeles, USAhttp://www.math.ucla.edu/⇠sjo/

Stanley Osher is a Professor of Mathematics, Computer Science, ChemicalEngineering and Electrical Engineering at UCLA. He is also an AssociateDirector of the NSF-funded Institute for Pure and Applied Mathematics atUCLA. He received his MS and PhD degrees in Mathematics from the CourantInstitute of NYU. Before joining the faculty at UCLA in 1977, he taught atSUNY Stony Brook, becoming professor in 1975. Professor Osher is one ofthe most highly cited researchers in both mathematics and computer scienceswith an h index of 100(according to google scholar). In recent years he hasaveraged one citation per hour. He has received numerous academic honorsand co-founded three successful companies, each based largely on his own(joint) research.

He has co-invented and/or co-developed the following widely used algorithms:

1. Essentially nonoscillatory (ENO), weighted essentially nonoscillatory (WENO) and other shock cap-turing schemes for hyperbolic systems of conservation laws and their analogues for Hamilton-Jacobiequations;

2. The level set method for capturing dynamic surface evolution;

3. Total variation and other partial di↵erential based methods for image processing;

4. Bregman iterative methods for L1 and related regularized problems which arise in compressive sensing,matrix completion, imaging and elsewhere;

5. Di↵usion generated motion by mean curvature and other threshold dynamics methods.

Professor Osher has been elected to the US National Academy of Science and the American Academy ofArts and Sciences. He was awarded the SIAM Pioneer Prize at the 2003 ICIAM conference and the Ralph E.Kleinman Prize in 2005. He was awarded honorary doctoral degrees by ENS Cachan, France, in 2006 and byHong Kong Baptist University in 2009. He is a SIAM and AMS Fellow. He gave a one hour plenary addressat the 2010 International Conference of Mathematicians. He also gave the John von Neumann Lecture atthe SIAM 2013 annual meeting. He is a Thomson-Reuters highly cited researcher-among the top 1% from2002-2012 in both Mathematics and Computer Science. In 2014 he received the Carl Friedrich Gauss Prizefrom the International Mathematics Union-this is regarded as the highest prize in applied mathematics.His current interests involve information science which includes optimization, image processing, compressedsensing and machine learning and applications of these techniques to the equations of physics, engineeringand elsewhere.

Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi (HJ) Equations Arisingin Control Theory and Elsewhere

AbstractIt is well known that certain HJ PDE’s play an important role in analyzing continuous dynamic games andcontrol theory problems. The cost of standard algorithms, and, in fact all PDE grid based approsimationsis exponential in the space dimension and time, with huge memory requirements. Here we propose and testmethods for solving a large class of HJ PDE relevant to optimal control without the use of grids or numericalapproximations. Rather we use rhe classical Hopf formulas for solving initial value problems for HJ PDE.We have noticed that if the Hamiltonian is convex and positively homogeneous of degree one that very fastmethods (related to those used in compressed sensing) exist to solve the resulting optimization problem.We seem to obtain methods which are polynomial in dimension. We can evaluate the solution in very highdimensions in between 10�4 and 10�8 seconds per evaluation on a laptop. The method requires very limitedmemory and is almost perfectly parallelizable. In addition, as a step often needed in this procedure, wehave developed a new and equally fast and e�cient method to find, in very high dimensions, the projectionof a point exterior to a compact set A onto A. We can also compute the distance to such sets much fasterthan fast marching or fast sweeping algorithms. The term “curse of dimensionality” was coined by RichardBellman in 1957 when he did his pioneering work on dynamic optimization.


Hal SmithArizona State University, USAhttp://math.la.asu.edu/⇠halsmith/

Hal L. Smith obtained his PhD in mathematics at the University of Iowa andis currently Professor of Mathematics at Arizona State University. He hasheld visiting professor positions at the University of Utah, Brown University,Georgia Institute of Technology, and the University of Minnesota. His re-search interests include nonlinear analysis, di↵erential equations, dynamicalsystems, and applications to the biological sciences.

Monotone Dynamical Systems: Reflections on New Advances & Applications

AbstractI o↵er some reflections on recent developments in a very select portion of the now vast subject of monotonedynamical systems. Continuous time dynamics generated by cooperative systems of ordinary di↵erentialequations, delay di↵erential equations, parabolic partial di↵erential equations, and control systems are themain focus. Results are included which the author feels have had a major impact in the applications. Theseinclude especially the theory of competition between two species or two teams and the theory of monotonecontrol systems.

Gang TianPeking University, Peoples Rep of China and Princeton University, USAhttp://bicmr.pku.edu.cn/⇠tian/http://www.math.princeton.edu/directory/gang-tian

Dr. Gang Tian has made fundamental contributions to geometric analysis,complex geometry and symplectic geometry. He did his undergraduate studyat Nanjing University in China, received his MS at Peking University andPhD at Harvard University. He is now a distinguished professor at Peking

University and an E. Higgins professor at Princeton University. Dr. Gang Tian established completely theexistence of Kahler-Einstein metrics on compact complex surfaces. He proved that the deformation of Calabi-Yau manifolds is unobstructed. Together with Ruan, he established a mathematical theory of the quantumcohomology and Gromov-Witten invariants. He was one of pioneers in constructing virtual cycles. WithLiu, he solved the Arnold Conjecture in the non-degenerate case . He introduced the K-stability which hasbeen further developed and has become a central notion in the theory of geometric stability. He initiated theAnalytical Minimal Model program through Kahler-Ricci flow, referred as the Song-Tian program. Togetherwith J. Morgan, amongst others, Dr. Gang Tian helped verifying the proof of the Poincare Conjecture andthe Geometrization Conjecture. In 2012, he gave a solution for the Yau-Tian-Donaldson Conjecture in thecase of Fano manifolds which has been a driving force in Kahler geometry for last few decades Dr. GangTian won Alan T. Waterman Award in 1994 and Veblen Prize in 1996. He spoke twice at the InternationalCongress of Mathematics in 1990 and 2002. He was elected to the National Academy of China in 2001 andthe American Academy of Arts and Science in 2004.

Einstein Equation and Kahler Geometry

AbstractIn this general talk, I will present some recent advances in Kahler geometry. The study of Kahler-Einsteinmetrics was initiated by E. Calabi in the 50’s. In the 70’s, Yau’s solution for the Calabi conjecture settles thecase when the scalar curvature is zero. Aubin and Yau solved the case when the scalar curvature is negative.Since then, it has been a very challenging problem to study the existence problem for Kahler-Einstein metricswith positive scalar curvature. Recently, a deep connection has been established between this existence anda variant of the geometric invariant theory I started in the 90s and advocated since then.


Special Session 1: Nonlinearity in Climate and the Geosciences, ASpecial Session Honoring Peter D. Lax

Michael Ghil, ENS, France and UCLA, USAMickael D. Chekroun, UCLA, USA

Shouhong Wang, Indiana University, USA

Dynamical systems have played a major role in the early development of modern, post-WWII climatedynamics, with the work of E.N. Lorenz, H.M. Stommel, G. Veronis and others. Today, this role extendsto other branches of the geosciences, and the emphasis is shifting from the use of autonomous dynamicalsystems to non-autonomous and random ones. This session will cover the applications of dynamical systems,in the broadest sense, to climate and the geosciences. The applications will span a hierarchy of models, fromthe simplest, autonomous systems of ordinary di↵erential equations to functional, partial and stochasticones. The session will be inspired by the elegance of Peter Lax’s contributions to many pertinent areas ofnonlinear mathematics, including but not restricted to numerical methods for evolution problems and thehighly nonlinear Korteweg-DeVries equation.

Rogue Waves

Jerry BonaUniversity of Illinois at Chicago, USAGustavo Ponce, Jean-Claude Saut, ChristofSparber

Rogue waves, or giant waves is the name given to acertain class of oceanic phenomena that involve sur-face waves which are large compared to man madeobjects such as ships or oil platforms. They seem tobe localized in both space and time. The lecture willfocus on one possible route to their generation.

Non-Markovian Reduced SDEs forMarkovian SPDEs

Mickael ChekrounUniversity of California, Los Angeles, USAHonghu Liu, Shouhong Wang

In this talk, a novel approach to deal with the pa-rameterization problem of the “small” spatial scalesby the “large” ones for stochastic partial di↵erentialequations (SPDEs), will be discussed. This approachrelies on stochastic parameterizing manifolds (PMs)which are random manifolds aiming to provide — in amean square sense — the optimal approximate small-scale parameterizations. Backward-forward systemswill be introduced to determine such PMs in practice.These auxiliary systems will be used for the analyticderivation of non-Markovian reduced stochastic dif-ferential equations (SDEs) with random coe�cients.It will be shown that these random coe�cients allowin certain circumstances for a striking restoring ofthe missing information due to the coarse-graining,namely to parameterize what is unresolved. Noise-induced transitions and stochastic bifurcations willserve as illustrations, including the case of additivenoise. This talk is based on a joint work with HonghuLiu and Shouhong Wang.

Discrete Stochastic Parametrizationfor Dynamical Systems

Alexandre ChorinUC Berkeley, USAFei Lu, Kevin Lin

Many problems in science are described by equationsthat cannot be solved in full because the equationsare either uncertain or overly complex, but whereone is interested in predicting only a small subset ofobserved variables. Under these conditions, it is nat-ural to formulate simple models for the variables ofinterest and correct them by terms inferred from theobservations. I will describe a fully discrete stochas-tic approach to this problem, which simplifies boththe inference from the observations and the solutionof the resulting equations. However, in the discreteapproach, like in any other, it is necessary to identifymodels that are parsimonious in computing e↵ort.I will present several ways of ensuring parsimony,based on ideas from di↵erential equations and frommachine learning. The examples include the Lorenz96 model and the Kuramoto-Sivashinski equation.

Detection and Tracking of CoherentStructures in Stochastic and Deter-ministic Models of the Atmosphereand Ocean

Gary FroylandUniversity of New South Wales, Australia

In complex, unsteady, geophysical flows, featuresthat persist on moderate timescales provide crucialinformation on dynamics and transport, which inturn is important for climate models. The non-autonomous and/or stochastic nature of many geo-physical models means that classical dynamical sys-tems approaches are inappropriate to diagnose coher-ent features in these flows. I will describe a generalmethod, based on global mixing minimisation, foridentifying coherent structures in non-autonomous,

SPECIAL SESSION 1 9

compressible, and possibly stochastic flows, with il-lustrations from atmosphere and ocean models in twoand three dimensions. I will discuss a second method,compatible with the first, but based on curve or sur-face minimisation, for deterministic flows.

The Wind-Driven Ocean Circula-tion: Applying Dynamical SystemsTheory to a Climate Problem

Michael GhilENS, Paris, and UCLA, France

The large-scale, near-surface flow of the mid-latitudeoceans is dominated by the presence of a larger, anti-cyclonic and a smaller, cyclonic gyre. The two gyresshare the eastward extension of western boundarycurrents, such as the Gulf Stream or Kuroshio, andare induced by the shear in the winds that cross therespective ocean basins.We study the low-frequency variability of this wind-driven, double-gyre circulation in mid-latitude oceanbasins, subject to time-constant, purely periodic andmore general forms of time-dependent wind stress.Both analytical and numerical methods of nonlineardynamics are applied in this study. Recent work hasfocused on the application of non-autonomous andrandom forcing to double-gyre models. We discussthe associated pullback and random attractors andthe non-uniqueness of the invariant measures that areobtained.This talk reflects collaborative work with a largeand still increasing number of people. Please visithttp://www.atmos.ucla.edu/tcd/ for their names, af-filiations, and respective contributions.

How to Uncover Coherent WaterMasses in an Unsteady Ocean?

George HallerETH Zurich, Switzerland

Transport by mesoscale oceanic eddies is broadlyviewed as important for climate modelling. The accu-rate assessment of this transport relies on the identi-fication of coherent water masses in otherwise turbu-lent velocity fields. Recent progress in nonlinear dy-namical systems enables such an identification by re-vealing analogues of generalized elliptic (or KAM) re-gions in non-autonomous, finite-time dynamical sys-tems. We show how these mathematical results pro-vide objective (i.e., frame-invariant) identification ofunsteady material vortices in the ocean. We illustratethe results by uncovering highly coherent Lagrangianeddies in two-dimensional satellite altimeter data andin three-dimensional numerical ocean models.

Books by and about Peter Lax

Reuben HershUniversity of New Mexico, USA

I will try to convince you to read books by and aboutPeter Lax.

Variational Principles for StochasticParameterisations in GeophysicalFluid Dynamics

Darryl HolmImperial College London, EnglandColin Cotter, Dan Crisan, Franco Flandoli

In recent work, Holm derived novel stochastic fluiddynamics equations aimed at geophysical fluid dy-namics applications by developing a new class ofstochastic variational principles. I will present someon-going research on the theoretical and numericalanalysis of the solutions of these nonlinear stochasticpartial di↵erential equations.

El Nino Dynamics: Its Beauty andChallenges

Fei-Fei JinUniversity of Hawaii at Manoa, USA

El Nino refers to major climate events that recur ev-ery few years, with ocean surface temperatures oftenreaching a few degrees Celsius warmer than normalover a vast area of the Tropical Pacific. These eventsreorganize the Earth’s weather and climate patterns,causing global environmental and socio-economic im-pacts. It was first recognized 50 years ago that ElNino results from an instability involving basin-scaledynamic interaction between two geophysical fluids,namely the atmosphere and oceans. In the last 30years, the dynamic system approach to the coupledatmosphere-ocean climate system has led to a muchdeeper understanding of this most influential, as wellas best observed, simulated, and now predicted cli-matic phenomenon. This approach has allowed theclimate dynamicist to unravel the basic mechanismsfor many key aspects of El Nino, including the natureof the instability that causes it, its phase locking withthe solar annual cycle, its striking asymmetry and itsfrequency cascade; it has also allowed us to uncoverEl Nino’s hidden mathematical beauty. The latterechoes Peter Lax’s statement that mathematics hasa mysterious unity which really connects seeminglydistinct parts, which is one of the glories of mathe-matics. In this talk honoring Peter Lax’s work andits impact on the climate sciences, I will present abrief review of the progress of El Nino dynamics andits remaining challenges.

On the Predictability Limit of Trop-ical Cyclone Intensity

Chanh KieuIndiana University, USA

Weather has long been projected to possess limitedpredictability due to the inherent chaotic nature ofthe atmosphere, which may result in an entirely dif-ferent state of the atmosphere after 2 weeks regard-less of how small initial errors are. Because of thecritical dependence of such a range of predictabilityon the underlying dynamics of atmospheric flows, a


natural question is how far in advance can we predicttropical cyclone (TC) intensity, given that the TC ro-tational dynamics is highly axisymmetric at the mesoscale ( 30-350 km)? In this study, we will examinethe growth of TC intensity errors at the maximumpotential intensity (MPI) limit in a phase space ofTC basic scales, using an axisymmetric TC model.It will be shown that there exists an MPI attractorat the maximum intensity limit that all TC orbitswill converge into, regardless of TC initial conditions.Direct estimation of the leading Lyapunov exponentshows that the MPI attractor is not only an attract-ing set but also chaotic in nature. This finding ofthe chaotic MPI attractor suggests an upper boundon the predictability limit of the TC intensity, whichprevents the 4-5 day absolute intensity errors in TCmodel forecasts from being reduced below a thresholdof 8-10 ms�1.

A Discrete-Time Approach toStochastic Model Reduction forChaotic Dynamical Systems

Kevin LinUniversity of Arizona, USAFei Lu, Alexandre Chorin

In computational modeling of complex dynamicalphenomena, it is often useful to be able to constructsimpler, reduced models that nevertheless capturekey dynamical features of interest. One well-studiedstrategy is to fit parametric families of stochasticmodels to data. In recent work, Chorin and Luproposed a novel, discrete-time approach that hascertain appealing features. I will report on applica-tions of this discrete-time approach to the Kuramoto-Sivashinsky PDE and other prototypical chaotic dy-namical systems, and discuss some of the issues thatarise and how they can be overcome.

Galerkin Approximations of Non-linear Delay Di↵erential EquationsRevisited

Honghu LiuVirginia Tech, USAMickael Chekroun, Michael Ghil, ShouhongWang

Delay di↵erential equations (DDEs) are widely usedin many fields such as the biosciences, climate dy-namics, control theory, and engineering. In particu-lar, certain DDEs or more general di↵erential equa-tions with retarded arguments can be derived fromhyperbolic partial di↵erential equations that supportwave propagation. In this talk, we revisit the ap-proximation problem of DDEs by ODE systems. Akey ingredient in our construction is a special poly-nomial basis whose elements are orthogonal with re-spect to a measure adjoining a point mass. The asso-ciated Galerkin scheme enjoys some nice propertiesthat help reduce the derivation of the correspond-ing convergence results to essentially very basic func-

tional analysis exercises. Analytic formulas are alsoavailable within this approach, which simplify the nu-merical treatment. The e�ciency of the method willbe illustrated on several examples, one of which hassolutions that recall Brownian motion.

Climate As a Problem of Nonequi-librium Statistical Mechanics

Valerio LucariniUniversity of Hamburg, Germany

The climate is a complex, chaotic, non-equilibriumsystem featuring a limited horizon of predictability,variability on a vast range of temporal and spatialscales, instabilities resulting into energy transforma-tions, and mixing and dissipative processes resultinginto entropy production. Despite great progresses,we still do not have a complete theory of climatedynamics able to encompass instabilities, equilibra-tion processes, response to changing parameters ofthe system, and coupling across scales of motion. Wewill outline some possible applications of the Ruelleresponse theory, showing how it allows for setting onfirm ground and on a coherent framework conceptslike climate sensitivity, climate response, climate tip-ping points, and for constructing parametrizations..We will show results for comprehensive global climatemodels.V. Lucarini, R. Blender, C. Herbert, F. Ragone,S. Pascale, J. Wouters, Mathematical and PhysicalIdeas for Climate Science, Reviews of Geophysics 52,809-859 (2014)

Modeling of the Humid Atmosphere

Roger TemamIndiana University, USAMakram Hamouda, Joseph Tribbia, XiaoyanWang

The humid atmosphere is a multi-phase system,made of air, water vapor, cloud-condensate, andrain water (and possibly ice / snow, aerosols andother components). The possible changes of phasedue to evaporation and condensation make the equa-tions nonlinear, non-continuous (and non-monotonein the framework of nonlinear partial di↵erentialequations). We will discuss some modeling aspects,and some issues of existence, uniqueness and regular-ity for the solutions of the considered problems, mak-ing use of convex analysis and variational inequali-ties.

The Impact of Threshold Nonlinear-ities on Atmospheric Predictability

Joseph TribbiaNCAR, USA

We consider the problem of atmospheric predictabil-ity in both the idealized circumstance of dry atmo-spheric dynamics and the more realistic situation inwhich threshold nonlinearity exists because of thepossibility of a phase change due to the presence

SPECIAL SESSION 1 11

of moisture. Recent research has reached conflictingconclusions as to whether such step function non-linearities substantially modify the classical implica-tions of quasi-geostrophic turbulence of exponentialpredictability. Alternatively, it has been posited thatmoist dynamics can result in a much more rapid cas-cade of error and thus result in a true “butterfly ef-fect” as originally suggested by Lorenz in analysis ofthe implications of his 1963 result on deterministicnon-periodic flow. A critical comparison of the non-linear cascade of error between the dry and moistcases will be presented.

The Low-Frequency Variability ofthe Ocean-Atmosphere CoupledSystem - a Dynamical Systems Per-spective

Stephane VannitsemRoyal Meteorological Institute of Belgium, Belgium

The low-frequency variability (LFV) of the atmo-sphere at mid-latitudes develops on a wide rangeof time scales. One particularly interesting indica-tor of this variability is the North Atlantic Oscilla-tion index measuring the fluctuations of predominantweather patterns in the course of the years over theAtlantic and Western Europe. The source of vari-ability is, however, controversial and several possibil-ities have been envisaged, including oceanic and cou-pled ocean-atmosphere variability and stratosphericwarming, possibly related to ENSO in the tropicalPacific.Recently we have demonstrated that genuinely cou-pled LFV can emerge in a very simple low-order, non-linear, coupled ocean-atmosphere model. This LFVconcentrates on and near a long-periodic, attractingorbit. This orbit combines atmospheric and oceanicmodes, and it arises for large values of the meridionalgradient of radiative input and of the frictional cou-pling. Chaotic behavior develops around this orbit asit loses its stability. This behavior is still dominatedby the LFV on decadal and multi-decadal time scalesthat is typical of oceanic processes. Furthermore, thisnatural coupled mode is still present as the numberof variables is increased in the model. In this talk,we will present the dynamics of this type of reduced-order models, together with their predictability.

Interplay Between Mathematics andPhysics

Shouhong WangIndiana University, USA

In this talk I shall present two examples demon-strating the symbiotic interplay between physics andmathematics. We start with a general principle thatdynamic transitions of all dissipative systems can beclassified into three categories: continuous, catas-trophic and random. We shall illustrate this princi-ple with an application of the theory to thermohalinecirculation. Then we present a basic principle: theprinciple of interaction dynamics (PID), which takesthe variation of the action functional under energy-momentum conservation constraint. With PID, wederive new gravitational field equations, providing aunified theory for dark matter and dark energy. Inaddition, the PID o↵ers a simple first principle ap-proach for introducing Higgs fields.

A Tree-Graph Approach to SelectedProblems of Nonlinear Dynamics

Ilya ZaliapinUniversity of Nevada Reno, USAYevgeniy Kovchegov, Maxim Arnold

Graph-theoretic (network) representation has provento be e↵ective in many areas of science, prominentlyincluding geophysics. This talk focuses on nonlin-ear dynamics problems addressed using the simplestnetworks - tree graphs. Trees provide a close approx-imation to many natural structures and processes,including river networks, spread of disease or infor-mation, transfer of gene characteristics, dynamics ofparticles with localized interactions, etc. This wouldsound like a trivial observation if not for the fol-lowing fact. Despite apparent diversity, a major-ity of rigorously studied branching structures belongto a two-parametric Tokunaga self-similarity classand exhibit Horton scaling. The Horton scaling is acounterpart of the power-law size distribution of sys-tem’s elements. The stronger Tokunaga constraintensures that di↵erent levels of a hierarchy have thesame probabilistic structure. I will introduce a dy-namic stochastic framework for studying self-similartrees and discuss so-called Horton-Smoluchowski dif-ferential equations that generalize Smoluchowski de-scription of coalescent phenomena in a tree-orientedframework. This provides a new characterization forthe classical Kingman’s coalescent process and sug-gests a novel view on general coalescence process. Ialso show how a tree representation facilitates stud-ies of self-similar solutions of the inviscid Burgers‘equation, and cluster dynamics in the systems of in-teracting particles.


Special Session 2: Emergence and Dynamics of Patterns in NonlinearPartial Di↵erential Equations

Danielle Hilhorst, Universite de Paris Sud, FranceYoshihisa Morita, Ryukoku University, JapanJunping Shi, College of William and Mary, USA

The solution structures of many nonlinear partial di↵erential equations reveal the emergence and the evo-lution of very exciting patterns. Such nonlinear models come from various fields of mathematical science,including material science as well as life sciences. In this session, we will bring together recent studies onsolutions of nonlinear partial di↵erential equations related to pattern formation, dynamics, and bifurca-tions, presenting new aspects of solutions capturing nonlinear phenomena together with underlying solutionstructures.

Nonlinear Elliptic Systems withMixed Interactions Between Com-ponents

Jaeyoung ByeonKAIST, Korea

For a nonlinear elliptic system where repulsive andattractive interactions coexist between components,the existence of a least energy vector solution and itspattern formation for large interactions will be intro-duced.

Convection and Linear Determinacy

Elaine CrooksSwansea University, WalesAmeera Al-Ki↵ai

In reaction-di↵usion-convection models, convectivee↵ects due to, for instance, movement of insects inwind, or the motion of chemotactic cells, etc, canclearly have major impact on the behaviour of so-lutions, and in particular, introduce asymmetry inspreading speeds and front propagation due to thepresence of first-order spatial derivatives. We willdiscuss su�cient conditions for linear determinacy,meaning that a spreading speed into an unstablestate equals a value predicted by the linearizationof the travelling-wave problem about that state, forclasses of monostable reaction-di↵usion-convectionequations and co-operative systems in one space di-mension. Separate conditions for spreading to theleft and to the right are clearly needed. Various ex-amples will be presented, illustrating the e↵ect onlinear determinacy of the interplay between the reac-tion and convection terms and di↵usion parameters,and the symmetry-breaking caused by convection.

Constant Solutions, Ground StateSolutions and Radial Terrace Solu-tions

Yihong DuUniversity of New England, Australia, Australia

In this talk, I will discuss the long-time behavior ofnonnegative solutions of the Cauchy problem

ut � �u = f(u)(x 2 RN , t > 0), u(x, 0) = u0

(x)(x 2 RN ),

where u0

is nonnegative with compact support, andf is a smooth function satisfying f(0) = 0. Sup-pose that u(x, t) is a globally bounded solution, wewill describe its behavior as t ! 1 in the spaceV1

:= L1loc(R

N ) and in the space V2

:= L1(RN ).Under rather general conditions on f , we show thatin V

1

the limit is a constant solution or a groundstate solution of the corresponding elliptic problemover RN , and in V

2

, the long-time behavior of u(·, t)is described by a radial terrace solution.This talk is based on joint works with Peter Polacikand with Hiroshi Matano.

Trend to Equilibrium for a Reaction-Di↵usion System Modelling EnzymeReaction

Jan EliasUniversity Paris-Sud, France

A spatio-temporal evolution of the species occur-ring in an enzyme reaction possessing some massconservation properties is considered. In particular,we study the large time behaviour of solutions of afour-component non-linear reaction-di↵usion systemto the unique steady state. The system is equippedwith the Laplacians for the di↵usive motion of thespecies and the reaction terms which are obtainedfrom the law of mass action kinetics applied to theenzyme reaction. A main tool used in the proposedlarge time analysis is an entropy method known fromkinetic theory which is capable to provide explicitlycomputable rates of the convergence.


Forced Traveling Waves of theFisher-KPP Equation in MovingEnvrionment

Jian FangHarbin Institute of Technology, Peoples Rep of China

In this talk, I will investigate the type, existence andmultiplicity of forced traveling waves for the followingFisher-KPP equation with a moving reaction term

ut = uxx + u(a(x� ct)� u), x 2 R

under certain conditions on continuous function a.This talk is based on joint works with Henri Beresty-cki and with Yijun Lou and Jianhong Wu.

On Certain Models of Liquid Crys-tals

Eduard FeireislCzech Academy of Sciences, Prague, Czech RepE.Rocca, G.Schimperna, A.Zarnescu

We discuss mathematical properties of a model of liq-uid crystals, where the equation for the Q tensor isof hyperbolic type containing second material deriva-tive. We show local solvability in the class of strongsolutions and discuss global existence of weak solu-tions with a defect measure. Finally, we show thatthese weak solutions must coincide with a strong so-lution emanating from the same initial data as longas the latter exists.

Convergence of a Finite VolumeScheme for a First Order Conser-vation Law Involving a Q-WienerProcess

Yueyuan GaoUniversity Paris-Sud, FranceTadahisa Funaki, Danielle Hilhorst, HendrikWeber

We study a time explicit finite volume method witha monotone scheme for a first order conservationlaw with a multiplicative source term involving a Q-Wiener process. We present some a priori estimatesincluding a weak BV estimate. After performing atime interpolation, we prove two entropy inequalitiessatisfied by the discrete solution and show that thediscrete solution converges up to a subsequence toa stochastic measure-valued entropy solution of theconservation law in the sense of Young measures asthe maximum diameter of the volume elements andthe time step tend to zero.

Localized States in Ohta-KawasakiModels

Nir GavishTechnion - IIT, IsraelArik Yochelis, Idan Versano

The Ohta-Kawasaki model is a nonlocal Cahn-Hilliard model for pattern formation driven by com-peting long- and short-range interactions. The modelis commonly used to describe pattern formation indiblock copolymer systems. Recently, we have de-veloped an Otha-Kawasaki type model for ionic liq-uids. This equation involves more general, possiblyasymmetric, long- and short-range interactions. Inthis talk, I will present a systematic study of patternformation in the classical and the extended Otha-Kawasaki model, which reveals the e↵ect of asymmet-ric interactions on pattern formation. Specifically, wefocus on new spatially localized states in 1D and 2Din both infinite and finite domain sizes. We showthat such states exist in both the classic and theextended Otha-Kawasaki model, and describe theirdependence upon domain size.

Numerical Simulations of the Prim-itive Equations with Humidity andSaturations Above Mountain

Youngjoon HongUniversity of Illinois, Chicago, USA

New avenues are explored to study the two dimen-sional inviscid primitive equations of the atmospherewith humidity and saturation, in presence of topog-raphy and subject to physically plausible boundaryconditions for the system of equations. The filteringof the gravity waves produces a compatibility con-dition similar to the condition of incompressibilityfor the Navier-Stokes equations, which we treat ina similar manner. In that respect, a version of theprojection method is introduced to enforce the com-patibility condition on the horizontal velocity field,which comes from the boundary conditions. The re-sulting scheme allows for a significant reduction ofthe errors near the topography when compared tomore standard finite volume schemes.

Traveling Waves in a Reaction-Di↵usion System Describing Smol-dering Combustion

Hirofumi IzuharaUniversity of Miyazaki, JapanEkeoma Rowland Ijioma, Masayasu Mimura

Combustion is a fast oxidation process and exhibitsdiverse behavior according to experimental condi-tions. When there is no natural convection of gassuch as experiments in the space shuttle and in avertically confined system, it is observed that un-expected finger-like smoldering combustion develops.In this talk, a reaction-di↵usion-advection system


that describes smoldering combustion is studied fromthe viewpoint of computer-aided analysis. In partic-ular, we focus on the traveling wave solutions of thesystem, which play a role of a characteristic propa-gation of combustion.

The Role of the Microenvironmentin Regulation of CSPG-DrivenTumor Growth: Invasive and Non-Invasive Gliomas

Yangjin KimKonkuk University, KoreaHyun-Geun Lee

Glioblastoma (GBM) is one of the most lethal type ofbrain cancer with poor survival time. GBM is char-acterized by infiltration of the cancer cells throughthe brain tissue while lower grade gliomas and othernon-neural metastatic types form self-contained non-invasive lesions. Glycosylated chondroitin sulfateproteoglycans (CSPGs), acting as critical regulatorsof the tumor microenvironment, dramatically governthe spatiotemporal status of resident reactive astro-cytes and activation of tumor-associated microglia.In this paper we develop a mathematical model toinvestigate the e↵ect of the CSPG distribution onregulation of a fundamental switch between two dis-tinct patterns: invasive and non-invasive tumors. Weshow that the model’s predictions agree with exper-imental results for a spherical glioma. The modelspecifically predicts that noninvasive tumor lesionsare highly associated with a thick extracellular ma-trix (ECM) containing rich CSPGs, while the absenceof glycosylated CSPGs results in di↵usely infiltra-tive tumors. It is also shown that heavy CSPGs candrive the exodus of resident reactive astrocytes fromthe main tumor mass, and direct inhibition of tu-mor invasion by the astrogliotic capsule, leading toencapsulation of non-invasive lesions. However, sta-ble residence of reactive astrocytes from GBM in theabsence or low level of CSPGs presents a microenvi-ronment favorable to di↵use infiltration due to lossof the primary (CSPG-induced cell-ECM bonding)and secondary (astrogliotic capsule) inhibitors. Themathematical model presents the clear role of the keytumor microenvironment in brain tumor invasion.

Construction of Dialogical Control

Ryo KobayashiHiroshima University, JapanAkio Ishiguro, Hitoshi Aonuma and KoichiOsuka

We introduce a challenge to construct a novel con-trol policy for mobile robots. Conventional controltheory made great successes by separating the plantand the environment and regarding the interactionbetween them as a disturbance. Such a treatmentworks well when the environment is well-known andwell-defined, such a situation is typically achievedin factories. But mobile robot is not the case, be-cause he/she must rush into the unknown environ-ment. Under such situations, interaction between

robots and environment can no more be regarded as adisturbance. We consider that some new control pol-icy is needed to make robots move around in the com-plex world like animals do. We are proposing a newcontrol policy named dialogical control, which meansthat the environment is not necessarily an enemy buta friend and we can have dialogue with an environ-ment. Our approach includes three basic concept -hierarchical control, Tegotae control and Ying-Yangcontrol. In this presentation, we mainly introduce aconcept of Tegotae control and its potentials.

Existence and Non-Existence ofNon-Constant Stationary Patternsfor Certain Prey-Predator TypeReaction Di↵usion Systems with aCross-Di↵usion E↵ect

Kazuhiro KurataTokyo Metropolitan University, JapanSohei Omata

In this talk, we consider a stationary problem forcertain prey-predator type reaction-di↵usion systemswith a cross-di↵usion e↵ect on a bounded domain inRn under homogeneous Neumann boundary condi-tion. This system does not have Turing’s di↵usion-driven instability, but has a cross-di↵usion-driven in-stability. So, we are interested in the cross-di↵usione↵ect and present results on the existence and non-existence of non-constant stationary patterns. Wealso study the limiting problem when the strengthof the cross-di↵usion parameter � tends to infinity.Numerically, we can see spiky patterns in the fullsystem and have some rigorous evidence at least inthe limiting system.

Traveling Wave Solutions of Lotka-Volterra Competition Systems withNonlocal Dispersal in PeriodicHabitats

Wan-Tong LiLabzhou University, Peoples Rep of ChinaXiongxiong Bao, Wenxian Shen

This talk is concerned with the existence, unique-ness and asymptotic stability of space periodic trav-eling wave solutions of the Lotka-Volterra competi-tion system with nonlocal dispersal and space pe-riodic dependence. Here we assume that the sys-tem admits two semitrivial space periodic equilibria(u⇤

1

(x), 0) and (0, u⇤2

(x)), where (u⇤1

(x), 0) is linearlyand globally stable and (0, u⇤

2

(x)) is linearly unstablewith respect to space periodic perturbations. Thisis the joint work with Xiongxiong Bao and WenxianShen.


Emergence and Transition ofClonal Selection Patterns in AcuteLeukemias

Anna Marciniak-CzochraUniversity of Heidelberg, Germany

Motivated by clonal selection observed in acutemyeloid leukemia (AML), we propose mathematicalmodels describing evolution of a multiclonal and hier-archical cell population. The models in a form of par-tial and integro-di↵erential equations are applied tostudy the role of self-renewal properties and growthkinetics during disease development and relapse. Itis shown how resulting nonlinear and nonlocal termsmay lead to a selection process and ultimately totherapy resistance. The solutions of the model tendasymptotically to Dirac measures multiplied by pos-itive constants. Additionally, we show stability ofthe model in the space of positive Radon measuresequipped with the flat metric (bounded Lipschitz dis-tance). The results are compared to AML patientdata. Model based interpretation of clinical data al-lows to assess parameters that cannot be measureddirectly. This might have clinical implications for fu-ture treatment and follow-up strategies.

The Phase-Field Crystal Modelwith Logarithmic Nonlinear Term

Alain MiranvilleUniversity of Poitiers, France

Our aim in this talk is to discuss the well-posednessof the phase-field crystal model with a logarithmicnonlinear term. In particular, we prove the existenceand uniqueness of variational solutions, based on avariational inequality.

Mathematics of Cell-Cell Adhesion:Experiments, Modeling and Analy-sis

Hideki MurakawaKyushu University, Japan

Cell adhesion is the binding of a cell to another cellor to an extracellular matrix component. This pro-cess is essential in organ formation during embryonicdevelopment and in maintaining multicellular struc-ture. Armstrong, Painter and Sherratt [J. Theor.Biol. 243 (2006), pp. 98-113] proposed a nonlocaladvection-di↵usion system as a possible continuousmathematical model for cell-cell adhesion. Althoughthe system is attractive and challenging, it gives bi-ologically unrealistic numerical solutions. We iden-tify the problems and change underlying idea of cellmovement from “cells move randomly to “cells movefrom high to low pressure regions. Then we providea modified continuous model for cell-cell adhesion.Numerical experiments illustrate that the modified

model is able to replicate not only the Steinberg’s cellsorting experiments but also some phenomena whichcan not be captured at all by Armstrong-Painter-Sherratt model. Furthermore, we give theoretical re-sults about the modified model.

Stability of Traveling Waves forSome Bistable Lattice DynamicalSystem

Ken-Ichi NakamuraKanazawa University, JapanToshiko Ogiwara

In this talk, we will study traveling waves for somebistable system on a one-dimensional lattice arisingin a model of competing species. We will discuss thestability of monotone traveling waves.

Spreading Fronts in the AnisotropicAllen-Cahn Equations on Rn

Mitsunori NaraIwate University, JapanHiroshi Matano, Yoichiro Mori

We consider the Cauchy problem for the anisotropicAllen-Cahn equation on Rn with n � 2 and study thelarge time behavior of the solutions with spreadingfronts. Our result states that, under some mild as-sumptions on the initial value, the solution developsa well-formed front whose position roughly coincideswith the spreading Wul↵ shape.

Large Time Behavior, LyapunovFunctionals and RearrangementTheory for a Nonlocal Di↵erentialEquation

Thanh Nam NguyenNational Institute for Mathematical Sciences, KoreaDanielle Hilhorst, Philippe Laurencot

We consider an initial-boundary value problem fora nonlocal parabolic equation of bistable type andstudy possible sharp transition layers that the so-lution may develop at a very early stage. It turnsout that such transition layers can be investigatedvia the structure of the omega limit set of the corre-sponding nonlocal ordinary di↵erential equation. Weprove that for a large class of initial functions, theomega limit set of the nonlocal ordinary di↵erentialequation only contains one element. Furthermore,that element is a step function taking at most twovalues. The proof bases on the rearrangement the-ory and the existence of infinitely many Lyapunovfunctionals.


Coupling Surface Di↵usion withGrain Boundary Migration: Wet-ting and Dewetting

Amy Novick-CohenTechnion-IIT, IsraelVadim Derkach, Eugen Rabkin, ArcadyVilenkin

Solid state wetting and dewetting can be reason-ably modeled by surface di↵usion, and dependingon the initial conditions and the parameters, can re-sult in the formation of a variety of patterns. Inthe context of thin nanocrystalline specimens, thesituation becomes more complicated since the wet-ting/dewetting surface is then coupled with a net-work of grain boundaries which can also migrate.Features of interest include: void formation, oscil-lation, acceleration, and fingering. In this lecturewe will report primarily on numerical studies, whichallow us to test various conjectures based on exper-iment. Our numerical studies open the door to avariety of analytical questions.

Nonlinear Fronts in the Swift-Hohenberg Equation As a Modelfor Phyllotaxis

Matt PennybackerUniversity of New Mexico, USAAlan C. Newell, Patrick Shipman

Some of the most spectacular patterns in the naturalworld can be found on members of the plant kingdom.Furthermore, the regular configurations of organs onplants, collectively called phyllotaxis, exhibit a re-markable predisposition for Fibonacci and Fibonacci-like progressions. Starting from a biochemical andmechanical growth model similar to the classic Swift-Hohenberg equation, we discuss the ways in whichnearly every property of phyllotaxis can be explainedas the propagation of a pushed pattern-forming front.

The Dynamics of Front-Like Solu-tions of Parabolic Equations on theReal Line

Peter PolacikUniversity of Minnesota, USA

We examine the behavior of bounded front-like solu-tions of reaction-di↵usion equations on the real line.First, we give a general result on the approach ofsuch solution to a propagating terrace, or, a stackedsystem of traveling fronts. Then we draw some con-sequences of this result, including in particular a qua-siconvergence property of the solutions.

Dynamics and Bifurcation of Multi-component Amphiphilic Membranes

Keith PromislowMichigan State University, USAQiliang Wu

Abstract: Polymer chains are typically hydropho-bic, the addition of functional groups to the back-bone adds regions of hydrophilicity. The amphiphilicmaterial (both hydrophobic and hydrophilic) has astrong a�nity for solvent, imbibing it to self assemblecharge-lined networks which serve as charge-selectiveion conductors in a host of energy conversion ap-plications. We present a continuum model for thefree energy of an amphiphilic mixture. The associ-ated gradient flows admit dynamic competition be-tween network morphologies of distinct co-dimension.We present a model for multicomponent amphiphilicmixtures that permits competitive geometric evolu-tion for co-dimension 1 bilayers and co-dimension twopore morphologies, present an analysis of the associ-ated spectral problems, and describe rigorous exis-tence results for pearled morphologies.

Multiscale Modelling and Analysisof the Mechanical Properties ofBiological Tissues

Mariya PtashnykUniversity of Dundee, ScotlandAndrey Piatnitski, Brian Seguin

In this talk the derivation and analysis of micro-scopic models for the mechanics of biological tis-sues that take into account the interactions be-tween the microscopic structure, mechanical prop-erties and the chemical processes will be presented.The strongly coupled systems of nonlinear reaction-di↵usion-convection equations for chemical processesand the equations of linear elasticity or poroelasticityfor deformations will constitute the microscopic mod-els. Using homogenization techniques we will derivethe macroscopic models for the mechanics of biologi-cal tissues. The analysis and numerical simulations ofthe macroscopic equations will demonstrate the pat-terns in the interactions between mechanical stressesand chemical processes.

The Existence of Fast-DecayGround States to a Weakly-CoupledElliptic System

Yuanwei QiUniversity of Central Florida, USAZhi Zheng, Xinfu Chen, Shoulin Zhou

This article studies the existence of positive solutionsto a weakly-coupled elliptic system in RN which orig-inates from self-similar solutions of a parabolic sys-tem ut = 4u+ vp, vt = 4v+ uq, where p, q > 1. Itis shown that there exist ground state solutions withexponential decay as |x| ! 1 as well as positive so-


lutions on a finite ball with zero Dirichlet boundarycondition when (p, q) is subcritical. On the otherhand, it is proved that when (p, q) is supercritical,both types of solutions do not exist under some tech-nical conditions.

Homogenization of Cahn-Hilliard-Type Equations Via Gradient Struc-tures

Sina ReicheltWeierstrass Institute Berlin, GermanyMatthias Liero

In the paper [1] we discuss two approaches to evolu-tionary �-convergence of gradient systems in Hilbertspaces. The formulation of the gradient system isbased on two functionals, namely the energy func-tional and the dissipation potential, which allows usto employ �-convergence methods. In the first ap-proach we consider families of uniformly convex en-ergy functionals such that the limit passage of thetime-dependent problems can be based on the theoryof evolutionary variational inequalities as developedby Daneri and Savare 2010. The second approachuses the equivalent formulation of the gradient sys-tem via the energy-dissipation principle and followsthe ideas of Sandier and Serfaty 2004. We applyboth approaches to rigorously derive homogenizationlimits for Cahn–Hilliard-type equations. Using themethod of weak and strong two-scale convergence viaperiodic unfolding, we show that the energy and dissi-pation functionals �-converge. In conclusion, we willgive specific examples for the applicability of each ofthe two approaches.

References

[1] M. Liero and S. Reichelt, Homogenization ofCahn–Hilliard-type equations via evolutionary�-convergence, WIAS Preprint No. 2114 (2015).

Exclusive Regions Connected byDi↵usion

Thomas SeidmanUMBC, USA

With 2 components u, v � 0 on t > 0, x 2 D theDE ut = �uv = vt leads to exclusive regions: extinc-tion of the (initially) smaller component. After theinitial transient in ut � ✏�u = �uv, etc. one con-siders the evolution of a spatial pattern of regions.[This is complicated even by the possibility of s sim-ple perturbation through other components (as, forthe 1-dim case with an additional perturbing com-ponent w with uv 7! w and uw 7! 0); in that caseit can be shown rigorously that this problem is wellposed.] More interesting geometry arises computa-tionally in higher dimensions. This represents col-laborations with A.Muntean and others.

Front Propagation and Symmetriza-tion for the Fractional Fisher-KPPEquation

Andrei TarfuleaUniversity of Chicago, USAJean-Michel Roquejo↵re

We prove strong gradient decay estimates for solu-tions to the multi-dimensional Fisher-KPP equationwith fractional di↵usion. It is known that this equa-tion exhibits exponentially advancing level sets withstrong qualitative upper and lower bounds on the so-lution. However, little has been shown concerning thegradient of the solution. We prove that, under mildconditions on the initial data, the first and secondderivatives of the solution obey a comparative expo-nential decay in time. We then use this estimate toprove a symmetrization result, which shows that thereaction front flattens and quantifiably circularizes,losing its initial structure.

Analysis and Optimization forEdge-Emitting Semiconductor Het-erostructures

Marita ThomasWeierstrass Institute Berlin, Germany

This contribution discusses results on the existenceof local-in-time classical solutions for edge-emittingsemiconductor heterostructures both in 2D and 3D.Electronics of the semiconductor is governed by thevan Roosbroeck system, consisting of the Poissonequation for the electrostatic potential and a systemof drift-di↵usion equations for the carrier transport.To describe lasing e↵ects of a semiconductor device,the van Roosbroeck system is coupled with the equa-tions of optics, given by a Helmholtz-type eigenvalueproblem and an ODE for the photon balance. Basedon this coupled system optimization strategies forthe light emission of a device are obtained. Thiscontribution is based on joint work with K. Disser,D. Peschka, J. Rehberg, and N. Rotundo (all WIASBerlin).

Numerical Analysis for ReactionDi↵usion Models of Spreading Phe-nomena with Free Boundary

Takeo UshijimaTokyo University of Science, Japan

In recent years, several reaction di↵usion models withfree boundary are proposed to describe the spreadingof species or the transmission of diseases. In thesesmodels the free boundary with Stefan like conditionis concerned and it describes the motion of the frontof the spreading phenomena. In this talk, we willdiscuss numerical methods for these models.


Nonplanar Traveling Fronts inReaction-Di↵usion Equations withCombustion and Non-KPP Monos-table Nonlinearities

Zhi-Cheng WangLanzhou University, Peoples Rep of China

In this talk we concern with nonplanar travelingfronts in reaction-di↵usion equations with combus-tion nonlinearity and non-KPP monostable nonlin-earity. Our study contains two parts: in the firstpart we establish the existence and stability of V-shaped traveling fronts in R2, and in the second partwe establish the existence and stability of travelingfronts of pyramidal shape in R3.

Analysis of Coupled DegenerateReaction-Di↵usion SIR Model De-scribing Population Dynamics ofFox Rabies

Fengqi YiHarbin Engineering University, Peoples Rep of China

A coupled degenerate reaction-di↵usion SIR modelfor the overall dynamics of the interaction betweenfox populations and rabies is considered. The modelhelps to explain the epidemiological patterns ob-served in Europe, including 3 to 5 year cycle in foxpopulations infected with rabies. We provide someglobal analyses of the model, including the asymp-totic behaviors of the system. In addition, we showthe existence of oscillatory behaviors due to Hopf bi-furcation in the homogeneous model described by aset of ordinary di↵erential equations. The existenceof traveling wave solutions of the reaction-di↵usionrabies epidemic model is also discussed.


Special Session 3: Nonlinear Evolution PDEs, Interfaces and Applications

Alain Miranville, University of Poitiers, FranceGunduz Caginalp, University of Pittsburgh, USAMaurizio Grasselli, Politecnico di Milano, Italy

Many issues in applied science can be formulated as interface problems which can be regarded as limitingcases of evolution equations exhibiting transition layers. The study of phase field or di↵usive interfaceproblems, Allen-Cahn and Cahn-Hilliard equations have been an active area for the past few decades. Thishas also been an additional motivation for studying general nonlinear evolution equations. This session willfocus on the mathematical properties (well-posedness, regularity, stability, asymptotic behavior of solutions,...) and applications of these equations.

Surface Tension, Higher OrderPhase Field Equations and Homo-geneous Equations

Gunduz CaginalpUniv of Pittsburgh, USA

Higher order phase field equations describe detailedstructure such as complex anisotropy. The surfacetension derived from these equations has some ba-sic properties that may provide insight into interfaceformation. Using dimensional analysis one can showthat surface tension has some homogeneity proper-ties. In this talk some open problems related to theseissues will be presented.

Analytical and Numerical Solutionsof the Phase Field Crystal Modelfor the Interface Dynamics

Peter GalenkoUniversity of Jena, GermanyI.G. Nizovtseva, K.R. Elder

Analytical and numerical methods are used to pre-dict the dynamic evolution of a crystal-liquid inter-face [1]. In this work, the dynamics of the phase fieldcrystal model is analyzed using complex amplitudewave representation. It is shown that if the phasesof the amplitudes are constant, the model reduces toa non-linear Allen-Cahn equation that can be solvedanalytically for traveling waves. Numerical solutionsof the phase field crystal model are compared withanalytically obtained traveling wave solutions. Thesesolutions can be used for the problem of crystal lat-tice selection at the front of crystalline phase invadinghomogeneous liquid state [2].

References

[1] P.K. Galenko, F.I. Sanches, and K.R. Elder,Traveling wave profiles for a crystalline front in-vading liquid states: analytical and numericalsolutions. Physica D: Nonlinear Phenomena 308(2015) 1-10.

[2] P.K. Galenko and K.R. Elder, Marginal stabilityanalysis of the phase field crystal model in onespatial dimension, Physical Review B 83 (2011)064113.

Generalized Regularized Long WaveEquation with White Noise Disper-sion

Olivier GoubetUniveriste de Picardie Jules Verne, FranceMin Chen, Youcef Mammeri

In this talk, we address the generalized BBM equa-tion with white noise dispersion which reads

du� duxx + ux � dW + upuxdt = 0,

in the Stratonovich formulation, where W (t) is astandard real valued Brownian motion. We first in-vestigate the initial value problem for this equation.We then prove theoretically and numerically that fora deterministic initial data, the expectation of the

L1x norm of the solutions decay to zero at O(t�

16 ) as

t approaches +1, by assuming that p > 8 and thatthe initial data is small in L1

x \H4

x. This decay ratematches the one for solution of the linear equationwith white noise dispersion.

Coupled Navier-Stokes and PhaseField Models for Interface Dynam-ics in Coating Process.

Ludovic GoudenegeCNRS, FranceIsabelle Cantat, Sebastian Minjeaud, JacopoSeiwert

A coating process, for instance the film depositionby withdrawing a substrate from a bath with solu-tion, is a complex phenomenon. In order to studyand simulate it, we need at least two equations. Onefor the dynamic of the fluid and one for the interfacebetween the di↵erent species.For this reason we present a phase-field model ofAllen-Cahn or Cahn-Hilliard type coupled with theNavier-Stokes equations. But we are concerned bythe dynamics of surfactant at the interface, so weadd a new equation to the coupled system in orderto take into account the e↵ects of surfactant.We will present several models for surfactant dynam-ics and study their impact on the evolution of the in-terface. In particular we will present the adequationbetween numerical simulations and experimentationswith respect to some classical well-known physicallaws.


On Micro-Macro Models for Two-Phase Flow with Dilute PolymericSolutions – Modeling and Analysis

Gunther GrunUniversity of Erlangen-Nurnberg, GermanyStefan Metzger

By methods from nonequilibrium thermodynamics,we derive a di↵use-interface model for two-phase flowof incompressible fluids with dissolved noninteract-ing polymers. The polymers are modeled by dumb-bells subjected to finitely extensible, nonlinear elastic(FENE) spring-force potentials. Their density andorientation are described by a Fokker-Planck-typeequation which is coupled to a Cahn-Hilliard and amomentum equation for phase-field and gross veloc-ity/pressure. Henry-type energy functionals are usedto describe di↵erent solubility properties of the poly-mers in the di↵erent phases or at the liquid-liquidinterface. Taking advantage of the underlying ener-getic/entropic structure of the system, we prove ex-istence of a weak solution globally in time. As aby-product in the case of Hookean spring potentials,we derive a macroscopic di↵use-interface model fortwo-phase flow of Oldroyd-B-type liquids. We dis-cuss extensions of the model to take the interactionbetween polymer and fluid interface orientation intoaccount (amphiphilic surfactant) and provide somenumerical experiments.

Mathematical Analysis of aParabolic System for Chemotac-tic E.coli Colonies

Danielle HilhorstCNRS and Univ. Paris-Sud Paris-Saclay, FranceR. Celinski, G. Karch, M. Mimura

We consider an initial-boundary value problem de-scribing the formation of colony patterns of bacte-ria Escherichia coli. This model consists of reaction-di↵usion equations coupled with the Keller-Segel sys-tem from the chemotaxis theory in a bounded do-main, supplemented with zero-flux boundary condi-tions and with nonnegative initial data. We answerquestions about the global in time existence of solu-tions as well as on their large time behavior. More-over, we show that the solutions of a related modelmay blow up in finite time.

Convergent Finite Di↵erenceScheme for Compressible Navier-Stokes Equations in Three Dimen-sions

Radim HosekCzech Academy of Sciences, Czech Rep

Motivated by works of Karper and co-authors wesuggest a numerical scheme for solving compressibleNavier-Stokes equations in three spatial dimensionsthat is based on finite di↵erence method. The scheme

is applied to a domain with simple cubic geometry.We mimick the existence proof developed by Lionsand Feireisl to show that a sequence of numericalsolutions on grids with decreasing discretization pa-rameter converges weakly (up to a subsequence) to aweak solution of the compressible Navier-Stokes sys-tem. The main di�culties of the proof comparedto the standard Karper’s FEM/DG scheme will bepointed out.

Global Dynamics of Bound-ary Droplets for the 2-D Mass-Conserving Allen-Cahn Equation

Jiayin JinGeorgia Institute of Technology, USAPeter W. Bates

In this talk I will present how to establish the exis-tence of an invariant manifold of bubble states for themass-conserving Allen-Cahn equation in two spacedimensions, and give the dynamics of the center ofthe bubbles.

A Di↵use Interface Tumour Modelwith Chemotaxis and Active Trans-port

Kei Fong LamUniversity of Regensburg, GermanyHarald Garcke, Emanuel Sitka, Vanessa Styles

We consider a thermodynamically consistent di↵useinterface model for tumour growth, which couples aCahn-Hilliard system and a reaction-di↵usion equa-tion. The system of PDEs models the growth of atumour in the presence of a nutrient and surroundedby host tissue. A new feature of the model is the in-clusion of transport mechanisms such as chemotaxisand active transport through specific choices of thefluxes. We will first give a heuristic derivation of asimplified model and discuss how chemotaxis and ac-tive transport are modelled, along with some resultsregarding the well-posedness of the system. Then,we will present the more general model, which is aCahn-Hilliard-Darcy system coupled to a convection-reaction-di↵usion equation for the nutrient. The ef-fects of including the transport mechanisms and fluidflow will be demonstrated with numerical computa-tions, and if time permitting, we will discuss some re-sults regarding the existence of weak solutions to thegeneral model. This is joint work with Harald Gar-cke, Emanuel Sitka (Regensburg) and Vanessa Styles(Sussex).


On the Viscous and Non-ViscousCahn-Hilliard and Allen-CahnEquations

Ahmad MakkiLMA- Universite de Poitiers, FranceAlain Miranville

We discuss the well-posedness and the asymptoticbehaviour of the viscous Allen-Cahn (resp. Cahn-Hilliard) and the non-viscous anisotropic Cahn-Hilliard (resp. Allen-Cahn) equations. In particu-lar, these models contain a regularization term, calledWillmore regularization. Also, we give some numeri-cal simulations which show the e↵ects of the viscosityterm on the anisotropic and isotropic Cahn-Hilliardequations.

A Phase-Field Model for ThermalBinary Alloys: Asymptotic Be-haviour

Pedro Marin-RubioUniversidad de Sevilla, SpainGabriela Planas, Jose Real

In this talk I will present some recent results onexistence, but with uniqueness issue unknown, to aphase-field model for a thermal binary alloy. Afterthat, using some techniques from multivalued anal-ysis, the long-time behaviour of dynamical systemsassociated the problem will be analysed. In par-ticular, existence and regularity of attractors andstructure of omega-limit sets will be discussed.

Existence of Pulses for a MonotoneReaction-Di↵usion System

Martine MarionEcole Centrale de Lyon, FranceV. Volpert

We discuss the existence of pulses for a monotonereaction-di↵usion system of two equations. The re-sult is applied to prove the existence of pulses forthe system of competition of species in populationdynamics.

On Stable Splitting Schemes forPhase-Field Models with Ion Trans-port

Stefan MetzgerFAU Erlangen-Nurnberg, GermanyGunther Grun

We present an energy-stable, decoupled discretescheme for a recent model (see [1]) supposed todescribe electrokinetic phenomena in two-phase flowwith general mass densities. This model permits todescribe the e↵ect of contact line hysteresis. It con-siders several kinds of species with di↵erent chargesand solubility properties, which may react with eachother. Exposing the fluids to an electric field inducesion motion and therefore influences the fluid flow.Topological changes in the droplet configuration arepractically important features to be captured by thisapproach.

As the model couples Nernst–Planck-equations withmomentum and Cahn–Hilliard phase-field equationsin a nonlinear way, the formulation of a stable, ef-ficient and fully practical numerical scheme is notobvious.

Using a subtle approximation of the velocities arisingin the convective terms, we manage to derive a stablesplitting scheme allowing to decouple the equationsfor the phase-field, the species, and the electro-staticpotential from the momentum equations (cf. [2]).

In the case of a fluid independent dielectric permittiv-ity, we end up with three blocks which can be treatedsequentially. By establishing a discrete counterpartof the continuous energy estimate, we will show thatthis splitting approach does not a↵ect the stability ofthe scheme. Finally, we shall present numerical sim-ulations showing species induced fluid motion anddroplet break-up to underline the full practicality ofthis approach.

References

[1] E. Campillo-Funollet, G. Grun, and F. Kling-beil, On Modeling and Simulation of Electroki-netic Phenomena in Two-Phase Flow with Gen-eral Mass Densities, SIAM J. Appl. Math., 72,1899 – 1925 (2012).

[2] S. Metzger, On numerical schemes for phase-field models for electrowetting with electrolytesolutions, PAMM, 15, 715–718 (2015).


Dynamic Boundary Conditions forAllen–Cahn Type Equations withSingularities

Ryota NakayashikiChiba University, JapanPierluigi Colli, Gianni Gilardi, Ken Shirakawa

This study is suported by CNR-JSPS Joint Project“Innovative variational methods for evolution equa-tions”. In this talk, we consider coupled systems ofnonlinear variational inequalities. Each system con-sists of an Allen-Cahn type equation in a boundedspatial domain⌦ , and another Allen-Cahn type equa-tion on the boundary � := @⌦, and also, these equa-tions are transmitted via a dynamic boundary con-dition. In particular, the equations in ⌦ are derivedfrom nonsmooth free-energies, and hence, the di↵u-sions in ⌦ are provided in quasi-linear forms withsingularities. The objective of this study is to builda mathematical method which enable to deal withthe quasi-linear and singular situations. On this ba-sis, we will set the focus of the discussion on the well-posedness of our systems in L2-based framework, andthe continuous associations among di↵erent systemsbrought by �-convergences of free-energies.

Coupled Surface Di↵usion andGrain Boundary Migration in ThreeGrain Systems

Amy Novick-CohenTechnion-IIT, IsraelVadim Derkach, John McCuan

The stability of thin nanocrystalline films is critical innumerous applications. We consider the relative mo-tion of three grains with prescribed volume, whichare constrained to lie within a triangular geometrywhich can be used to produce a thin film “tiling” ofthe plane via mirror refection. In our model, the ex-terior surface of the grains evolve by surface di↵usionmotion and the boundaries between the grains evolveby motion by mean curvature. Drawing conclusionsbased on existence of steady states, relative energies,and the results of numerical simulations, we demon-strate that many phenomena are truly nonlinear.

On the Viscous Cahn-Hilliard-Navier-Stokes Equations with Dy-namic Boundary Conditions

Madalina PetcuUniversity of Poitiers, FranceLaurence Cherfils

The presentation concentrates on the study of theviscous Cahn-Hilliard-Navier-Stokes model, endowedwith dynamic boundary conditions, from the theo-retical and numerical point of view. We start bydeducing results on the existence, uniqueness andregularity of the solutions for the continuous prob-lem. Then we propose a space semi-discrete finite

element approximation of the model and we studythe convergence of the approximate scheme. We alsoprove the stability and convergence of a fully dis-cretized scheme, obtained using the semi-implicit Eu-ler scheme applied to the space semi-discretizationproposed previously. Numerical simulations are alsopresented to illustrate the theoretical results.

Convergence to Equilibrium forFinite Element Discretizations ofthe Modified Phase Field CrystalEquation

Morgan PierreUniversity of Poitiers, FranceMaurizio Grasselli

The phase field crystal equation is a conservativeSwift-Hohenberg equation which has been employedto model and simulate the dynamics of crystallinematerials. The modified version considered here canaccount for elastic interactions. We propose a fullydiscrete approximation of this model based on aGalerkin approximation in H1 for the phase func-tion and a second-order energy stable time discretiza-tion. We analyze the properties of this approxima-tion (well-posedness, stability, convergence as the dis-cretization parameters tend to 0). In particular, wewill prove that the discrete solution converges to asteady state as time goes to infinity.

Optimal Control of the DendriteStructure Using Magnetic Field

Amer RasheedLahore University of Management Sciences (LUMS),PakistanAziz Belmiloudi

We present the optimal control of a phase field model,recently developed by A. Rasheed and A. Belmiloudi[1], which represents the e↵ect of magnetic field onthe evolution of dendrite during the solidification pro-cess of a binary alloy in an isothermal environment.The aim of this study is to control the desired dynam-ics of the dendrite by using magnetic field as a con-trol variable. In the control problem, the cost func-tional measures the distance between the calculatedand desired dynamics. We have established the ex-istence results and optimality conditions along withthe adjoint problem.

References

[1] A. Rasheed and A. Belmiloudi, Modeling andsimulations of dendrite growth using phase-fieldmethod with magnetic field e↵ect, Communica-tions in Computational Physics, 14(2), pp. 477- 508, 2013.


A Convergent, Energy Stable,and E�cient Hexagonal FiniteDi↵erence Scheme for the PhaseField Crystal Amplitude ExpansionModel

Steven WiseThe University of Tennessee, USAZhen Guan, Vili Heinonen, John Lowengrub,Cheng Wang

The phase field crystal model is a sixth order, non-linear parabolic PDE describing crystal dynamics atthe atomic length scale, but on di↵usion time scales.It can describe point defects, grain boundaries, andelastically mediated phase transformations. I will de-scribe a reformulation of the model using the tech-nique of amplitude expansions, which reduces themodel to a system of fourth-order equations, but in-troduces several complexities. I will briefly describesome properties of the PDE solutions and will theninvestigate an energy stable and convergent numeri-cal method for the amplitude model based on a non-standard finite di↵erence discretization of space. Iwill show a number of numerical examples demon-strating the advantages of the scheme. This is jointwork with Zhen Guan, Vili Heinonen, John Lowen-grub, and Cheng Wang.

Long-Time Behaviour of the PlateEquation with Nonlocal Nonlinear-ity

Sema YaylaHacettepe University, TurkeyZehra Arat, Azer Khanmamedov

We consider Cauchy problems for the semilin-ear plate equations with nonlocal nonlinearitiesf(krukL2

(Rn

)

)�u and f(ku (t)kLp

(Rn

)

) |u|p�2 u . In

the case of nonlocal nonlinearity f(krukL2(Rn

)

)�u,we prove the existence of the global attractor andestablish the regularity and finite dimensionality ofthis attractor under the strict positivity condition onthe damping coe�cient. For the plate equation withnonlinearity f(ku (t)kLp

(Rn

)

) |u|p�2 u, we prove theexistence and regularity of the global attractor un-der the weaker condition on the damping coe�cient.Namely, in this case we require the strict positivitycondition on the damping coe�cient only in the ex-terior of some ball.


Special Session 4: Classical and Geophysical Fluid Dynamics

Youngjoon Hong, University of Illinois at Chicago, USAMadalina Petcu, Universite de Poitiers, France

Roger Temam, Indiana University, USA

Over the past decades, mathematics and geophysics have found successful interactions through the study ofthe Euler, Navier-Stokes and convection-di↵usion equations, and their geophysical counterparts the Boussi-nesq and Primitive Equations. The main purpose of this workshop is to bring together specialists in the fieldand to exchange ideas on these problems. This special session focuses on recent progresses in the develop-ment and applications of classical and geophysical fluid dynamics. In particular, the session aims to addresssome important issues such as the properties of solutions, existence, regularity, stability and asymptoticbehaviors arising from nonlinear di↵erential equations.

Some Dynamical Problems Arisingin Blood Flow

Jerry BonaUniversity of Illinois at Chicago, USAChun-Hsiung Hsia, Daniela Jasso-Valdez

Pulmonary arterial hypertension is an insidious dis-ease that currently has poor prospects for treatment.In an e↵ort to better understand this disease, a modelis developed for blood flow in the relevant part of thebody. The model is shown to have at least some pre-dictive capability. A plan is then outlined for usingthe model in conjunction with laboratory data to be-gin fathoming the remodeling that is a hallmark ofthe condition, and which is oftentimes the cause ofpremature death.

The Role of the Pressure in thePartial Regularity Theory for WeakSolutions of the Navier-StokesEquations

Diego ChamorroUniversite d‘Evry, FrancePierre-Gilles Lemari‘e-Rieusset, KawtherMayoufi

We study the role of the pressure in the partial reg-ularity theory for the weak solutions of the Navier-Stokes equations. By introducing the notion of dissi-pative solutions, due to Duchon and Robert, we willprovide a generalization of the Ca↵arelli, Kohn andNirenberg theory. Our approach gives a new enlight-enment of the role of the pressure in this theory inconnection to Serrin’s local regularity criterion.

Low-Dimensional Approximate So-lutions of Hamilton-Jacobi-BellmanEquations, and Applications to theOptimal control of nonlinear evolu-tion equations

Mickael ChekrounUniversity of California, Los Angeles, USAAxel Kroner, Honghu Liu

In this talk a new approach to approximate the solu-tions of Hamilton-Jacobi-Bellman (HJB) equations,will be presented. The focus of our study is con-cerned with the optimal control of nonlinear evolu-tion equations in Hilbert spaces. The approach fitsinto the long tradition of seeking for slaving relation-ships between the small scales and the large ones butdi↵er by the introduction of a new type of manifoldsto do so, namely the finite-horizon parameterizingmanifolds introduced by Chekroun and Liu (ACTAAppl. Math., 2015, 135, pp. 81-144). A control infeedback form will be derived from the reduced HJBequations associated with the corresponding optimalcontrol problem for the surrogate system. Rigorouserror estimates between the solutions of the corre-sponding HJB equations and the approximate oneswill be also derived and application to fluid problemsdiscussed. This talk is based on a joint work withAxel Kroner and Honghu Liu.

The Barotropic Quasi-GeostrophicEquation Under a Free Surface

Qingshan ChenClemson University, USA

When the length scale of the flow is on the same or-der of the Rossby deformation radius, the classicalrigid-lid assumption is no longer valid, the impactof the free surface deformation on the the vorticityfield is no longer negligible, and therefore it has to beaccounted for in the model. In this talk, we presentsome new results concerning the well-posedness of thebarotropic quasi-geostrophic equation under a freesurface. The connection of this model to other morecomplex and more realistic models will also be dis-cussed.


Two-Point Boundary Value Prob-lem for a Higher Order NonlinearDispersive Wave Equation

Hongqiu ChenUniversity of Memphis, USAShu-Ming Sun, Juan-Ming Yuan

The focus of the talk is the higher order nonlinear dis-persive equation which models unidirectional propa-gation of small amplitude long waves in dispersivemedia. The specific interest is in the initial, twopoint-boundary value problem. With proper require-ment on initial and boundary condition, we show lo-cal and global well posedness.

Determining Wavenumber for FluidEquations

Alexey CheskidovUIC, USA

We introduce a determining wavenumber for theforced 3D Navier-Stokes equations (NSE) defined foreach individual solution. Even though this wavenum-ber blows up if the solution blows up, its time aver-age is uniformly bounded by Kolmogorov‘s dissipa-tion wavenumber for all solutions on the global at-tractor. A similar approach in the two-dimensionalcase is consistent with the prediction of Kraichnan‘stheory of turbulence.

Determining Modes for the SurfaceQuasi-Geostrophic Equations

Mimi DaiUniversity of Illinois Chicago, USAAlexey Cheskidov

We introduce a determining wavenumber for the sur-face quasi-geostrophic (SQG) equationdefined for each individual trajectory and then studyits dependence on the force. While in the subcriti-cal and critical cases this wavenumber has a uniformupper bound, it may blow upwhen the equation is supercritical. A bound onthe determining wavenumber provides determiningmodes, which in some sense measure the number ofdegrees of freedom of the flow, or resolution neededto describe a solution to the SQG equation.

Optimal Ship Forms Based onMichell’s Wave Resistance: Numer-ical Aspects

Julien DambrineUniversity of Poitiers, FranceMorgan Pierre, Germain Rousseaux

The slender body approximation, first introduced byRankine for the study of a potential flows arounda thin obstacle in an infinite domain states that athin obstacle has the same e↵ect on the flow as a

distribution of source/well doublets on a line. Thismodel has been extended later for linear free surfaceflows and lead to the explicit formulas of Michell forthe computation of the wave making resistance ofa slender ship moving with an uniform velocity indeep waters. This formula provides the wave makingresistance as a quadratic function of the hull o↵setfunction. Although very well known and widely usedfor the prediction of the wave-making resistance ofsimple ships, very few attempts of hull optimisationhave been made through this model. The goal of thistalk is to illustrate the theoretical issues of ship hulloptimisation with some numerical calculations.

Boundary Layers of the Navier-Stokes Equations

Gung-Min GieUniversity of Louisville, USA

In this talk, we discuss some recent progresses onboundary layer analysis of the Navier-Stokes equa-tions.

Energy and Potential Enstro-phy Flux Constraints in Quasi-Geostrophic Models

Eleftherios GkioulekasUniversity of Texas Rio Grande Valley, USA

We investigate an inequality constraining the energyand potential enstrophy flux spectra in two-layer andmulti-layer quasi-geostrophic models. Its physicalsignificance is that it can diagnose whether any givenmulti-layer model that allows co-existing downscalecascades of energy and potential enstrophy can allowthe downscale energy flux to become large enough toyield a mixed energy spectrum where the dominantk�3 scaling is overtaken by a subdominant k�5/3 con-tribution beyond a transition wavenumber kt situ-ated in the inertial range. The validity of the fluxinequality implies that this scaling transition can-not occur within the inertial range, whereas a viola-tion of the flux inequality beyond some wavenumberkt implies the existence of a scaling transition nearthat wavenumber. This flux inequality holds uncon-ditionally in two-dimensional Navier-Stokes turbu-lence, however, it is far from obvious that it contin-ues to hold in multi-layer quasi-geostrophic models,because the dissipation rate spectra for energy andpotential enstrophy no longer relate in a trivial way,as in two-dimensional Navier-Stokes. We derive thegeneral form of the energy and potential enstrophydissipation rate spectra for a generalized symmetri-cally coupled multi-layer model. From this result,we prove that in a symmetrically coupled multi-layerquasi-geostrophic model, where the dissipation termsfor each layer consist of the same Fourier-diagonallinear operator applied on the streamfunction field ofonly the same layer, the flux inequality continues tohold. It follows that a necessary condition to violatethe flux inequality is the use of asymmetric dissipa-tion where di↵erent operators are used on di↵erent


layers. We explore dissipation asymmetry further inthe context of a two-layer quasi-geostrophic modeland derive upper bounds on the asymmetry that willallow the flux inequality to continue to hold. Asym-metry is introduced both via an extrapolated Ekmanterm, based on a 1980 model by Salmon, and via dif-ferential small-scale dissipation. The results given aremathematically rigorous and require no phenomeno-logical assumptions about the inertial range. Su�-cient conditions for violating the flux inequality, onthe other hand, require phenomenological hypothe-ses, and will be explored in future work.

Rotating Fluids and Boundary Lay-ers

Makram HamoudaIndiana University, USASoumaya Ben Chaabane, Mahdi Tekitek

We investigate in this talk the boundary layers ap-pearing for a fluid under moderate rotation whenthe viscosity is small. The fluid is modeled bytime-dependent rotating Stokes equations also knownas the Stokes-Coriolis equations. The fluid is con-sidered in an infinite channel with periodicity onthe lateral boundary and Dirichlet boundary condi-tions on the top and bottom of the channel. First,we analytically derive the correctors which describethe sharp variations at large Reynolds number i.e.small viscosity). Second, thanks to a new finite vol-ume method (NFVM) we give the numerical solu-tions of the Stokes-Coriolis system at small viscosity(10�2�10�10). The NFVM can be applied to a largeclass of singular perturbation problems.

Optimal Ship Forms Based onMichell’s Wave Resistance: Theo-retical Aspects

Morgan PierreUniversity of Poitiers, FranceJulien Dambrine, Germain Rousseaux

In 1898, J.H. Michell provided a formula for the waveresistance of a thin ship moving at constant speed incalm water. By adding a simple viscous resistanceterm (proportional to the area of the immerged hulland to the square of the speed), we obtain a modelfor the total resistance of water to the motion of theship. This allows us to seek for optimal ship hulls ex-pressed in a parametric form, with parameters in theregion of the vertical plane of symmetry. We inves-tigate some mathematical properties of the optimalhull, first when the domain of parameters is fixed,and then when this domain varies.

Two-Dimensional Bifurcation in theWhitham Equation with SurfaceTension

Filippo RemonatoNTNU - Norwegian University of Science andTechnology, NorwayHenrik Kalisch, Mats Ehrnstrom, MatthewJohnson

The capillary Whitham equation

ut + u2 +K ⇤ u = 0,

with the convolution kernel defined as

K(x) = F�1 (mT (⇠)) (x),

mT (⇠) =

s(1 + T ⇠2) tanh(⇠)

⇠,

models small-amplitude shallow water waves withsurface tension. Here T 2 R represents the capil-larity.In 2011, Ehrnstrom, Escher and Wahlen proved theexistence of small amplitude sheets of steady solu-tions bifurcating from a two-dimensional kernel forthe water-wave problem with vorticity.A similar result has been given for irrotational waves,but with surface tension, by Toland and Jones.In this talk we show that an analogous result holds forthe capillary Whitham equation, establishing globalone-dimensional and local two-dimensional bifurca-tion analytically. In addition, we present several nu-merical results showing the profile of the waves alongdi↵erent branches.

On Nontraditional Quasi-Geostrophic Equations

Antoine RousseauInria Chile, FranceCarine Lucas, James C. McWilliams

We present nontraditional models where the so-calledtraditional approximation on the Coriolis force is re-moved. In the derivation of the quasi-geostrophicequations, we obtain new terms in �/", where �(aspect ratio) and " (Rossby number) are bothsmall numbers. We provide here some rigorouscrossed-asymptotics with regards to these parame-ters, prove some mathematical results and providephysical properties on the nontraditional models, andsituate them among traditional ones.


Dynamic Transitions of Quasi-Geostrophic Channel Flow

Taylan SengulMarmara University, TurkeyHenk Dijkstra, Jie Shen, Shouhong Wang

The main aim of this talk is to describe the dy-namic transitions in flows described by the two-dimensional, barotropic vorticity equation in a peri-odic zonal channel, one of the cornerstone dynamicalmodel of the ocean and atmospheric circulation.The equation admits a steady state solution whichrepresents a zonal jet. In this talk, the recent ad-vances in this problem which addresses the stabilityproblem of the bifurcated periodic solutions will beconsidered. In particular, it will be shown that themodelled flow exhibits either a continuous or catas-trophic transition as the basic zonal jet loses its sta-bility.

Precipitating Quasi-GeostrophicEquations

Leslie SmithUniversity of Wisconsin, Madison, USA

A persistent challenge in atmospheric science is tounderstand the dynamical role of water substance,tightly linked with the complicated dynamics ofclouds and convection. This work presents a frame-work for theory and modeling of precipitating con-vection on mid-latitude large scales, analogous to thedry quasi-geostrophic (QG) equations. A precipitat-ing QG approximation is derived from a distinguishedlimit of parent equations combining asymptoticallyfast cloud microphysics and a Boussinesq core. Thepresence of phase changes from water vapor to liquid

water and vice versa leads to important di↵erencesfrom the dry QG case: (i) the appropriately definedpotential vorticity (PV) changes across phase bound-aries, (ii) the evolution for the PV must be coupledto an evolution for a thermodynamic quantity com-bining temperature and water, and (iii) the inversionto find the pressure involves a nonlinear elliptic op-erator. Some examples will be discussed.

Numerical Algorithms for Sta-tionary Statistical Properties ofDissipative Dynamical Systems

Xiaoming WangFlorida State University, USA

It is well-known that physical laws for large chaoticdynamical systems are revealed statistically.The main concern of this manuscript is numericalmethods for dissipative chaotic infinite-dimensionaldynamical systems that are able to capture the sta-tionary statistical properties of the underlying dy-namical systems.We show that the main ingredients in ensuring theconvergence of the long time statistical properties ofthe numerical schemes are: (1) uniform dissipativ-ity of the scheme in the sense that the union of theglobal attractors of the numerical approximations ispre-compact in the phase space; (2) convergence ofthe solutions of the numerical scheme to the solutionof the continuous system on the unit time interval[0,1] modulo an initial layer, uniformly with respectto initial data from the union of the global attractors.The two conditions are reminiscent of the Lax equiv-alence theorem where stability and consistency areneeded for the convergence of a numerical scheme.Applications to the complex Ginzburg-Landau equa-tion and the two-dimensional Navier-Stokes equa-tions in a periodic box will be discussed.


Special Session 5: Spatial and Evolutionary Aspects in Ecology andEpidemiology

Yuan Lou, Ohio State University, USA and Renmin University, Peoples Rep of ChinaKing-Yeung Lam, Ohio State University, USA

Steve Cantrell, Chris Cosner, University of Miami, USA

Spatial heterogeneity is indispensable in explaining many important phenomena in ecology and epidemiology,e.g. the persistence of complicated foodwebs and the definition of the basic reproduction number in a spatialhabitat. A natural and challenging question is to understand the e↵ect of space in the evolution of variousprocesses including animal behavior and the spread of diseases. Prompted by these considerations, a varietyof mathematical models has been proposed and studied, including PDE models, patch models, integral-di↵erential models, etc. In this special session we will bring together people across di↵erent modelingframeworks both to review the current state-of-the-art of the area, and to discuss the emerging theoreticalquestions and their mathematical challenges.

On Competition of Species withDi↵erent Di↵usion Strategies

Elena BravermanUniversity of Calgary, CanadaMd. Kamrujjaman, L. Korobenko

We study the interaction between two species choos-ing di↵erent types of dispersal in a heterogeneous en-vironment, where in addition intrinsic growth ratesand carrying capacities can be di↵erent. As a par-ticular case we assume that one of them moves inthe direction of most per capita available resourceswhile the other choose a di↵erent type of di↵usion.Then, only the spatial profile of the intrinsic growthrate matters: if they coincide, together with spatiallyheterogeneous carrying capacities, then competitiveexclusion of the latter population is inevitable. How-ever, the situation may change if intrinsic growthrates for the two populations have di↵erent spatialforms, or carrying capacities are di↵erent.We also study a Lotka system describing two popu-lations which can compete or cooperate, and each ofthem chooses its di↵usion strategy as the tendencyto have a distribution proportional to a certain posi-tive prescribed function. We prove that higher di↵u-sion coe�cients are detrimental while higher growthrates, as well as lower resources sharing, are benefi-cial for population survival, either for similar di↵u-sion strategies or whenever one of the species choosesthe carrying capacity driven dispersal. We considerthe case when the choice of dispersal strategies guar-antees coexistence, and compare di↵erent di↵usionstrategies. The case of several species is discussed(work in progress).

Population Genetic PDE Modelsof Dispersal and Spatially VaryingSelection

Reinhard BuergerUniversity of Vienna, AustriaLinlin Su

In population genetics we study how the genetic com-position of biological populations is shaped by ecolog-ical and genetic factors such as selection, mutation,recombination, or migration. This talk will focus on

PDE models of a spatially distributed population dis-persing in a heterogeneous environment. We extendclassical and recent results, which assume a singlegene locus under selection, by considering two re-combining loci under selection. This does not onlyadd biological realism, but also substantial mathe-matical di�culties caused by the more complicatedstructure of the underlying PDEs. In particular, theequilibrium structure depends strongly on the recom-bination rate. We use perturbation techniques to ex-plore the existence, stability and other properties ofstationary solutions, called clines, for various limit-ing cases. Several open problems will be mentioned.This talk will be based on joint work with Dr. LinlinSu.

Spatial Sorting: Travelling Wavesand Accelerating Fronts

Vincent CalvezCNRS & ENS de Lyon, France

I will present several aspects of dispersal evolutionduring expansion of an invasive species: 1) existenceof travelling waves for a minimal reaction-di↵usion-selection model in the case of bounded dispersalrates; 2) front acceleration in the case of unboundeddispersal rates. Interestingly, analysis is performedfor a population dynamics equation which is struc-tured with respect to both space and dispersal rate.

Resident-Invader Dynamics in Infi-nite Dimensional Systems

Robert Stephen CantrellUniversity of Miami, USAChris Cosner, King-Yeung Lam

We discuss an extension of the Tube Theorem fromadaptive dynamics to infinite dimensional contexts,including that of reaction-di↵usion equations.


The Reduction Principle, the IdealFree Distribution, and The Evolu-tion of Dispersal Strategies

Chris CosnerUniversity of Miami, USA

The problem of understanding the evolution of dis-persal has attracted much attention from mathemati-cians and biologists in recent years. For reaction-di↵usion models and their nonlocal and discrete ana-logues, in environments that vary in space but notin time, the strategy of not moving at all is oftenconvergence stable within many classes of strategies.This is related to a “reduction principle which statesthat in general dispersal reduces population growthrates. However, when the class of feasible strategiesincludes strategies that generate an ideal free popula-tion distribution at equilibrium (all individuals haveequal fitness, with no net movement), such strate-gies are known to be evolutionarily stable in variouscases. Much of the work in this area involves us-ing ideas from dynamical systems theory and partialdi↵erential equations to analyze pairwise invasibilityproblems, which are motivated by ideas from adap-tive dynamics and ultimately game theory. The talkwill describe some past results and current work onthese topics.

Propagation Phenomena in Nonlo-cal Monostable Equation: TravellingWave Vs Acceleration

Jerome CovilleINRA, FranceM. Alfaro

I will present some recent results on existence/non-existence of travelling wave solution for the homoge-neous nonlocal equation

@tu(t, x) = J ? u(t, x)� u(t, x) + f(u(t, x)), R+ ⇥R

when f is a monostable degenerate function and Ja probability density, i.e. f such that f(0) = 0 =f ‘(0) = f(1) and f > 0 in (0,1). In this situation,the existence of a travelling wave is conditioned bythe tailed of J and the behaviour of f near 0. Whenf(s) = sp(1 � s), I will present the curve that de-scribes the region where Travelling waves exists ornot. I will also present an optimal criteria for theexistence of TW when f is of ignition type.

A Patched Malaria Model: Implica-tions for Control

Patrick de LeenheerOregon State University, USA

We consider the dynamics of a mosquito-transmittedpathogen in a multi-patch Ross-Macdonald malariamodel with mobile human hosts, mobile vectors, anda heterogeneous environment. We show the existenceof a globally stable steady state, and a threshold

that determines whether a pathogen is either absentfrom all patches, or endemic and present at somelevel in all patches. Each patch is characterized bya local basic reproduction number, whose value pre-dicts whether the disease is cleared or not when thepatch is isolated: patches are known as “demographicsinks“ if they have a local basic reproduction num-ber less than one, and hence would clear the dis-ease if isolated; patches with a basic reproductionnumber above one would sustain endemic infectionin isolation, and become “demographic sources“ ofparasites when connected to other patches. Sourcesare also considered focal areas of transmission forthe larger landscape, as they export excess parasitesto other areas and can sustain parasite populations.We show how to determine the various basic repro-duction numbers from steady state estimates in thepatched network and knowledge of additional modelparameters, hereby identifying parasite sources in theprocess. This is useful in the context of control ofthe infection on natural landscapes, because a com-monly suggested strategy is to target focal areas, inorder to make their corresponding basic reproduc-tion numbers less than one, e↵ectively turning theminto sinks. We show that this is indeed a successfulcontrol strategy -albeit a conservative and possiblyexpensive onein case either the human host, or thevector does not move. However, we also show thatwhen both humans and vectors move, this strategymay fail, depending on the specific movement pat-terns exhibited by hosts and vectors.

E↵ects of Di↵usion on Total Biomassin Heterogeneous Continuous andDiscrete-Patch Systems

Donald DeangelisU. S. Geological Survey, USAWei-Ming Ni, Bo Zhang

Theoretical models of populations on a system of twoconnected patches previously have shown that, whenthe two patches di↵er in maximum growth rate andcarrying capacity, and in the limit of high di↵usion,conditions exist for which the total population sizeat equilibrium exceeds that of the Ideal Free Dis-tribution, which predicts that the total populationwould equal the total carrying capacity of the twopatches. However, this result has only been shownfor the Pearl-Verhulst growth function on two patchesand for a single-parameter growth function in contin-uous space. Here we provide a general criterion fortotal population size to exceed total carrying capac-ity for three commonly used population growth ratesfor both heterogeneous continuous and multi-patchheterogeneous landscapes with high population dif-fusion. We show that a necessary condition for thissituation is that there is a convex positive relation-ship between the maximum growth rate and the car-rying capacity, as both vary across a spatial region.This relationship occurs in some biological popula-tions, though not in others, so the result has ecolog-ical implications.


Hawks, Doves, and Population Dy-namics

Theodore GalanthayIthaca College, USAVlastimil Krivan

The hawk-dove model describes a game of conflictwhere two players (perhaps members of a large pop-ulation) compete for some benefit. We have createdan ordinary di↵erential equations model that incor-porates population dynamics, and we approach thestudy of this game through the lens of dynamicalsystems. Hawks and doves are consumers who com-pete for a common resource. We subdivide hawksinto three classes: searchers, handlers, and fighters.Doves are subdivided into two classes: searchers andhandlers. In this talk, I will present preliminary re-sults of the analysis of this model system and discussits ecological importance.

Disease Persistence for a Ross-Macdonald Model in an Environ-ment with Identical Patches

Daozhou GaoShanghai Normal University, Peoples Rep of ChinaChris Cosner, P. van den Driessche

Based on a multipatch Ross-Macdonald model, westudy the role of human and/or mosquito movementon malaria spread between an arbitrary number ofidentical patches. By using a theorem on line-sumsymmetric matrix, we establish an eigenvalue in-equality on the product of a class of nonnegativematrices and then apply it to prove that the basicreproduction number of the multipatch model, R

0

,is always greater than or equal to that of the modelof each patch in isolation. Biologically, this meansthat host and/or vector movement promotes the per-sistence of malaria in an environment with identicalpatches.

Spatial Dynamics for Lattice Dif-ferential Equations with a ShiftingHabitat

Changbing HuUniversity of Louisville, USABingtuan Li

We study a lattice di↵erential model that describesthe growth and spread of a species in a shifting habi-tat. We show that the long term behavior of solutionsdepends on the speed of the shifting habitat edge cand a number c⇤(1) that is determined by the max-imum linearized growth rate and the di↵usion coef-ficient. We demonstrate that if c > c⇤(1) then thespecies will become extinct in the habitat.

Spatiotemporal Virulence DynamicsDuring an Epidemic in the Presenceof Multiple Evolutionary Trade-O↵s

Paul HurtadoUniversity of Nevada, Reno, USA

In recent years, empirical studies have fueled devel-opment of mathematical theory that describes theevolutionary dynamics of pathogen characteristics asthey spread among their hosts. Virulence, or thedegree to which a pathogen causes host morbidityor mortality, is often the primary parasite trait ofinterest. One of the primary mechanisms drivingvirulence dynamics is the well known Transmission-Virulence trade-o↵ in which parasite strains exhibit astrong correlation between how sick they make theirhosts (virulence) and how easily they are transmit-ted. Here I will present results describing how asecond, newly described Movement-Virulence trade-o↵ (between host movement and pathogen virulence)can alter those spatiotemporal virulence dynamics.These results have important implications for study-ing spatiotemporal epidemic data and may help ex-plain virulence dynamics during the epidemic phaseof many emerging infectious diseases.

Population Dynamics of Honeybeeswith Varroa Destructor As Parasiteand Virus Vector: the PotentialE↵ects of foraging behavior

Yun KangArizona State University, USAKomi Messan, Krystal Blanco, Talia Davis,Ying Wang, Gloria DeGrandi-Ho↵man

The worldwide decline in honeybee colonies duringthe past 50 years has often been linked to the spreadof the parasitic mite Varroa destructor and its in-teraction with certain honeybee viruses carried byVarroa mites. In this talk, we study a honeybee-mite-virus model and a honeybee-mite model withdispersal to have a better understanding of the syn-ergistic e↵ects of parasitism and virus infections aswell as the foraging behavior of honeybees on hon-eybee population dynamics and its persistence. In-teresting findings from our work include: (a) Dueto Allee e↵ects experienced by the honeybee popula-tion, initial conditions are essential for the survival ofthe colony. (b) Low adult honeybees to brood ratioshave destabilizing e↵ects on the system which gener-ate fluctuating dynamics that lead to a catastrophicevent where both honeybees and mites suddenly be-come extinct. This catastrophic event could be po-tentially linked to Colony Collapse Disorder (CCD)of honeybee colonies. (c) Virus infections may havestabilizing e↵ects on the system, and parasitic mitescould make disease more persistent. (d) How mayforaging behavior of honeybees stabilize/destabilizethe honeybee population dynamics in the presence ofparasitic mites. Our model illustrates how the syn-ergy between the parasitic mites and virus infections


consequently generates rich dynamics including mul-tiple attractors where all species can coexist or goextinct depending on initial conditions. Our findingsmay provide important insights on honeybee virusesand parasites and how to best control them.

Dirac Concentrations in an Integro-PDE Arising from Evolution ofDispersal.

King-yeung LamThe Ohio State University, USAWenrui Hao, Yuan Lou

We consider an integro-PDE model for a populationstructured by the spatial variables and a trait vari-able a↵ecting the dispersal coe�cients. Competitionfor resource is local in spatial variables, but nonlo-cal in the trait variable. We focus on the asymptoticprofile of positive steady state solutions. Our resultshows that in the limit of small mutation rate, thesolution remains regular in the spatial variables andyet concentrates in the trait variable and forms Diracconcetrations (i) at one boundary point; (ii) the in-terior; or (iii) at both boundary points. The maintechniques are the perturbed test function approachfor viscosity solution, a Liouville result on a cylin-der, and elliptic DeGiorgi-Nash-Moser estimates forthe obligue derivative problem. Finally, connectionsto notions and concepts in evolutionary game theorywill be discussed. This is joint work with Wenrui Hao(MBI) and Yuan Lou(Ohio State).

Cross Di↵usion Models in Popula-tion Biology

Yuan LouRenmin University/Ohio State, Peoples Rep ofChinaWei-Ming Ni, Michael Winkler, Shoji Yotsu-tani

Cross-di↵usion system is an important class ofreaction-di↵usion problems. At the individual level,the basic underlying assumption for cross-di↵usionis that the transition probability only depends upondeparture conditions, e.g., population density and en-vironmental condition at the departure location. Wewill discuss the Shigesada-Kawasaki-Teromoto modelfor two competing species. This talk is based on jointworks with Wei-Ming Ni, Michael Winkler, ShojiYotsutani.

Special Solutions of the Kirkpatrick-Barton System

Judith MillerGeorgetown University, USA

In an influential study, Kirkpatrick and Barton de-veloped and numerically solved a partial di↵erentialequation model for the joint evolution of populationdensity and the mean of a quantitative trait in timeand space. Stationary localized solutions, or pinned

states, of this system model the setting of speciesrange limits through “genetic swamping“; travelingwaves model the advance of an invasive species sub-ject to spatially heterogeneous selection. We proveexistence of both types of solutions and present nu-merical evidence for a spreading speed.

Stray Cat Population Dynamics

Andrew NevaiUniversity of Central Florida, USAJe↵ Sharpe

Stray cat populations cause ecological destructionand spread many diseases in places that people live.Here, we describe a mathematical model for theirpopulation dynamics. The gender-based model in-cludes kittens, adult females and adult males. Anet reproduction number R

0

distinguishes betweeenpopulation extinction (R

0

< 1) and population per-sistence (R

0

> 1). In a separate talk, the modelwill be extended to include the spatial movement ofadult males between patches and the spread of felineleukemia.

Algae-Herbivore Interactions withAllee E↵ect and Chemical Defense

Samares PalUniversity of Kalyani, IndiaJoydeb Bhattacharyya

Macroalgae exhibit a variety of characteristics thatprovide a degree of protection from herbivores. Onecharacteristic is the production of chemicals that aretoxic to herbivores. The toxic e↵ect of macro-algaeon herbivorous reef fish is studied by means of aspatio-temporal model of population dynamics with anon-monotonic toxin-determined functional responseof herbivores. The growth rate of macro-algae is me-diated by Allee e↵ect. We see that under certainconditions the system is uniformly persistent. Con-ditions for local stability of the system are obtainedwith weak and strong Allee e↵ects. We observethat in presence of Allee e↵ect on macro-algae, thesystem exhibits complex dynamics including Hopf-bifurcation and saddle-node bifurcation.

E↵ects of Asymmetric Movementon Population Dynamics

Zhisheng ShuaiUniversity of Central Florida, USA

Spatial heterogeneity and spatial movement play animportant role in population dynamics. Spatialmovement in heterogeneous environment or networkscould be symmetric or asymmetric (biased). Thee↵ects of symmetric and asymmetric movement onpopulation dynamics will be investigated using sev-eral ecological and epidemiological models from theliterature.


A Nonlocal Spatial PopulationModel on Continuous Time andSpace

Fan ZhangFlorida Atlantic University, USAChris Cosner

We formulate an integro-di↵erential model based onLevins‘ metapopulation model, on a continuous het-erogeneous environment. Nonlocal dispersal can beused to describe the spatial movement of seeds, ju-

veniles and o↵spring produced by adult individualssuch as sessile animals and plants. The dispersalkernel allows long distance movement on continuumhabitat. In this model, we are interested in the co-existence status when assuming both species use thesame resource which is the space that an individualcould occupy. It is a noticeable e↵ect that the spa-tially heterogeneity could promote coexistence whileassuming one competitor is stronger than the otherone. We are also interested in the evolution of disper-sal strategies between two similar species and showthat the ideal free dispersal strategy is evolutionarilystable in this model.


Special Session 6: Numerical Approximation of Fractional and IntegralDi↵erential Equations

Jie Shen, Purdue University, USAChuanju Xu, Xiamen University, Peoples Rep of China

Progress in the last two decades have demonstrated that many phenomena in various fields of science,mathematics, engineering, bioengineering, and economics are more accurately described by using fractionalderivatives and/or integral di↵erential equations, and they are emerging as a new powerful tool for mod-eling many di�cult type of complex systems. However, due to their non-local and singular nature, it is atremendous challenge to design accurate and e�cient numerical methods for solving fractional and integraldi↵erential equations. The main objective of this special session is to bring together experts for fractionaland integral di↵erential equations to present their latest advances and discuss future directions.

Analysis of Two-Grid Methods forMiscible Displacement Problem byMixed Finite Element Methods

Yanping ChenSouth China Normal University, Peoples Rep ofChina

The miscible displacement of one incompressible fluidby another in a porous medium is governed by a sys-tem of two equations. One is elliptic form equationfor the pressure and the other is parabolic form equa-tion for the concentration of one of the fluids. Sinceonly the velocity and not the pressure appears explic-itly in the concentration equation, we use a mixedfinite element method for the approximation of thepressure equation. In order to find a stable finiteelement discretization method method, we use di↵er-ent discretization method for the concentration equa-tion, such as finite element method with characteris-tic; mixed finite element method with characteristic;expanded mixed finite element method with charac-teristic etc. To linearize the discretized equations, weuse one (two) Newton iterations on the fine grid inour methods. Firstly, we solve an original non-linearcoupling problem. Then, solve a linear system on thefine grid and while in second method we make a cor-rection on the coarse grid between one (two) Newtoniterations on the fine grid. We obtain the error es-timates of two-grid method, it is shown that coarsespace can be extremely coarse and we achieve asymp-totically optimal approximation. Finally, numericalexperiment indicates that two-grid algorithm is verye↵ective.

Deriving the PDE Governing theFunctional Distribution of theAnomalous Paths and Its Multires-olution Galerkin method

Weihua DengLanzhou University, Peoples Rep of ChinaXiaochao Wu, Eli Barkai

. Anomalous dynamics widely appear in cell migra-tion, the motion of mRNA molecules and lipid gran-ules in live cells. Since the pioneering work of Mon-troll and Weiss in 1965, CTRW models have becomea pillar of statistical physics, especially in charac-terizing the anomalous dynamics. When the jump

length and/or waiting times of the CTRW obey(s)the distribution of power law (with divergent first orsecond moment), it describes the anomalous di↵u-sion, i.e., the second moment of the particle’s trajec-tories is a nonlinear function of time t. For perform-ing more deep research, more often the jump lengthsand/or waiting times need to be tempered. This isbecause that the lifetime is finite and physical space isbounded; sometimes it is the reason that we expect tomodel the observed experimental phenomena of thetransition between anomalous dynamics and normalones. In this talk, we will derive the PDE governingthe functional distribution of the tempered anoma-lous paths, and discuss the multiresolution Galerkinmethod of the derived PDE.

Spectral Collocation Methods forFractional Boundary Value Prob-lems

Zeng FanhaiBrown University, USAGeorge Em Karniadakis, Zhiping Mao

We develop spectral collocation methods for variable-order fractional di↵erential equations with two end-point singularities. In order to develop the spec-tral collocation methods, we derive three-term recur-rence relations for both integrals and derivatives ofthe weighted Jacobi polynomials of the form (1 +x)µ1(1 � x)µ2P a,b

j (x) (a, b, µ1

, µ2

> �1), which leadsto the desired di↵erentiation matrices. We applythese new di↵erentiation matrices to construct col-location methods to solve fractional boundary valueproblems and fractional partial di↵erential equationswith two endpoint singularities. Some theoretical ex-planations are investigated to illustrate that the newmethods can achieve better accuracy than the exist-ing numerical methods. We demonstrate that thesingular basis enhances greatly the accuracy of thenumerical solutions by properly tuning the parame-ters µ

1

and µ2

, even for cases that we do not knowexplicitly the form of singularities in the solution atthe boundaries.


Memory Independent Predictor-Corrector Method for Solving Dif-ferential Equations of FractionalOrder

Bongsoo JangUNIST, KoreaThien Binh Nguyen

An accurate and e�cient new class of predictor-corrector schemes are proposed for solving nonlin-ear di↵erential equations of fractional order. By in-troducing a new prediction method which is explicitand of the same accuracy order as that of the correc-tion stage, the new schemes achieve a uniform accu-racy order regardless of the values of fractional order↵. In cases of 0 < ↵ < 1, the new schemes sig-nificantly improve the numerical accuracy compar-ing with other predictor-corrector methods whose ac-curacy depends on ↵. Furthermore, by computingthe memory term just once for both the predictionand correction stages, the new schemes reduce thecomputational cost of the so-called memory e↵ect,which make numerical schemes for fractional di↵er-ential equations expensive in general. Both 2nd-orderscheme with linear interpolation and the high-order3rd-order one with quadratic interpolation are devel-oped and show their advantages over other comparingschemes via various numerical tests.

Well-Conditioned Fractional Col-location Methods Using FractionalBirkho↵ Interpolation Basis

Yujian JiaoShanghai Normal University, Peoples Rep of ChinaLi-Lian Wang, Can Huang

The purpose of this talk is twofold. Firstly, we pro-vide explicit and compact formulas for computingboth Caputo and Riemann-Liouville fractional pseu-dospectral di↵erentiation matrices(F-PSDM) of anyorder at general Jacobi-Gauss-Lobatto points. Weshow that in the Caputo case, it su�ces to computeF-PSDM of order µ 2 (0, 1) to compute that of anyorder k+µ with integer k � 0, while in the modifiedRL case, it is only necessary to evaluate a fractionalintegral matrix of order µ 2 (0, 1). Secondly, we in-troduce suitable fractional JGL Birkho↵ interpola-tion problems leading to new interpolation polyno-mial basis functions with remarkable properties: (i)the matrix generated from the new basis yields theexact inverse of F-PSDM at interior JGL points; (ii)the matrix of the highest fractional derivative in acollocation scheme under the new basis is diagonal;and (iii) the resulted linear system is well-conditionedin the Caputo case, while in the modified RL case, theeigenvalues of the coe�cient matrix are highly con-centrated. In both cases, the linear systems of thecollocation schemes using the new basis can solvedby an iterative solver within a few iterations.

Time Stepping Schemes for Frac-tional Di↵usion

Bangti JinUniversity College London, EnglandRaytcho Lazarov, Zhi Zhou

In this talk we discuss the time stepping schemesfor fractional di↵usion, which involves a fractionalderivative in time. The nonlocality of the fractionalderivative poses significant challenge in the develop-ment and analysis of robust numerical schemes fore�ciently simulating fractional di↵usion. I shall de-scribe recent progresses, e.g., the L1 scheme and con-volution quadrature, together with finite element inspace. Throughout numerical results will be pre-sented to illustrate the convergence theory.

Fourier Spectral Method for Frac-tional Cahn-Hilliard Equation

Zhiping MaoBrown University, USAMark Ainsworth

In this presentation, I am going to present a frac-tional Cahn-Hilliard equation(FCHE) which possessmass conservation law and energy law. The stabil-ity and L1 boundness are studied. Fourier Galerkinmethod is applied to solve FCHE and the conver-gence is analyzed. Numerical tests are showed tostudy the convergence of the scheme and the behav-ior of the FCHE.

Extended Spectral Method for Solv-ing Singular Problems

Jie ShenPurdue University, USASheng Chen

The usual spectral methods will provide high-orderaccuracy for problems with smooth solutions. How-ever, they may not work well for problems with singu-lar solutions due to various facts such as corner sin-gularities, non-matching boundary conditions, non-smooth coe�cients. For many singular problems, itis possible to determine the forms of a few leading sin-gular terms. It is expected that we can improve theconvergence rate by adding these singular terms intothe spectral basis. However, the new system withadded singular terms is usually ill conditioned andhard to solve. We present a new extended spectral-Galerkin method which allows us to split it into twoseparate problems: one is to find an approximationfor the smooth part by a usual spectral method, theother is to determine an approximation to the sin-gular part with k terms by solving a k ⇥ k system.So the new method is very easy to implement, verye�cient and is capable of providing very accurate ap-proximations for a class of singular problems.


A Direct Spectral Method for Op-timal Control Problems Governedby the Time Fractional Di↵usionEquation

Shengzhu ShiHarbin Institute of Technology (HIT), Peoples Repof ChinaBoying Wu, Jiebao Sun, Wenjuan Yao

This paper is devoted to designing and analyzingan e�cient numerical method for unconstrained con-vex distributed optimal control problems governed bytime fractional di↵usion equation. Unlike the generalmethod focusing on the first order optimality condi-tion, we approximate the optimal control problemdirectly with polynomials and turn the problem intoa nonlinear problem (NLP). The error estimate forthe spectral approximation is derived and some nu-merical experiments are presented.

A Fractional Phase-Field Modelfor Two-Phase Flows with TunableSharpness: Algorithms and Simula-tions

Fangying SongBrown University, USAChuanju Xu, George Em Karniadakis

We develop a fractional extension of a mass-conserving Allen-Cahn phase-field model that de-scribes the mixture of two incompressible fluids. Thefractional order controls the sharpness of the in-terface, which is typically di↵usive in integer-orderphase-field models. The model is derived based on anenergy variational formulation. An additional con-straint is employed to make the Allen-Cahn formu-lation mass-conserving and comparable to the Cahn-

Hilliard formulation but at reduced cost. The spatialdiscretization is based on a Petrov-Galerkin spec-tral method whereas the temporal discretization isbased on a stabilized ADI scheme both for the phase-field equation and for the Navier-Stokes equation.We demonstrate the spectral accuracy of the methodwith fabricated smooth solutions and also the abilityto control the interface thickness between two fluidswith di↵erent viscosity and density in simulations oftwo-phase flow in a pipe and of a rising bubble. Wealso demonstrate that using a formulation with vari-able fractional order we can deal simultaneously withboth erroneous boundary e↵ects and sharpening ofthe interface at no extra resolution.

Fast and Accurate Numerical Meth-ods for Fractional Partial Di↵eren-tial Equations

Hong WangUniversity of South Carolina, USA

Fractional PDEs provide an adequate and accu-rate description of transport processes that exhibitanomalous di↵usion and long-range space-time inter-actions. Computationally, because of the nonlocalproperty of these models, the numerical methods of-ten generate dense sti↵ness matrices. Traditionally,direct methods were used to solve these problems,which require O(N3) computations (per time step)and O(N2) mememy, where N is the number of un-knowns. We go over the development of accurate ande�cient numerical methods for FPDEs, which hasan optimal order storage and almost linear computa-tional complexity. These methods were developed byutilizing the structure of the sti↵ness matrices. Nolossy compression or approximation was used. Hence,these methods retaining the same accuracy and ap-proximation/conservation property of the underlyingnumerical methods.


Special Session 8: New Trends in Calculus of Variations and PartialDi↵erential Equations

Irene Fonseca, Carnegie Mellon University, USAGiovanni Leoni, Carnegie Mellon University, USA

Variational formulations of physical and biological questions are central in applied mathematics. The goalof this minisymposium is to provide a forum to discuss recent progress and promising directions in calculusof variations, geometric measure theory, partial di↵erential equations, and applications to science and engi-neering. Thematic areas of focus will include: variational problems involving material defects, multi-scaleproblems, thin structures, homogenization and mixing, and interfacial phenomena.

Cohesive Behaviour Arising in Ho-mogenization of Mumford-ShahType Functionals

Marco BarchiesiUniversita di Napoli Federico II, ItalyGiuliano Lazzaroni, Caterina Ida Zeppieri

I will discuss the e↵ective properties of a compos-ite material whose micro-structure is constituted bya brittle material with periodically distributed softinclusions. In particular, I will show that the pres-ence of the soft inclusions gives rise, in the limit, toa cohesive behaviour and a toughening phenomenon.

The Rigidity Problem for Sym-metrization Inequalities

Filippo CagnettiUniversity of Sussex, England

We will discuss several symmetrizations (Steiner,Ehrhard, and spherical symmetrization) that areknown not to increase the perimeter. We will showhow it is possible to characterize those sets whoseperimeter remains unchanged under symmetrization.We will also characterize rigidity of equality cases.By rigidity, we mean the situation when those setswhose perimeter remains unchanged under sym-metrization, are trivially obtained through a rigidmotion of the (Steiner, Ehrhard or spherical) sym-metral.We will achieve this through the introduction of asuitable measure-theoretic notion of connectedness,and through a fine analysis of the barycenter functionfor a special class of sets. These results are obtainedtogether with several collaborators (Maria Colombo,Guido De Philippis, Francesco Maggi, Matteo Perug-ini, Dominik Stoger).

Lower Semicontinuity of a Class ofIntegral Functionals on the Space ofFunctions of Bounded Deformation

Gianni dal MasoSISSA, ItalyGianluca Orlano, Rodica Toader

We study the lower semicontinuity of some free dis-continuity functionals with linear growth defined onthe space of functions with bounded deformation.The volume term is convex and depends only on theEuclidean norm of the symmetrized gradient. We in-troduce a suitable class of surface terms, which makethe functional lower semicontinuous with respect toL1 convergence.

Homogenization in the Frameworkof A-Quasiconvexity with VariableCoe�cients

Elisa DavoliUniversity of Vienna, AustriaIrene Fonseca

A homogenization result for a family of integral ener-gies will be presented, where the field under consider-ation are subjected to periodic first order oscillatingdi↵erential constraints in divergence form. The talkis based on the theory of A-quasiconvexity with vari-able coe�cients and on two-scale convergence tec-niques.

Recent Applications of QuantitativeStability to Convergence to Equilib-rium

Alessio FigalliUT Austin, USA

Geometric and functional inequalities play a crucialrole in several PDE problems. In view of applicationsto PDEs, a natural question is the following: sup-pose that a function almost solve the Euler-Lagrangeequation associated to some functional inequality. Isthis function close to one of the minimizers? In thissetting, one has to face the presence of bubbling phe-


nomena. In this talk I’ll give a overview of thesegeneral questions using some concrete examples, andthen present some recent applications to the asymp-totic behavior for some fast di↵usion equation relatedto the Yamabe flow.

Domain Formation in MembranesNear the Onset of Instability

Gurgen HayrapetyanOhio University, USAI. Fonseca, G. Leoni, B. Zwicknagl

The formation of microdomains, also called rafts, inbiomembranes is believed to be attributed to the sur-face tension of the membrane. In order to model thisphenomenon, a family of energy functionals involvinga coupling between the local composition and the lo-cal curvature was proposed by Seul and Andelman.We discuss the �-convergence of this family of en-ergy functionals, involving nonlocal as well as neg-ative terms in the regime of strong surface tension.

Formulas for Relaxed Disarrange-ment Densities

Marco MorandottiSISSA, Trieste, ItalyA. C. Barroso, J. Matias, D. R. Owen

Structured deformations provide a multiscale geome-try that captures the contributions at the macrolevelof both smooth geometrical changes and non-smoothgeometrical changes (disarrangements) at submacro-scopic levels. Recently, Owen and Paroni evaluatedexplicitly some relaxed energy densities arising inChoksi and Fonseca’s energetics of structured defor-mations. In this talk, we will show that a di↵erent ap-proach to the energetics of structured deformations,that due to Baıa, Matias, and Santos, confirms theroles of the relaxed densities established by Owen andParoni. In doing so, we give an alternative, shorterproof of Owen and Paroni’s results, and we estab-lish additional explicit formulas for other measuresof disarrangements.

Large Deviations, Gradient Flows,and Taking Limits

Mark PeletierTU Eindhoven, NetherlandsGiovanni Bonaschi, Giuseppe Savare

It is now well understood that there is a strongconnection between gradient flows on one hand andlarge-deviation principles on the other hand. In asense, this connection takes the form of a single func-tional that characterizes both the large deviationsand the gradient-flow behaviour.

In this talk, which is work with Giovanni Bonaschiand Giuseppe Savare, I will show how this insightproduces a unification of both structure and method.I will focus on a very simple stochastic system, andshow how the large-deviation rate functional is re-lated to a generalized gradient flow - with a param-eter. By taking various limits in this parameter werecover both linear gradient-flow behaviour and rate-independent behaviour.The unification lies in the fact that we can base ourentire discussion on this one functional. It charac-terizes the structure and is also the main actor inthe limit-taking, leading both to compactness and tocharacterization of the limit. This work shows howthe connection between large deviations and gradientflows is not only philosophically interesting but alsoprovides tools for analysis.

A Variational Approach to Exis-tence and Uniqueness of CrystallineCurvature Flows

Marcello PonsiglioneSapienza University of Rome, ItalyAntonin Chambolle, Massimiliano Morini

In this seminar I will describe a new approach to ex-istence and uniqueness of crystalline curvature flows.The results are valid in any dimension and for arbi-trary, possibly unbounded, initial closed sets. Thecomparison principle is obtained by means of a suit-able weak (distributional) formulation of the flow,while the existence of a global-in-time solution fol-lows via a minimizing movements approach.

Slow Motion for the One-Dimensional Swift-HohenbergEquation

Matteo RinaldiCarnegie Mellon University, USAGurgen Hayrapetyan

The behavior of solutions of the Swift–Hohenbergequation in a bounded interval I ⇢ R with periodicboundary conditions is studied. Combining resultsfrom �–convergence and ODE theory it is shown thatsolutions that start L1–close to a jump function v, re-main close to v. This can be achieved by regardingthe equation as the L2–gradient flow of a given en-ergy functional and studying the asymptotic behav-ior of solutions of its Euler–Lagrange equation. Thelinearization of such equation provides almost sharpestimates on the tail of the associated energy.


On the Structure of PDE-Constrained Measures and Ap-plications

Filip RindlerUniversity of Warwick, EnglandGuido De Philippis

Vector-valued measures satisfying a PDE constraintappear in various areas of nonlinear PDE theory andthe calculus of variations. Often, the shape of singu-larities that may be contained in these measures, suchas jumps or fractal parts, is of particular interest. Inthis talk, I will first motivate how variational prob-lems in crystal plasticity naturally lead to such PDE-constrained measures and how their shape is physi-cally relevant. Then, I will present a recent generalstructure theorem for the singular part of any vector-valued measure satisfying a linear PDE constraint.As applications, we obtain a simple new proof of Al-berti’s Rank-One Theorem on the shape of deriva-tives of functions with bounded variation (BV), anextension of this theorem to functions of bounded de-formation (BD), and a structure theorem for familiesof normal currents. Further, our structure result forcurrents implies the solution to the conjecture thatif every Lipschitz function is di↵erentiable almost ev-

erywhere with respect to some positive measure (i.e.the Rademacher theorem holds with respect to thatmeasure), then this measure has to be absolutely con-tinuous relative to Lebesgue measure. This is jointwork with Guido De Philippis (SISSA Trieste).

On the Shape of the D-DimensionalCahn-Hilliard Energy Landscape

Maria WestdickenbergRWTH Aachen, GermanyMichael Gelantalis and Alfred Wagner

For mean values close to �1, it is easy to see thatthe constant state is a local energy minimizer ofthe Cahn-Hilliard energy. As already described inthe seminal work of Cahn and Hilliard, stochas-tic fluctuations lead to nucleation of small, droplet-shaped regions of +1, which may then grow and co-alesce. Moreover, whether the regions of +1 growor shrink should depend on whether their mass islarge enough to form a so-called critical nucleus. Wedescribe recent (deterministic) work on the Cahn-Hilliard energy landscape in the regime of mean valueclose to �1 and large system size. Our work leadsto quantitative bounds on the volume and approx-imate droplet-shape of a candidate for the criticalnucleus. Our methods involve Gamma-limits, quan-titative isoperimetric inequalities, and Steiner sym-metrization.


Special Session 9: Stochastic Modeling in Fluid Dynamics: Theory andApproximation

Chuntian Wang, University of California, Los Angeles, USAHonghu Liu, Virginia Tech, USA

Roger Temam, Institute for Scientific Computing and Applied Mathematics,Indiana University Bloomington, USA

Stochastic models of fluid dynamics play a significant role in science, with numerous applications in e.g.climate science, geophysics and engineering. Moreover, sitting at the interface between probability theory,mathematical analysis and theory of parabolic and hyperbolic partial di↵erential equations, these problemsprovide interesting and challenging mathematical complications. Motivated by the need from both theoutside and inside of the mathematical community, our session will focus on the recent advances in stochasticfluid dynamics modeling, especially those derived from mathematical physics like plasma physics, fluiddynamics and thermomechanics. We aim to bring together researchers to discuss such models from boththe theoretical and applied points of view, with topics including the regularity behavior of solutions, thestochastic dynamics and the stochastic numerical implementations.

Abridged Determining Parametersand Data Assimilation for the MHDEquations with Applications to Sta-tistical Solutions

Animikh BiswasUMBC, USA

We establish the existence of determining modes,nodes and data assimilation technique for the twodimensional MHD equations when only one compo-nent of the velocity and magnetic fields are observed.We discuss applications to the determination of sta-tistical solutions inspired by the recent work of Foias,Mondaini and Titi.

Multi-Locality and Fusion Ruleson the Generalized Structure Func-tions in Two-Dimensional andThree-Dimensional Navier-StokesTurbulence

Eleftherios GkioulekasUniversity of Texas Rio Grande Valley, USA

Using the fusion rules hypothesis for three-dimensional and two-dimensional Navier-Stokes tur-bulence, we generalize a previous non-perturbativelocality proof to multiple applications of the non-linear interactions operator on generalized structurefunctions of velocity di↵erences. We shall call thisgeneralization of non-perturbative locality to multi-ple applications of the nonlinear interactions operator“multilocality“. The resulting cross-terms pose a newchallenge requiring a new argument and the introduc-tion of a new fusion rule that takes advantage of rota-tional symmetry. Our main result is that the fusionrules hypothesis implies both locality and multilocal-ity in both the IR and UV limits for the downscale en-ergy cascade of three-dimensional Navier-Stokes tur-bulence and the downscale enstrophy cascade andinverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate

to non-perturbative locality of generalized structurefunctions on all orders, and not the term by term per-turbative locality of diagrammatic theories or closuremodels that involve only two-point correlation andresponse functions.

Stochastic Swift-Hohenberg Equa-tion with Degenerate Linear Multi-plicative Noise

Marco HernandezIndiana University, USAKiah Wah Ong

We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear multiplica-tive noise acting on a finite set of modes of the domi-nant linear flow is introduced. Existence of a stochas-tic flow and a local stochastic invariant manifold forthis stochastic form of SHE are both adressed in thiswork. We show that the approximate reduced systemcorrespoding to the invariant manifold undergoes astochastic pitchfork bifurcation, and obtain numer-ical evidence suggesting that this picture is a goodapproximation for the full system as well.

Compressible Fluid Flows Drivenby Stochastic Forcing

Martina HofmanovaTechnical University Berlin, Germany

We study the Navier-Stokes equations governing themotion of an isentropic compressible fluid in three di-mensions driven by a multiplicative stochastic forc-ing. In particular, we consider a stochastic perturba-tion of the system as a function of momentum anddensity. We establish existence of a finite energyweak martingale solution as well as a weak-stronguniqueness principle and several singular limit re-sults. The talk is based on joint works with DominicBreit and Eduard Feireisl.


Non-Markovian Reduced Equationsfor Stochastic PDEs

Honghu LiuVirginia Tech, USAMickael Chekroun, Shouhong Wang

In this talk, a new approach to deal with the param-eterization problem of the small spatial scales by thelarge ones for stochastic partial di↵erential equationswill be discussed. This approach relies on stochas-tic parameterizing manifolds (PMs) which are ran-dom manifolds aiming to provide—in a mean squaresense—approximate parameterizations of the smallscales by the large ones. Backward-forward systemswill be introduced to give access to such PMs aspullback limits depending on the time history of cer-tain approximations of the low modes dynamics. Itwill be shown that the corresponding pullback limitscan be e�ciently determined in practice, leading inturn to an operational procedure for the derivationof non-Markovian reduced equations able to achievegood modeling performances. Examples will then bepresented to illustrate that the memory e↵ects con-veyed by such reduced systems can play a key role tocapture noise-induced transitions or large excursionscaused by the noise.

Stochastic Systems of Di↵usionEquations with Polynomial Reac-tion Terms.

Phuong NguyenIndiana University, USADu Pham

In this work, we consider the stochastic version of thedi↵usion equations with polynomial reaction termsof arbitrary degrees forced by a multiplicative noise. We first establish the existence local in time ofthe martingale solution and then derive the pathwiseuniqueness to imply the existence of a pathwise so-lution for a limited period of time. The proofs relyon the Skorohod representation, the Gyongy-Krylovtheorem and a stopping time argument.

Martingale and Pathwise Solu-tions to the Stochastic BinghamFluid Equations with MultiplicativeNoise.

Du PhamUTSA, USA

In this talk, the martingale and pathwise solutionsof the stochastic Bingham fluid are studied. We alsofurther show that in two dimension, when the yieldlimit of the stochastic Bingham fluid goes to 0 thenthe pathwise solutions of the Bingham equations willconvergence to the pathwise solutions of the NavierStokes equations in probability.

Invariant Measure Concentratedon C1(T) for the Benjamin-OnoEquation

Mouhamadou SyUniversite de Cergy Pontoise, France

The existence of an invariant measure for a PDEallows to describe its large time dynamics via thePoincare recurrence theorem. In the context ofthe Benjamin-Ono equation, such a measure wasconstructed in any Sobolev space (Deng-Tvzetkov-Visciglia). This allows to establish a reccurence intime property of the periodic in space BO solutionsbelonging to Sobolev spaces. However, these mea-sures neglect the space C1. In this talk we con-struct a measure on H3 invariant under the flowof the BO equation and concentrated on the spaceC1. We give then a description of the long timebehavior of the infinitely smooth in space BO solu-tions. We developp for this equation the stochastic-perturbation/dissipation approach already used inother contexts. We will discuss some qualitativeproperties of this measure. At the end, we willpresent briefly a measure for the Klein-Gordon equa-tion.

CLT and Large Deviation in a ToyQuadratic Model: Whether to Usea Gaussian Approximation?

Molei TaoGeorgia Tech, USA

We consider a simple system evolved under both dis-sipation and quadratic inputs from a fast Ornstein-Uhlenbeck process. A Gaussian approximation canbe obtained via multiscale analysis, and the result isanalogous to central limit theorem and has its owndomain of applicability. However, this approximationfails to correctly characterize rare events in the slow-fast system. An appropriate characterization will bedemonstrated by the establishment of a large devia-tion principle. Joint work with Eric Vanden-Eijnden.

The Stampacchia Maximum Princi-ple for Stochastic Partial Di↵eren-tial Equations and Applications

Roger TemamIndiana University, USAMickael Chekroun, Eunhee Park

Using the truncation technique of Stampacchia, weestablish the positivity of the solutions of certainstochastic partial di↵erential equations, and otherproperties as appropriate. Among the applications,we show how to prove the existence of positive solu-tions of equations in nonlinear population dynamics,where the nonlinearity is an even power of the solu-tion u.


Time Discrete Approximation ofWeak Solutions for StochasticEquations of Geophysical FluidDynamics and Applications

Chuntian WangUniversity of California, Los Angeles, USANathan Glatt-Holtz, Roger Temam

As a first step towards the numerical analysis of thestochastic primitive equations of the atmosphere andthe oceans, we study here their time discretizationby an implicit Euler scheme. From the determin-

istic point of view the 3D Primitive Equations arestudied in their full form on a general domain andwith physically realistic boundary conditions. Fromthe probabilistic viewpoint we consider a wide classof nonlinear, state dependent, white noise forcingswhich may be interpreted in either the Ito or theStratonovich sense. The proof of convergence of theEuler scheme, which is carried out within an abstractframework covers the equations for the oceans, theatmosphere, the coupled oceanic-atmospheric systemas well as other related geophysical equations. Weobtain the existence of solutions which are weak inboth the PDE and probabilistic sense, a result whichis new by itself to the best of our knowledge.


Special Session 10: Complex Systems and Nonlinear Dynamics

Jung-Chao Ban, National Dong Hwa University, TaiwanKuo-Chang Chen, National Tsing Hua University, Taiwan

The Hamiltonian systems, ODE and PDE provide powerful frameworks for the modelling of the complexsystems. Meanwhile, the nonlinear phenomena of complex systems also inspire many interesting problemsin mathematical analysis. This special session is concerned with the nonlinear analysis on various fields ofcomplex dynamical systems. Related topics including the analysis on fractal geometry, N-body problem, andtraveling waves solutions in partial di↵erential equations. The aim of the proposed session is to exchangeour ideas and discuss recent advances in such fields.

Entropy of Tree Shifts of FiniteType

Jung-Chao BanNational Dong Hwa University, Taiwan

This talk studies the entropy of tree shifts of finitetype with and without boundary conditions. Wedemonstrate that computing the entropy of a treeshift of finite type is equivalent to solving a systemof nonlinear recurrence equations. Furthermore, theentropy of binary Markov tree shifts defined on twosymbols is either 0 or ln2. Meanwhile, the realizationof entropy of one-dimensional shifts of finite type iselaborated, which indicates that tree shifts are ca-pable of rich phenomena. Considering the influenceof three di↵erent types of boundary conditions, say,the periodic, Dirichlet, and Neumann boundary con-ditions, the necessary and su�cient condition for thecoincidence of entropy with and without boundarycondition are addressed.

On the Topology of Tree-Shifts

Chih-Hung ChangNational University of Kaohsiung, Taiwan

Topological behavior, such as chaos, irreducibility,and mixing of a one-sided shift of finite type, is wellelucidated. Meanwhile, the investigation of multidi-mensional shifts, for instance, textile systems is dif-ficult and only a few results have been obtained sofar. This paper studies shifts defined on infinite trees,which are called tree-shifts. Infinite trees have a nat-ural structure of one-sided symbolic dynamical sys-tems equipped with multiple shift maps and consti-tute an intermediate class in between one-sided shiftsand multidimensional shifts. We have shown not onlyan irreducible tree-shift of finite type, but also a mix-ing tree-shift that are chaotic in the sense of Devaney.Furthermore, the graph and labeled graph represen-tations of tree-shifts are revealed so that the verifi-cation of irreducibility and mixing of a tree-shift isequivalent to determining the irreducibility and mix-ing of matrices, respectively. This extends the classi-cal results of one-sided symbolic dynamics. A neces-sary and su�cient condition for the irreducibility andmixing of tree-shifts of finite type is demonstrated.Most important of all, the examination can be donein finite steps with an upper bound.

Syzygies of the N-Center Problem

Kuo-chang ChenNational Tsing Hua University, TaiwanGuowei Yu

In this talk we consider the N -center problem withcollinear centers and identify syzygy sequences whichcan be realized by minimizers of the Lagrangian ac-tion functional. In particular, we show that the num-ber of such realizable syzygy sequences of length L forthe 3-center problem is at least FL+2

�2, where Fn isthe Fibonacci sequence. Moreover, with fixed lengthL, the density of such realizable syzygy sequences oflength L for the N -center problem approaches 1 asN goes to infinity. This is a joint work with GuoweiYu.

Lie Symmetry to Nonlinear Oscilla-tor Systems

Zhaosheng FengUniversity of Texas-Rio Grande Valley, USA

In this talk, we apply the method of Lie point sym-metry to study a generalized second-order nonlinearordinary di↵erential equation, which includes sev-eral physically important nonlinear oscillators such asforce-free Helmholtz oscillator, force-free Du�ng andDu�ng-van der Pol oscillators, modified Emden-typeequation and its hierarchy, generalized Du�ng-Vander Pol oscillator equation hierarchy, and so on, andinvestigate the integrability properties of this rathergeneral equation. We identify and classify severalnew integrable cases for arbitrary values of expo-nents, which determines the tangent vector as wellas the infinitesimal generator. Using the Lie pointsymmetry reduction method, we find the infinitesi-mal generator and canonical coordinates. Combiningthem with the inverse transformation, we obtain thefirst integrals of the second-order nonlinear ODEs un-der certain parametric conditions. Our results showthat many classical integrable nonlinear oscillatorscan be derived as subcases of our results and signif-icantly enlarge the list of integrable equations thatexists in the contemporary literature.


A Note on Zeros of the First KindBessel Functions

Cheng-hsiung HsuNational Central University, Taiwan

In this talk, we consider the zeros‘ distribution ofthe first kind Bessel functions J⌫(z) of order ⌫ � 1.The problem arises from the conjecture given inour previous work which considered the existenceof smooth solutions for one-dimensional compressibleEuler equation with gravity. Expanding the smoothsolution as a power series of parameters, we were con-fronted with the investigation of the zeros‘ distribu-tion of J⌫(z). This is a joint work with Chi-Ru Yang.

Global Structure and Regularity ofSolutions to the Eikonal Equation

Tianhong LiChinese Academy of Sciences, Peoples Rep of China

The Eikonal equation is regarded as a very importantequation in geometric optics. It is derivable fromMaxwell’s equations of electromagnetics by WKBmethods, and provides a link between physical (wave)optics and geometric (ray) optics, and also has manyapplications in optimal control, path planing and etc.It is a Hamilton-Jacobi equation with HamiltonianH(P ) = |P |, which is not strictly convex and smooth.The regularizing e↵ect of Hamiltonian for the Eikonalequation is much weaker than strictly convex Hamil-tonians. There are many new phenomena, for in-stance, the appearance of contact discontinuity. Inthis talk, the set of nondi↵erentiable points is stud-ied. We also discuss the regularity of u in the com-pliment of the set of nondi↵erentiable points.

Periodic Solutions of a Prescribed-Energy Problem for a SingularHamiltonian System

Mitsuru ShibayamaKyoto University, Japan

We study the existence of periodic solutions for aprescribed-energy problem of a Hamiltonian systemwhose potential function has a singularity at the ori-gin like 1/|q|↵(q 2 RN ).It is known that there exists a generalized periodicsolution which may have collisions, and the numberof possible collisions has been estimated. In this talkwe obtain new estimation of the number of collisions.Especially we show that the obtained solution has nocollision if N � 2 and ↵ > 1.

Birkho↵ Normal Form for NullForm Wave Equations

Chi-Ru YangMcMaster University, CanadaWalter Craig, Amanda French

In this talk, we will construct third order Birkho↵normal forms transformations for the class of waveequations on Rn for n � 3 which are both Hamilto-nian PDEs and null forms. We will identify the nullcondition as the vanishing of the three-wave interac-tion coe�cients on the cubic order resonant variety.The main point of the construction is that the nor-mal forms transformation is a continuous mappingof an appropriate Sobolev space which removes thequadratic nonlinear terms of the equation, and this inturn gives a new proof via canonical transformationsof the global in time existence theorems of S. Klain-erman and J. Shatah for null form wave equationsfor small data. This is a progress report with WalterCraig (McMaster University and the Fields Institute)and Amanda French (Haverford).


Special Session 11: PDEs with Applications in Biology, Fluid Mechanicsand Material Sciences

Yi Li, California State University at Northridge, USAYuanwei Qi, University of Central Florida, USA

Jack Xin, UC Irvine, USA

This session will bring experts of theoretical and numerical PDEs fromseveral areas of applications to communicate up to date progress on evolutionary equations modeling TuringPattern Formation, Population Dynamics and Infection Disease, Chemical Waves and Combustion, Turbu-lent Flame and G-equation and Surface Reaction in Material Sciences. In particular, both theoretical andnumerical results will be presented to stimulate collaboration to study challenge problems as teams in future.

Bifurcation for a Free BoundaryProblem Modeling the GrowthTumors with the Nonlinear Prolif-eration/death Rate

Fengjie LiChina University of Petroleum, Peoples Rep of China

In this paper, we study bifurcations for a free bound-ary problem modeling the growth of tumors underthe action of inhibitors. We introduce the e↵ect of thedrug into the net proliferation / death rate which isnonlinear for the unknown function in the tumor re-gion. Symmetry-breaking solutions bifurcating fromthe radially symmetric stationary solutions are ob-tained.

Analysis of Some Nonlinear EllipticSystems

Congming LiShanghai Jiao Tong University, Peoples Rep ofChinaWenxiong Chen, Ze Cheng, Genggeng Huang,Yutian Lei, Chao Ma, John Villabvert

This talk is on the well-known Hardy-Littlewood-Sobolev and the related Schrodinger and Riesz typesystems. We present a brief survey on some im-portant known results, a short introduction of someof our recent results, and also a brief description ofsome basic problems that we are interested. Beyondthe basic qualitative properties such as the existence,non-existence, and classification of positive solutions,we are also interested in the integrability, asymptoticat infinite, and symmetries of positive solutions.

The Stability of Travelling Wavesfor General Autocatalytic ReactionSystems

Yi LiCalifornia State University, Northridge, USAYaping Wu

We investigate the following autocatalytic reactionsystem

(ut = d4u� uqvp

vt = 4v + uqvp.(1)

System (1) can describe the following autocatalyticstep in isothermal, autocatalytic chemical reactionschemes

qA+pB ! (p+q)B, isothermal reaction rate kuqvp

where the autocatalysis B is stable and does notdecay to further product of reaction,

u: concentration of reactant A,v: concentration of autocatalyst B.

Cubic autocatalytic step:

2A+B ! 3B, rate k1

a2b,

System (1) also arises in thermal-di↵usive combus-tion problems, and mathematical biology (e.g. epi-demic models).

Turbulent Flame Speeds in G-Equation Models and ABC Flow

Yu-Yu LiuNational Cheng Kung University, Taiwan

In turbulent combustion theory, the G-equation mod-els are level set Hamilton-Jacobi equations where themotion of the flame front is described by a prescribedflow velocity and a laminar velocity. Besides theconstant laminar flame speed, the curvature and thestrain terms may be added into the laminar velocityfor the flame stretching e↵ect. I will report the cur-rent numerical study on G-equation models in threedimensional space with flow velocity chosen as theArnold-Beltrami-Childress (ABC) flows. The numer-ical methods are finite di↵erence WENO schemes onmonotone Hamiltonian.


Soret and Dufour E↵ects on Nat-ural Convetion from a Cone inViscoelastic Fluid by the Lineariza-tion Method

Gilbert MakandaCentral University of Technology, So AfricaPreciuos Sibanda

The paper applies non-perturbation successive lin-earization method to solve the nonlinear coupledboundary value problem on Soret and Dufour e↵ectsfrom a cone in viscoelastic fluid. The partial dif-ferential equations are transformed into a system ofordinary di↵erential equations which are then solvedusing the successive linearization method (SLM). Theboundary layer velocity, temperature and concentra-tion profiles are numerically computed for di↵erentviscoelastic, concentration, buoyancy, Soret and Du-four numbers. The numerical method is based on theChebyshev spectral collocation technique. Many re-searches have been carried out on Dufour and Sorete↵ects but did not consider natural convection inviscoelastic fluids under linear surface temperature.The results are computed and compared with otherresults in the literature showing a strong agreementand e�ciency of the SLM. These comparisons showthe robustness of the successive linearization methodand that it can be used as an alternative in solvingnonlinear boundary value problems.

Bifurcation Curves for 1D Pre-scribed Mean Curvature Equations

Hongjing PanSouth China Normal University, Peoples Rep ofChinaRuixiang Xing

In this talk we will discuss some recent progress on bi-furcation curves for one-dimensional prescribed meancurvature problems. Notice that the well-knownBernstein-Nagumo condition is violated in this case.The special structure of the mean curvature oper-ator produces some remarkable e↵ects. The lengthparameter plays a key role. By various examples, weshow the rich diversity of bifurcation curves for thequasilinear problem, di↵erent from semilinear cases.

Stochastic Partial Functional Dif-ferential Equations with Delay

Peter PangNational University of Singapore, Singapore

We will present our recent results on the existenceand uniqueness of strong solutions to stochastic par-tial functional di↵erential equations (SPFDEs) withlocally monotone coe�cients, locally Lipschitz non-linearity, and time delay. We note that, whileSPFDEs have important applications, they are farless studied than SPDEs and SFDEs. Our resultsextend and widen the applicability of those of Liu-

Rocker (2010), Caraballo et al (2000), and Taniguchiet al (2002). We illustrate the applicability of ourresults by applying them to a stochastic 2D Navier-Stokes equation with time delay, and a stochasticNicholson’s blowflies equation with time delay.

Traveling Wave to Gray-Scott Sys-tems

Yuanwei QiUniversity of Central Florida, USAXinfu Chen, Yajing Zhang, Zhi Zheng

In this talk I shall present some recent works onreaction-di↵usion systems which arise fromGray-Scott type of models of pattern formation. Weshow the existence, multiplicity and special featuressuch as oscillation of the traveling wave solutions.Applications to models in population dynamics andinfection diseases will also be discussed.

A Multi-Scale Stochastic FiniteElement Method for Random Het-erogeneous Materials and Its Val-idation Using Immersed InterfaceFinite Element Method

Lihua ShenUniversity of Calgary, CanadaXi Xu, Changfeng Wang, Wenting Jin

In this talk, a multi-scale stochastic finite elementmethod for random heterogeneous materials is pre-sented. To show its validation, Monte Carlo methodcombined with the immersed interface finite elementmethod is used to simulate a type of simple ran-dom heterogeneous materials. The numerical re-sults from the multi-scale stochastic finite elementmethod, standard finite element method and the im-mersed interface finite element method are compared.

Symmetry-Breaking Bifurcation fora Free Boundary Problem ModelingSolid Tumour Growth with ECMand MDE Interactions

Ruixiang XingSun Yat-sen University, Peoples Rep of ChinaHongjing Pan

In this talk, we deal with a free boundary problemmodeling solid tumour growth with ECM and MDEinteractions. This problem has a unique radially sym-metric stationary solution. We show that symmetry-breaking solutions bifurcate from the radially sym-metric stationary solutions.


Motion by Mean Curvature for aSecond Order Gradient Theory

Nung Kwan (Aaron) YipPurdue University, USADrew Swartz

We prove in a radially symmetric geometry, the con-vergence in the sharp interfacial limit, to motion bymean curvature of a second order gradient modelfor phase transition. This is in spirit similar to theclassical Allen-Cahn theory of phase boundary mo-tion. However the corresponding dynamical equationis fourth order thus creating some challenging di�cul-ties for its analysis. A characterization and stabilityanalysis of the optimal profile are performed whichare in turn used in the proof of convergence of anasymptotic expansion.

Flame Propagation Enhanced bythe ABC Flow

Yifeng YuUC Irvine, USAJack Xin, Andrej Zlatos

G-equation is a well known model in turbulent com-bustion. its simplest form is a convex but not co-ercive Hamilton-Jacobi equation. The associated ef-fective Hamiltonian can be viewed as the turbulentflame speed. A significant problem is to understandthe dependence of the turbulent flame speed on thevelocity of the ambient fluid. In this talk, I will showthat if the flow velocity is given by the 1-1-1 Arnold-Beltrami-Childress (ABC) flow, the turbulent flamespeed grows linearly with respect to the flow veloc-ity. This is based on a joint work with Jack Xin andAndrej Zlatos.

Classification of Blowup Solutionsfor a Degenerate Parabolic Equa-tion with Nonlinear Gradient Terms

Zhengce ZhangXian Jiaotong University, Peoples Rep of ChinaYan Li

In this talk, we will present our recent results onL1 blowup and gradient blowup. We study the de-generate parabolic equations with nonlinear gradi-ent terms. In the di↵erent ranges of reaction expo-nents, we give the complete classification of blowupresults including L1 blowup, gradient blowup andthe global-in-time existence.

The Study the Global Stabilities ofDi↵usive Predator-Prey Systems ofHolling-Tanner Type and of LesileType

Yi ZhuUniversity of Central Florida, USAYuanwei Qi

In this paper we study the global stabilities of di↵u-sive predator-prey systems of Holling-TannerType and of Lesile type in a bounded domain⌦ ⇢RN with no-flux boundary condition. By using anovel approach, we establish much improved globalasymptotic stability of the unique positive equilib-rium solutions. We also show the result can be ex-tended to more general type of systems with het-erogeneous environment and/or other kind of kineticterms.


Special Session 12: Propagation Phenomena in Reaction-Di↵usionSystems

Hirokazu Ninomiya, Meiji University, JapanMasaharu Taniguchi, Okayama University, Japan

This special session is concerned with mathematical analysis on propagation phenomena and pattern forma-tion in reaction-di↵usion systems. Related topics are equilibrium states, traveling waves, spiral waves andasymptotic behavior of solutions in reaction-di↵usion systems as well as the related free boundary problems.The aim of this special session is to exchange our ideas and promote our knowledge and understanding onthis subject.

Inside Dynamics of Positive Solu-tions in Some Non-Local Equations

Jerome CovilleINRA, FranceOlivier Bonnefon, Jimmy Garnier, LionelRoques

In this talk, I will present some joint work withOlivier Bonnefon, Jimmy Garnier and Lionel Roquesconcerning the inside dynamics of positive solutionsof some non-local equations. The notion of insidedynamics has been recently used to characterise thegenetic structure of a colonizing population modeledby a traveling wave solution of a homogeneous re-action di↵usion equation. The notions of pushedand pulled fronts were at this purpose introduced.I will present some extensions of the notions ofpushed/pulled fronts to the case of positive solu-tions of some homogeneous non-local reaction di↵u-sion equations and the classification that we were ableto achieve in a monostable situation.

On Existence of Wavefront Solu-tions in Mixed Monotone Reaction-Di↵usion Systems with Time Delays

Wei FengUniversity of North Carolina Wilmington, USAWeihua Ruan, Xin Lu

We study the existence of traveling wave solutionsin a general class of mixed quasi-monotone reaction-di↵usion systems with time delays. First, by apply-ing the Schauder Fixed Point Theorem, we prove theexistence of a traveling wave solution between clas-sically defined upper and lower solutions. For betterapplications of the upper-lower solution method onvarious real-life models, the existence result is fur-ther extended under weak form or piecewise smoothupper-lower solutions. In several reaction-di↵usionsystems with time delays, we apply our main resultto establish the existence of traveling wave solutionsflowing towards the positive or coexistent states un-der reasonable conditions on ecological parameters.

Stability of Standing Planar SpotSolutions in Three-ComponentFitzHugh-Nagumo Systems

Hideo IkedaUniversity of Toyama, Japan

Various localized planar patterns are observed inmany reaction-di↵usion systems. Especially, two-component systems are well studied so far and severalmathematical results are obtained. But, even if suchstationary localized planar solutions (stationary spotsolutions) exist stably, one do not get stable travelingspot solutions which are bifurcated from stationaryspot solutions. This implies that such traveling spotsolutions seem to be unstable in two-component sys-tems if they exist. In this talk, we will show theexistence of stationary spot solutions and the stabil-ity properties of them in three-component FitzHugh-Nagumo systems and consider the possibility of thesupercritical drift bifurcation. Note that these resultsare already obtained by Heijster and Sandstede viathe formal analysis (Physica D 275(2014),19-34).

Reduction Approach to a Reaction-Di↵usion System for CollectiveMotions of Camphor Boats

Kota IkedaMeiji University, JapanEi Shin-Ichiro

The collective motion of camphor boats in the waterchannel exhibits both a homogeneous and an inho-mogeneous state, depending on the number of boats.The motion of each camphor boat is described bya traveling pulse in a reaction-di↵usion model pro-posed in Nagayama et al. (2004), in which camphorboats are assumed to be interact each other by thechange of surface tension by di↵usive molecules onthe water surface. In order to verify the inhomoge-neous motion of camphor boats, we have to study thelinearized eigenvalue problem and see the destabiliza-tion of the homogeneous flow. However, the eigen-value problem is too di�cult to analyze even if thenumber of camphor boats is 2. Then we would liketo derive a reduced system from the original modeland analyze it by applying the center manifold theo-rem. Several reaction-di↵usion systems can generatea solution with a pulse shape. Pulse-pulse interac-tion is treated mathematically in Ei et al. (2002), inwhich a reduced system of an ODE form is derived


from a reaction-di↵usion model by applying a centermanifold theorem. Since the delta functions natu-rally arise in our model, the theory established in L2-framework cannot be applied directly. In this talk,we modify the previous results in Ei et al. (2002) andpropose a new approach of reduction to systems withthe delta function.

Exponential Stability of a TravelingWave for an Area Preserving Cur-vature Motion with Two EndpointsMoving Freely on a line

Shimojo MasahikoOkayama University of Science, JapanKagaya Takashi

The asymptotic behavior of solutions to an area-preserving curvature flow of planar curves in the up-per half plane is concerned. Two endpoints of thecurve slide along the horizontal axis with prescribedfixed contact angles. By establishing an isoperimet-ric inequality, we prove the global existence of thesolution. We then study the asymptotic behavior ofsolutions with concave initial data near a travelingwave.

Spreading Speed and AsymptoticProfiles of Solutions for NonlinearFree Boundary Problems in HighSpace Dimension.

Hiroshi MatsuzawaNational Institute of Technology, Numazu College,JapanYihong Du, Maolin Zhou

In this talk we consider nonlinear di↵usion equationsut = �u + f(u) with Stefan free boundary condi-tions. When the nonlinear term f(u) is of monos-table, bistable or combustion type, these problemsare used to describe the spreading of a biological orchemical species with the free boundary representingthe expanding front. In this talk, we consider radiallysymmetric solutions. Then the Stefan free boundaryconditions are reduced to h‘(t) = �µur(t, h(t)) where{x 2 RN : |x| = h(t)} is the free boundary. We showthat when spreading occurs, a logarithmic shiftingappears in sharp estimate of h(t) when N � 2, whilesuch a logarithmic shifting does not appear whenN = 1.

Existence of Traveling Wave So-lutions to Curvature Flows withExternal Driving Force

Harunori MonobeMeiji University, JapanHirokazu Ninomiya

Mean curvature flow is a geometric flow of hypersur-faces. In 2, 3-dimensional Euclidean spaces, meancurvature flow with external driving force is natu-rally appeared as a mathematical model describing

various physical and biological phenomena, e.g., soapfilms, grain boundaries, ray optics and singular limitof population dynamics. In this talk, we consider theexistence of traveling wave solutions, for curvatureflow with external driving force, which is a Jordancurve in 2-dimensional space. When there exist trav-eling wave solutions, we intend to make reference tothe shape of traveling waves too.

Large Time Behavior of the Solu-tions with Spreading Fronts in theAllen-Cahn Equation

Mitsunori NaraIwate University, JapanHiroshi Matano

In this talk, we consider the Cauchy problem for theAllen-Cahn equation ut = �u + f(u) on Rn withn � 2, where the nonlinearity f is of bistable andunbalanced type. We show that, under some assump-tions on the inital value, the solution develops a well-fromed front whose coutours are nearly shperical andare spreading as time goes to infinity.

Mean Curvature Flow of EntireGraphs Evolving Away from theHeat Flow

Xuan Hien NguyenIowa State Univ., USAGregory Drugan

We present two initial graphs over the entire Rn,n � 2 for which the mean curvature flow behaves dif-ferently from the heat flow. In the first example, thetwo flows stabilize at di↵erent heights. With our sec-ond example, the mean curvature flow oscillates in-definitely while the heat flow stabilizes. These resultshighlight the di↵erence between dimensions n � 2and dimension n = 1, where Nara and Taniguchiproved that entire graphs in C2,↵(R) converge to so-lutions to the heat equation with the same initialdata.

Layered Interface Systems and ItsDynamics

Hirokazu NinomiyaMeiji University, JapanH. Mitake and K. Todoroki

The dynamics of three dimensional interface motionis often complicated. To reduce the dimension willbe helpful to understand it. For this purpose, we willintroduce a new system of two dimensional interfaceequations. We call it layered interface system. In thetalk I will explain how to derive it and its dynamicsespecially for mean curvature flow.


Micro Phase Separation in HigherDimensions

Yoshihito OshitaOkayama University, Japan

We study the equation for the distribution of theparticle radii obtained by taking the homogenizationlimit of the evolution equation describing the microphase separation phenomenon in higher dimensions.

Traveling Wave Solutions of Quasi-linear Reaction-Di↵usion Systemswith Mixed Quasi-Monotonicity

Weihua RuanPurdue University Calumet, USAWei Feng, Xin Lu

We study the existence of traveling wave solutionsin a general class of mixed quasi-monotone reaction-di↵usion systems with nonlinear di↵usions in theform

@ui/@t = r · (Di (ui)rui) + fi (u) , �1 < x < 1, t > 0

for i = 1, . . . , n. Each function fi is quasi-monotoneincreasing for some components of u =(u

1

, . . . , un)and decreasing for other components of u. Such sys-tems model reaction-di↵usion processes with a den-sity driven di↵usion mechanism. Under certain gen-eral conditions we prove the existence of a travelingwave solution that is between a pair of coupled upperand lower traveling wave solutions. An example ofquasilinear Lotka-Volterra population model is givenas an illustration of application.

Asymptotic Behavior of Solutionsof Virus Dynamics Models withDi↵usion

Toru SasakiOkayama University, JapanTakashi Suzuki

In this talk, we consider the asymptotic behavior ofsolutions of virus dynamics systems with di↵usion,for example,

@u1

@t= d

1

�u1

+ ��m�1

� �u1

u3

,

@u2

@t= d

2

�u2

+ �u1

u3

� au2

,

@u3

@t= d

3

�u3

+ aru2

� bu3

,

with ui = ui(x, t), t � 0, x 2 ⌦, where ⌦ is abounded set in R3, with smooth boundary. Hereu1

, u2

, and u3

denote the populations of uninfectedcells, of infected cells, and of infectious agents, re-spectively. On the boundary of the domain, we im-pose the homogenuous Neumann condition. We canuse the Lyapunov function proposed in Korobeinikov(Bull. Math. Biol., 2004), which deals with the ODEsystem for spatially homogeneous solutions.

An (N-1)-Dimensional Con-vex Compact Set Gives an N-Dimensional Traveling Front

Masaharu TaniguchiOkayama University, Japan

We study traveling fronts to the unbalanced Allen–Cahn equation or cooperation-di↵usion systems inthe N-dimensional Euclidean space for N that islarger or equals to 3. We consider (N-2)-dimensionalsmooth surfaces as boundaries of strictly convexcompact sets in in the (N-1)-dimensional Euclideanspace, and define an equivalence relation betweenthem. We prove that there exists a traveling frontassociated with a given surface and show its stability.The associated traveling fronts coincide up to phasetransition if and only if the given surfaces satisfy theequivalence relation.

Traveling Waves and Stability ofReaction Di↵usion Equations

Rong YuanBeijing Normal University, Peoples Rep of China

In this talk, we would like introduce some resultsabout the asymptotically stability and the conver-gence rate of the pyramidal traveling fronts.


Special Session 13: Chemotactic Cross-Di↵usion in Complex Frameworks

Michael Winkler, University of Paderborn, GermanyDariusz Wrzosek, University of Warsaw, Poland

Chemotaxis is a ubiquitously observed mechanism of interaction between individuals (cells or organisms)in biological systems. Typically the chemotactic interaction is mediated be a chemical agent whose densitygradient determines the direction of movement of the individuals. Chemotaxis mechanisms are known to playoutstanding roles in numerous processes of self organization at all levels of complexity of biological systems.According to various modeling approaches, an adequate mathematical description thereof requires the studyof parabolic PDE systems involving certain cross-di↵usive terms as their most characteristic ingredient.Important results in the literature identify situations in which in the framework of basic models, suchchemotactic cross-di↵usion indeed enforces spontaneous generation of structures or space-time patterns e.g.in the sense of stabilization toward non-constant equilibria, or even in the sense of singularity formation.The goal of the proposed special session is to present up-to-date results in the analysis of chemotaxissystems, with emphasis on qualitative properties of solutions such as boundedness and asymptotic behavioror the occurrence of blow-up or pattern formation. According to recent developments in both modelingand analysis, a particular focus will be on models for chemotactic cross-di↵usion in complex frameworks,including couplings to further mechanisms such as cell proliferation, haptotaxis, or also interaction with asurrounding fluid. By planning to bring together experienced and young researchers working in these areas,we intend to stimulate a fruitful sharing of ideas, and thereby to initiate new directions of future research.

Concentration Waves of Bacteria atthe Mesoscopic Scale

Vincent CalvezCNRS & ENS de Lyon, France

Concentration waves of swimming bacteria Es-cherichia coli were described in his seminal paper byAdler (Science 1966). These experiments gave riseto intensive PDE modelling and analysis, after theoriginal model by Keller and Segel (J. Theor. Biol.1971), and the work of Alt and co-authors in the80s. Together with Bournaveas, Perthame, Raouland Schmeiser, we have revisited this old problemfrom the point of view of kinetic transport equations.This framework is very much adapted to the so-calledrun-and-tumble motion, in which any bacteria mod-ulate the frequency of reorientation (tumble) – andthus the duration of free runs – depending on chem-ical variations in its environment.In this talk, I will present existence results for soli-tary waves both at the macroscopic scale, and at themesoscopic scale. The macroscopic problem consistsof a drift-di↵usion equation derived from the kineticequation after a suitable di↵usive rescaling, coupledto two reaction-di↵usion equations. Mathematicaldi�culties arise at the mesoscopic scale, where theproof of existence of travelling waves require a re-fined description of spatial and velocity profiles.I will also present numerical simulations done in col-laboration with Gosse and Twarogowska, in orderto illustrate some unexpected behavior of the meso-scopic problem.

A New Approach to the Large TimeBehavior in Chemotaxis LogisticModel

Xinru CaoRemin University of China, Peoples Rep of China

In this talk, we present a new approach towards thelarge time behavior of chemotaxis model with logisticsource.

Reaction Enhancement by Chemo-taxis

Elio EspejoUniversidad Nacional de Colombia, ColombiaRamon Plaza; Carlos Malaga, Takashi Suzuki

An interesting problem arising in many contexts ofmath-biology is the study of the relevance of chemo-taxis in reaction-di↵usion processes. I will approachthis problem through a mathematical model repre-senting the fertilization process of corals. As a re-sult we obtain a system of partial di↵erential equa-tions describing the cell dynamics being a↵ected bythree basic phenomena: di↵usion, chemotaxis anda surrounded flux. First we prove that our modelhas in general global classical solutions. Next, wecompare the assymptotic behavior with and with-out chemotaxis. We show the relevance of chemo-taxis after making a systematic adaptation of thewell-known moments technique on bounded domains(usually used for proving blow-up in Keller-Segel sys-tems), to analyze the behavior of the cell dynamicswhen the chemotactic signal increases. The proposedmethod seems to be readily adaptable to many othermodels specially when working in two dimensions.In the context of the proposed model, our analysisshows that in the two dimensional case the remainingfraction of unfertilized eggs at any given time t > 0becomes arbitrarily small if the chemotaxis signal issu�ciently large.


Global Solvability in 2D Keller-Segel-Stokes Systems with DecayingSensitivity

Kentarou FujieTokyo University of Science, Japan

In this talk, we consider two-dimensional Keller-Segel-Stokes systems with decaying sensitivity (e.g.logarithmic sensitivity). It is shown that for all rea-sonable regular initial data, the system has a globaland bounded solution. The proof is based on thelocalization method, which was established in Fujie-Senba (2016).

Global Existence of Smooth Solu-tions to the SKT System in HighDimensional Spaces

Luan HoangTexas Tech University, USATruyen V. Nguyen, Tuoc V. Phan

We investigate the global time existence ofsmooth solutions for the Shigesada-Kawasaki-Teramoto (SKT) system of cross-di↵usion equationsof two competing species in population dynamics. Ifthere are self-di↵usion in one species and no cross-di↵usion in the other, we show that the system hasa unique smooth solution for all time in boundeddomains of any dimension. We obtain this re-sult by deriving global W 1,p-estimates of Calderon-Zygmund type for a class of nonlinear reaction-di↵usion equations with self-di↵usion. These es-timates are achieved by employing Ca↵arelli-Peralperturbation technique together with a new two-parameter scaling argument.

Boundedness and Large Time Be-havior of an Attraction-RepulsionChemotaxis Model with LogisticSource

Haiyang JinSouth China University of Technology, Peoples Repof ChinaShishi Jie

In this talk, we will study an attraction-repulsionKeller-Segel chemotaxis model with logistic source.When repulsion cancels attraction, the relation of thedamping parameters ✓ and the space dimension n isestablished to ensure the existence of classical solu-tion with uniform-in-time bound. Furthermore, forthe classical logistic source f(u) = µu(1�u), we alsostudy the large time behavior of solution under someconditions on the parameters.

Long-Term Behaviour in aChemotaxis-Fluid System withLogistic Source

Johannes LankeitPaderborn University, Germany

We consider the coupled chemotaxis Navier-Stokesmodel with logistic source terms

nt + u ·rn = �n� �r · (nrc) + n� µn2

ct + u ·rc = �c� nc

ut + (u ·r)u = �u+rP + nr� + f, r · u = 0

in a bounded, smooth domain⌦ ⇢ R3 under ho-mogeneous Neumann boundary conditions for n andc and homogeneous Dirichlet boundary conditionsfor u and with given functions f 2 L1(⌦ ⇥ (0,1))satisfying certain decay conditions and� 2 C1+�(⌦)for some � 2 (0, 1).

We construct weak solutions and prove that aftersome waiting time they become smooth and finallyconverge to the semi-trivial steady state (

µ, 0, 0).

Decay Rates of Solutions for 2-DChemotaxis-Navier-Stokes System

Yuxiang LiSoutheast University, Peoples Rep of ChinaZhang Qingshan

We consider an initial-boundary value problem forthe incompressible chemotaxis-Navier-Stokes equa-tion on a bounded domain of two dimension, whichdescribes the spatio-temporal evolution of popula-tions of swimming aerobic bacteria. It is shown byWinkler [ARMA 2014] that the solution converges toa constant stationary state. In the present talk, weshow that the convergence is exponential.

Blow-Up in Finite Time for Solu-tions in Chemotaxis Systems with aSource Term

Monica MarrasUniversity of Cagliari, ItalyS.Vernier-Piro, G.Viglialoro

We investigate a class of parabolic-parabolic Keller-Segel type system in a bounded domain in RN , withN = 2, 3, under di↵erent boundary conditions, withtime dependent coe�cients, a nonlinear cross di↵u-sion and a positive source term. The solutions mayblow up in finite time T and under appropriate as-sumptions on data, we derive an explicit lower boundfor blow-up time.


On a Chemotaxis System with Sin-gular Chemotactic Sensitivity andA Non-Di↵usible Chemical

Cristian Morales-RodrigoUniv. Sevilla, SpainM. Winkler

In this talk we consider, in higher dimensions, a sys-tem of partial di↵erential equations with singularchemotactic sensitivity. We will prove the existenceof global weak solutions and their asymptotic behav-ior.

Quantized Blowup Mechanism for2D Smoluchowski-Poisson Equa-tion: Stationary, Infinite Time, andFinite time

Takashi SuzukiOsaka University, Japan

We study the Smoluchowski-Poisson equation in 2Dbounded domain with smooth boundary. The so-lution is positive, provided with total mass conser-vatin and free energy decreasing. Its stationary stateis the Boltzmann-Poisson equation, where the quan-tized blowup mechanism is observed with the singularlimits described by the Hamiltonian of point vortices.First, blowup in infinite time arises with this singu-lar limit. Concerning the blowup in finite time, next,collapse mass quantization arises with possible colli-sion of sub-collases. Furthermore, total free energyis bounded if and only if any blowup point is simple,which is equivalent to the localized type II blowuprate.

On a Chemotaxis System with Non-local Terms

J. Ignacio TelloUniversidad Politecnica de Madrid, Spain

We will present result of a system of partial dif-ferential equations describing the evolution of apopulation under chemotactic e↵ects with nonlo-cal reaction terms. By introducing global competi-tive/cooperative factors in terms of the mass of thepopulation, we obtain, for a range of parameters, re-sults of the existence of solutions and the asymptoticbehavior.

Explicit Lower Bound of Blow-UpTime to a Parabolic ChemotaxisSystem with Nonlinear Cross-Di↵usion

Stella Vernier-PiroUNICA, ItalyYoushan Tao

Let us consider the chemotaxis model

⇢ut = �u� �r · (h(u)rv), x 2 B

1

(0), t > 0,vt = �v � v + u, x 2 B

1

(0), t > 0,

under homogeneous Neumann boundary conditionsin a unit ball B

1

(0) ⇢ R3 centered at the origin, with� > 0.Under the assumption that h(u) = (u(u + 1)m�1,m 2 R, (u(x, 0), v(x, 0)) = (u

0

(|x|), v0

(|x|)) 2C0(B

1

(0))⇥W 1,1(B1

(0)), whenever m 2 [ 23

, 2], theblow-up time of a classical solution to the corre-sponding initial-boundary problem has an explicitlower bound measured in terms of �,

RB1(0)

up0

andRB1(0)

|rv0

|2q for appropriate p > 1 and q > 1. The

global classical solution exists bounded if m

Some Results on a Parabolic-Parabolic Keller-Segel System withLogistic Term

Giuseppe ViglialoroUniversity of Cagliari, Italy

This talk is concerned with a fully parabolic Keller-Segel system, modeling chemotaxis phenomena,which is defined in a n-dimensional (bounded andsmooth) domain and whose source term is controlledby a sort of logistic function. Precisely, in line withthe contribution “Chemotaxis with logistic source:Very weak global solutions and their boundednessproperties (Winkler, J. Math. Anal. Appl., 348,(2008), 708-729)“, dealing with the parabolic-ellipticversion of the aforementioned problem, our main goalis to give a result of existence of solutions for the pre-sented system and discuss some boundedness prop-erties of such solutions. In addition, we also presentnumerical simulations which inspire further questionsrelated to this model.

Boundary Layers Arising fromChemotaxis Models

Zhian WangHong Kong Polytechnic University, Hong KongHongyun Peng, Changjiang Zhu

In this work, we consider an initial-boundary valueproblem of a hyperbolic system which was derivedfrom a chemotaxis system with singular sensitivitymodeling the initiation of angiogenesis. By imposing(time-dependent) Dirichlet boundary conditions, weshow that as the chemical di↵usion vanishes, the so-lution becomes very sharp around the boundary andhence exhibits the boundary layer phenomenon. Wealso numerically illustrate that the boundary layer isonly a transient behavior: exists in short-time butvanishes in large-time.


Interspecies Competition andChemorepulsion

Dariusz WrzosekWarsaw University, Poland

Classical Lotka-Volterra model of competition is ex-tended to account for the random dispersal of indi-viduals and for their capability to avoid encounterswith competitors by means of a chemo-sensory re-action to the smell of rivals ( chemorepulsion). Weconsider the case of di↵using and non-di↵using re-pellent and study the existence of global in time so-lutions, existence of non-constant steady states andlong time-behaviour. The models are compared withother models of interspecies competition.

How Strong a Logistic DampingCan Prevent Blow-Up for the Mini-mal Keller-Segel Model?

Tian XiangRenmin University of China, Peoples Rep of China

We study a general class of fully parabolic Keller-Segel chemotaxis systems with growth sources, un-der homogeneous Neumann boundary conditions ina multi-dimensional bounded domain. It is recentlyknown that blowup is possible even in the presence ofsuperlinear growth restrictions. Here, we first deriveseveral characterizations on the growth versus bound-edness.. Then we apply these criteria to the minimalKS model with a logistic source u�µu2 and obtain aquantitative description of the competition betweenchemotactic aggregation and logistic damping, and,in particular, obtain an explicit formula independentof the size of the domain, initial data, and Sobolevembedding constants for the logistic damping rate µ

0

such that blow-up is completely ruled out if µ > µ0

.This o↵ers a quantized e↵ect of the logistic sourceon the prevention of blow-ups and hence improvesexisting results in this regard. Moreover, µ

0

tendsto infinity when either di↵usion coe�cient of cells

or chemicals shrinks to zero. Therefore, small di↵u-sion, especially, degenerate or nonlinear di↵usion, en-hances the possibility of the occurrence of unboundedsolutions. This gives a clue on how to produce blow-up solutions for Keller-Segel chemotaxis models withlogistic sources.

Boundedness in a QuasilinearParabolic-Parabolic Keller-SegelSystem Via Maximal Regularity

Tomomi YokotaTokyo University of Science, JapanSachiko Ishida

We consider a quasilinear degenerate parabolic-parabolic Keller-Segel system. Global existence ofweak solutions to the system on the whole spacewas studied by Sugiyama-Kunii (JDE, 2006) and theresult was improved by Ishida-Yokota (JDE, 2012),while global existence and boundedness of solutionsto the nondegenerate system on bounded convex do-mains was established by Tao-Winkler (JDE, 2012)and the convexity assumption was removed by Ishida-Seki-Yokota (JDE, 2014) not only in the case of non-degenerate di↵usion but also in the case of degeneratedi↵usion. The purpose of this talk is to present a sim-ple way to prove global existence and boundedness ofsolutions via maximal regularity.

Global Solutions of a ChemotaxisSystem Without Gradient-Sensing.

Changwook YoonYonsei University, KoreaYong-Jung Kim

We consider a Keller-Segel model describing cell ag-gregation phenomena. The model is formulated un-der the assumption that microscopic scale bacteriadoes not sense the macroscopic scale concentrationgradient of chemical. We focus on the global ex-istence, uniform boundedness of solutions and thesteady states having aggregation structures.


Special Session 14: Nonlinear Evolution Equations and Related Topics

Mitsuharu Otani, Waseda University, JapanTohru Ozawa, Waseda University, Japan

This session will focus on the recent developments in the theory of Nonlinear Evolution Equations andRelated Topics including the theory of abstract evolution equations in Banach spaces as well as the studies(the existence , regularity and asymptotic behaviour of solutions) of various types of Nonlinear PartialDi↵erential Equations.

Properties of Solutions of NonlinearEvolution Equations

Zhivko AthanassovBulgarian Academy of Sciences, Bulgaria

We establish results on the existence, uniqueness,and regularity of solutions of (E) du(t)/dt = Au(t)+f(t, u(t)) in a Banach space. We are concerned withthe case when A does not generate a semi-group.Our assumptions are motivated by the applicationsto parabolic boundary value problems of higher orderin t. The higher order equation is reduced to a firstorder system of the form (E) by introduction the t -derivatives as new unknowns. The approach adoptedin this talk has been employed in the linear case byS. Agmon and L. Nirenberg.

Existence of Three Nontrivial So-lutions for a Nonlinear FractionalLaplacian Problem

Fatma Gamze DuzgunHacettepe University, Turkey

We consider the problem

⇢(��)s u = f(x, u) in ⌦u = 0 in ⌦c,

(1.1)

where⌦ ⇢ RN (N > 1) is a bounded domain witha C2 boundary, s 2 (0, 1), and f : ⌦ ⇥ R ! R is aCaratheodory function. The fractional Laplacian op-erator is defined for any su�ciently smooth functionu : RN ! R and all x 2 R by

(��)s u(x) = CN,s lim"!0

+

Z

RN\B"

(x)

u(x)� u(y)|x� y|N+2s

dy,

(1.2)where B"(x) is the open ball of radius " > 0 centeredat x and CN,s > 0 is a suitable normalization con-stant. For the existence of three nontrivial solutionsof problem (1.1), we make use of the second deforma-tion theorem and some spectral properties of (��)s

if f(x, ·) is sublinear at infinity and make use of themountain pass theorem and Morse theory if f(x, ·) issuperlinear at infinity.

Lifespan of Strong Solutions tothe Periodic Nonlinear SchrodingerEquations Without Gauge Invari-ance

Kazumasa FujiwaraWaseda University, Department of Pure and AppliedPhysics, JapanTohru Ozawa

A lifespan estimate and sharp condition of the initialdata for finite time blowup for the periodic nonlinearSchrodinger equations without gauge invariance arepresented. The condition for finite time blowup issharp from a viewpoint of the average of initial dataand a simple and direct explanation how the condi-tion comes into play in blowup in a general settingsare given. Moreover, an explicit upper bound of thelife span of solutions is obtained.

Well-Posedness for a GeneralizedDerivative Nonlinear SchrodingerEquation

Masayuki HayashiWaseda University, JapanTohru Ozawa

We study the Cauchy problem for a generalizedderivative nonlinear Schrodinger equation with theDirichlet boundary condition. We establish the lo-cal well-posedness results in the Sobolev spaces H1

and H2. Solutions are constructed as a limit of ap-proximate solutions by a method independent of acompactness argument. We also discuss the globalexistence of solutions in the energy space H1.

Analytic Smoothing E↵ect for aQuadratic System of SchrodingerEquations in the Framework ofMass Sub Critical

Gaku HoshinoWaseda University, Japan

We consider the Cauchy problem for a system ofnonlinear Schrodinger equations in the framework ofmass sub critical setting.In particular, we study the analytic smoothing e↵ectfor the solutions to the Cauchy problem for a systemof nonlinear Schrodinger equations under the massresonance condition in the framework of mass subcritical setting with large data.


On the Cauchy Problem for theNonlinear Schrodinger Equationwith Initial Data with Infinite L2-Norm

Ryosuke HyakunaWaseda University, Japan

In this talk, we consider the initial value problem forthe nonlinear Schrodinger equation:

iut + �u+ |u|↵�1u = 0, u(0, x) = u0

(x) 2 Lp(Rn).

We are interested in the problem of the solvability ofthis Cauchy problem for classes of initial date withinfinite L2-norm. In this talk, we focus our attentionon Lp-space, and FLp-spaces and investigate well-posedness and existence of global solutions.

On Some Nonlinear Evolution Equa-tions and Application to Mathemat-ical Models

Akisato KuboFujita Health University, Japan

In this talk we consider initial-Neumann bound-ary value problem of nonlinear evolution equationswith strong dissipation and proliferation arising frommathematical biology and physics formulated as

(NE)

8>>>>>>>>>><

>>>>>>>>>>:

utt = Dr2ut

+r · (�1

(ut, e�u)�

2

(u,ru)ru)+µ

1

ut(1� ut)for (x, t) 2 ⌦⇥ (0,1) (1)

@@⌫

u|@⌦ = 0 on @⌦⇥ (0,1) (2)

u(x, 0) = u0

(x), ut(x, 0) = u1

(x) in ⌦ (3)

where constants D,µ1

are positive, ⌦ is a boundeddomain in Rn with a smooth boundary @⌦ and ⌫is the outer unit normal vector. Under some regu-larity and boundedness conditions of the coe�cient�i(ut, e

�u), i = 1, 2 of (1), we derive the energy esti-mate of (NE), which enables us to show the globalexistence in time and asymptotic behavior of the so-lution. We apply our result to related mathematicalmodels and discuss the property of solutions of them.Ebihara [1] considered (1) in a more general form andproved a local existence in time of the solution. In [2]we deal with (NE) in case �

2

(ut, e�u) ⌘ 1and show

global existence in time of solutions.

References

[1] Y. Ebihara, On some nonlinear evolution equa-tions with the strong dissipation, II, J. Di↵eren-tial Equations, 34, 339-352, 1979.

[2] A. Kubo, H. Hoshino and K. Kimura, Globalexistence and asymptotic behaviour of solutionsfor nonlinear evolution equations related to tu-mour invasion, AIMS Proceedings 2015, 733-744, 2015.

Solvability of the ComplexGinzburg-Landau Equation

Takanori KurodaWaseda University, JapanMitsuharu Otani

We are concerned with the existence of a solution tothe initial value problem for the complex Ginzburg-Landau equation with the Dirichlet boundary valuecondition of the form:8>>><

>>>:

ut � (�+ i↵)�u+ (+ i�)|u|q�2u� �u = f

(t, x) 2 [0, T ]⇥ ⌦,

u(t, x) = 0 (t, x) 2 [0, T ]⇥ @⌦,

u(0, x) = u0

(x) x 2 ⌦.

Here⌦ ⇢ RN is a bounded domain with smoothboundary, T > 0, u is an unknown complex-valuedfunction and u

0

is a given complex-valued function.Given parameters are � > 0, ,↵,�,� 2 R, 2 < qand f is an external force. In this talk, we mainlydiscuss the case where < 0.


Asymptotic Behavior of SomeGradient-Like Systems

Yoji KurosakiWaseda University, JapanMitsuharu Otani

We consider the following second order gradient-likesystem:

u(t) + g(u(t)) +rF (u(t)) = f(t), (1)

where F : RN ! R is analytic, g : RN ! RN is locallyLipschitz and coercive with g(0) = 0 and f : R+ ! Rquickly decays to zero as t tends to infinity. We give asu�cient condition to assure that any bounded globalsolution u of (1) converge to an equilibrium point as ttends to infinity. In this proof we use the Lojasiewiczinequality.

A Priori Decay Estimates for Trav-eling Wave Solutions to a NonlocalDispersive Model for Shallow Water

Long PeiNorwegian University of Science and Technology,NorwayGabriele Bruell and Mats Ehrnstrom

For the Whitham equation, as a model for shallowwater, we prove that if its traveling wave solutionswith supercritical wave speed are continuous on Rand tend to 0 at infinity, then these solutions willdecay exponentially fast at infinity. Based on thea priori decay estimates and the method of movingplanes, we can further prove that those traveling wavesolutions are symmetric solutions with only one crestover the real line.

Global Attractors and Weak Ex-ponential Attractors for StronglyDamped Wave Equations withNonlinear Hyperbolic DynamicBoundary Conditions

Joseph ShombergProvidence College, USAP. Jameson Graber

We discuss the well-posedness and asymptotic behav-ior of a strongly damped semilinear wave equationequipped with nonlinear hyperbolic dynamic bound-ary conditions. Well-posedness is due to some recentsemigroup results, which here were carried out in thepresence of a parameter distinguishing whether theunderlying operator is analytic, ↵ > 0, or only ofGevrey class, ↵ = 0. For ↵ 2 [0, 1], we establish theexistence of a global attractor; the family of globalattractors is upper-semicontinuous as ↵ ! 0+. Wealso prove the existence of a weak exponential at-tractor (a finite dimensional compact attracting setin the weak topology of the phase space). This resultensures the corresponding global attractor also pos-

sesses finite fractal dimension in the weak topology;moreover, the dimension is independent of the per-turbation parameter ↵. In both settings, attractorsare found under minimal assumptions on the nonlin-ear terms.

Stepanov-Like Weighted Asymp-totic Behavior of Solutions to SomeStochastic Di↵erential Equations inHilbert spaces

Chao TangSichuan University, Peoples Rep of ChinaY. K. Chang

we first introduce the notation and properties ofS2-weighted pseudo almost automorphy for stochas-tic processes. And then, we apply the results ob-tained to consider the existence and uniqueness ofS2-weighted pseudo almost automorphic solutions tosome stochastic di↵erential equations in a real sep-arable Hilbert space under global Lipschitz condi-tions. Moreover, we also investigate asymptotic be-havior of solutions to a stochastic di↵erential equa-tion under S2-weighted pseudo almost automorphiccoe�cients without global Lipschitz conditions. Ourmain results extend some known ones in the senseof square-mean weighted pseudo almost automorphyor S2- pseudo almost automorphy for stochastic pro-cesses.

Time Periodic Problem of a SystemDescribing Double-Di↵usive Con-vection Phenomena in the WholeSpace

Shun UchidaWaseda University, JapanMitsuharu Otani

We consider the time periodic problem of the follow-ing equations which describe double-di↵usive convec-tion phenomena in some porous medium.

8>>><

>>>:

@t~u = ⌫�~u� a~u�rp+ ~gT + ~hC + ~f1

@tT + ~u·rT = �T + f2

@tC + ~u·rC = �C + ⇢�T + f3

r·~u = 0

in RN ⇥ [0, S],

where N = 3, 4 and S > 0 denotes the time period.Unknown functions are ~u, T , C and p, which repre-sent the fluid velocity, the temperature, the concen-tration of solute and the pressure respectively.As for given data, a, ⌫, ⇢ are positive constants and~g, ~h are constant vectors. Moreover, ~f

1

, f2

, f3

desig-nate given external forces.The main purpose of this talk is to construct a pe-riodic solution of the system without the smallnesscondition of given data ~f

1

, f2

, f3

via the conver-gence of solutions for some approximate equationsin bounded domains.


Uniqueness of Limit Flow for a Classof Quasi-Linear Parabolic Equations

Tatsuya WatanabeKyoto Sangyo University, JapanMarco Squassina

In this talk, the issue of uniqueness of the limit flowfor a class of quasi-linear parabolic equations

iut + �u+ u�u2 � u+ |u|p�1u = 0 in RN ⇥ (0,1)

is discussed. We shall investigate conditions whichguarantee that the global solutions decay at infin-ity uniformly in time and their entire trajectory ap-proaches a single steady state as time goes to infinity.We also consider a characterization of solutions whichblow-up, vanish or converge to a stationary state forinitial data of the form �'

0

.


Special Session 15: Special Session on Monotone Dynamical Systems andApplications

Hal L. Smith, Arizona State University, USAJanusz Mierczynski, Wroclaw University of Technology, Poland

The theory of monotone dynamical systems grew out of the much earlier and well-developed monotonemethods and comparison theory largely through the work of M.W. Hirsch and H. Matano in the 1980s.Essentially, the theory focuses on the implications of order-preservation of the (semi-) flow map on theasymptotic behavior of solutions. Typically, the asymptotic behavior of order-preserving dynamical systemsis much simpler than for generic systems. Applications of the theory to systems of ordinary di↵erentialequations, delay di↵erential equations, parabolic partial di↵erential equations, and to the discrete-time dy-namics generated by monotone maps and systems of di↵erence equations have grown rapidly. Mathematicalmodels in biology and chemical reaction dynamics are a rich source of applications because state variablesare often intrinsically nonnegative and therefore the requirement that the dynamics be order-preserving isless restrictive. Recently, the theory has been extended to a control theory setting that includes systeminputs and outputs and to random dynamical systems as well to stochastic games. New applications havearisen by identifying novel order relations preserved by special classes of dynamical systems. This specialsession will focus on these recent developments in both theory and applications.

Carrying Simplicies of Continuousand Discrete-Time Systems

Stephen BaigentUCLL, England

I will describe the invariant carrying simplex of com-petitive ecological systems and how it may be used tostudy the stability of fixed points. Both continuous-and discrete-time models will be discussed. I willshow how, in some cases, the carrying simplex canbe shown to be the graph of a convex or concavefunction. Lastly, I will show that an analogue of thecarrying simplex exists for ecological models that arenot necessarily monotone.

Persistence of Aquatic Insect Popu-lations Subject to Flooding

Patrick de LeenheerOregon State University, USA

We consider a logistically growing aquatic insect pop-ulation that is subject to flooding events which followa Poisson process. Each flooding event has two oppo-site e↵ects. On the one hand, the insect populationis decreased instantaneously due to washout; this ismodeled by multiplying the current population levelby a random factor taking values in [0, 1]. On theother hand, the flood also increases the carrying ca-pacity -also modeled by a random factor, but thisone taking values in [1,1). Following this increase,the carrying capacity relaxes back to a baseline. Thismodel is motivated by the fact that right after eachflood, the habitat available for species re-growth hasincreased. But with time, the flood plains dry out un-til a new flooding event occurs. The central questionaddressed here is how various ecological parameters,in conjunction with the random disturbance charac-teristics, a↵ect the persistence of the aquatic species.

Bistable Traveling Waves for Mono-tone Semiflows

Jian FangHarbin Institute of Technology, Peoples Rep ofChinaXiao-Qiang Zhao

In this talk, I will first recall some classical results onthe existence of traveling waves connecting two stablesteady states for typical reaction-di↵usion equationsand their analogues. Such an existence result is thenestablished for a class of bistable evolution systemsin homogeneous/periodic environment from a mono-tone dynamical system point of view. Finally, theobtained results are illustrated with concrete modelsthat may arise from population ecology.

Open Di↵erential Positive Systems:Attractors and Interconnection

Fulvio ForniUniversity of Cambridge, England

The novel notion of di↵erential positivity extendslinear positivity to the nonlinear setting: a nonlin-ear system is di↵erentially positive if its lineariza-tion along trajectories leaves a cone (field) invariant.A monotone system is di↵erentially positive with re-spect to a constant cone. Di↵erential positivity ex-tends monotonicity to manifolds, replacing order re-lations with local orders. The property strongly re-stricts the asymptotic nonlinear behavior, throughsuitable extensions of Perron-Frobenius theory andof results on contraction of the Hilbert metric. Weillustrate these technical points by revisiting classi-cal properties of monotone systems and by provid-ing novel results for systems on manifolds. We alsodiscuss the notion of di↵erential positivity for opensystems. We study the dependence of the system at-tractors on (locally) ordered inputs. We illustratethe notion of positivity preserving interconnections,a key step towards a robust interconnection theoryfor networks of di↵erentially positive systems.


Group Actions on Monotone Skew-Product Semiflows

Mats GyllenbergUniversity of Helsinki, Finland

We discuss a general framework of monotone skew-product semiflows under a connected group action.In a prior work, a compact connected group G-actionhas been considered on a strongly monotone skew-product semiflow. Here we relax the strong mono-tonicity and compactness requirements, and estab-lish a theory concerning symmetry or monotonicityproperties of uniformly stable 1-cover minimal sets.We then apply this theory to show rotational sym-metry of certain stable entire solutions for a class ofnonautonomous reaction-di↵usion equations on Rn,as well as monotonicity of stable traveling waves ofsome nonlinear di↵usion equations in timerecurrentstructures including almost periodicity and almostautomorphy.The talk is based on joint work with Feng Cao andYi Wang.

Monotone Semidynamical Systemswith Dense Periodic Points

Morris HirschUniversity of Wisconsin, Madison, USA

Let Rn be ordered by a solid, pointed closed convexcone K. Let F := {Ft} be an order-preserving semi-dynamical system in a connected open set X ⇢ Rn,with either continuous time (t 2 R

+

) or discrete time(t 2 N). Assume henceforth: Periodic points aredense in X. This implies F is nonchaotic in a strongsense:

Proposition Every point has a neighborhood inwhich F is uniformly equicontinuous.But I think there is a much stronger result:

Conjecture F is globally periodic.This has been verified in several settings:

Theorem The conjecture holds in the followingcases: n 3 and time is discrete, n 4 and timeis continuous, K is polyhedral.

In particular, the conjecture holds when K is an or-thant. Since a globally periodic flow in a polyhedralcone is trivial, we have:

Corollary For a nontrivial cooperative or compet-itive vector field in the positive orthant, there is anonempty open set containing no periodic point.

Competition for Two Essential Re-sources with Internal Storage andPeriodic Input

Sze-Bi HsuNational Tsing-Hua University, TaiwanFeng-Bin Wang, Xiao-Qiang Zhao

We study a mathematical model of two species com-peting in a chemostat for two internally stored essen-tial nutrients , where the nutrients are added to theculture vessel by way of periodic forcing functions.For the case of single species growth, we apply thetheory of monotone dynamical systems to show thatthe population is washed out if a sub-threshold cri-terion holds, while there is a global stable positiveperiodic solution if a super-threshold criterion holds.For the case of competition of two species, we provethat when there is a mutual invasion of both semi-trivial periodic solution , both uniform persistenceand existence of periodic coexistence state are estab-lished.

The Decomposition Formula forStochastic Lotka-Volterra Systemswith Identical Intrinsic GrowthRate and Its Applications to Sta-tionary Motions

Jifa JiangShanghai Normal University, Peoples Rep of ChinaChen Lifeng, Dong Zhao, Zhai Jianliang

We exploit the long-run behavior for stochasticLotka-Volterra systems (SLVS) with identical intrin-sic growth rate both in pull-back trajectory and instationary motion, which is motivated by convec-tion in a rotating layer. It is first proved so-calledthe decomposition formula : every solution processfor SLVS is expressed in terms of a solution for thedeterministic Lotka-Volterra system (DLVS) withoutnoise perturbation multiplied by an appropriate so-lution process of the scalar logistic equation with thesame type noise perturbation. By virtue of this for-mula, it is verified that every pull-back omega-limitset is an omega-limit set for DLVS multiplied by therandom equilibrium of the scalar stochastic logisticequation. We still investigate the weak convergencefor the transition probability function of solution pro-cess as the time tends to infinity. It is shown that abounded orbit for the DLVS deduces the existenceof stationary measure for SLVS supported in a conedecided by the closure of this orbit. This makes us toconstruct many examples to possess a continuum ofergodic stationary measures or multiple isolated er-godic stationary measures. Suppose that the DLVSis dissipative. Then we prove that the set of sta-tionary measures with small noise intensity is tight,and that their limiting measures in weak topologyare invariant with respect to the DLVS as the noiseintensity tends to zero, whose supports are containedin the Birkho↵ center of the DLVS. Finally we pro-vide the complete classification for three dimensionalcompetitive SLVS with identical intrinsic growth rate


in both stationary motions and trajectories. Everyomega limit set for DLVS produces an ergodic sta-tionary measure for SLVS, which makes us to getall ergodic stationary measures and furthermore allstationary measures. Precisely, there are exactly 37dynamic scenrios in terms of competitive coe�cients.Among them, each pull-back trajectory in 34 classesis asymptotically stationary, but possibly di↵erentstationary solution for di↵erent trajectory in sameclass. In each of these 34 classes, every equilibriumfor DLVS produces an ergodic stationary measure,all stationary measures are obtained by all convexcombination of all such ergodic stationary measures.Two of the remain classes possess stochastic closedorbits, and a continuum of ergodic stationary mea-sures supported in cones decided by periodic orbitsfor DLVS respectively, which weakly converge to theHaar measures of periodic orbits as the noise inten-sity tends to zero. In the final class, almost everypull-back trajectory cyclically oscillates around theboundary of the stochastic carrying simplex which ischaracterized by three unstable stationary solutions,the realized trajectories produce a family of station-ary measures which weakly converge to an invariantmeasure supported in the three unstable axial equi-libria. This rigorously proves that a stochastic ver-sion for so called statistical limit cycle exists and thatthe turbulence in a fluid layer heated from below androtating about a vertical axis is robust under stochas-tic disturbances.

A Remark on Global Dynamics ofCompetition Systems in OrderedBanach Spaces

King-Yeung LamThe Ohio State University, USADaniel Munther

A well-known result in [Hsu-Smith-Waltman, Trans.AMS (1996)] states that in a competitive semiflow de-fined on X

+

= X1,+⇥X

2,+, the product of two conesin respective Banach spaces, if (u⇤, 0) and (0, v⇤) arethe global attractors in X

1,+ ⇥ {0} and {0} ⇥ X2,+

respectively, then one of the following three outcomesis possible for the two competitors: either there is atleast one coexistence steady state, or one of (u⇤, 0),(0, v⇤) attracts all trajectories initiating in the or-der interval I = [0, u⇤] ⇥ [0, v⇤]. However, it wasdemonstrated by an example that in some cases nei-ther (u⇤, 0) nor (0, v⇤) is globally asymptotically sta-ble if we broaden our scope to all of X

+

. In thispaper, we give two su�cient conditions that guaran-tee, in the absence of coexistence steady states, theglobal asymptotic stability of one of (u⇤, 0) or (0, v⇤)among all trajectories in X

+

. Namely, one of (u⇤, 0)or (0, v⇤) is (i) linearly unstable, or (ii) is linearlyneutrally stable but zero is a simple eigenvalue. Ourresults complement the counter-example mentionedin the above paper as well as applications that arisein practice.

Mean Field Dynamics in SocialNetworks

Chjan LimRensselaer Polytechnic Institute, USAWeituo Zhang

This talk will present several interesting dynamicsand bifurcations that arise in social networks. Meanfield models are often in the form of simple dynam-ical systems with well defined center manifolds. Alarge subclass have monotonic properties which haveimplications for social consensus.

Dispersal in Advective Environ-ments

Yuan LouRenmin University/Ohio State, Peoples Rep ofChinaKing-Yeung Lam, Frithjof Lutscher, PengZhou

We consider some mathematical models in advec-tive environments, where individuals are exposed tounidirectional flow, with the possibility of being lostthrough the boundary. We study the persistence andrange for a single species. We also consider the evolu-tion of dispersal in such advective environments. Ouranalysis suggests that, in contrast to the case of noadvection, slow dispersal is generally selected againstin advective environments, and fast or intermediatedispersal rate will be favored.

Criteria for the Existence of Prin-cipal Eigenvalue of Time PeriodicCooperative Linear Systems withNonlocal Dispersal

Wenxian ShenAuburn University, USAXiongxiong Bao

This talk is concerned with the criteria for the exis-tence of principal eigenvalues of time periodic coop-erative linear nonlocal dispersal systems with Dirich-let type, Neumann type or periodic type boundaryconditions. It first introduces the definition of prin-cipal eigenvalues of such cooperative systems. Next,it shows that a time periodic cooperative linear non-local dispersal system has a principal eigenvalue inthe following cases: the nonlocal dispersal distance issu�ciently small; the spatial inhomogeneity satisfiesthe so called vanishing condition; or the spatial inho-mogeneity is nearly globally homogeneous. It shouldbe pointed out that a cooperative linear nonlocal dis-persal system may not have a principal eigenvalue.Finally, it discusses some applications of the estab-lished principal eigenvalue theory.


On Heteroclinic Cycles of Competi-tive Maps Via Carrying Simplices

Yi WangUniversity of Sci&Tech of China, Peoples Rep ofChinaJifa Jiang, Lei Niu

In this talk, we concentrate on the e↵ects of hete-roclinic cycles and the interplay of heteroclinic at-tractors or repellers on the boundary of the carryingsimplices for low-dimensional discrete-time compet-itive systems. We first present an alternative theo-rem on the existence of the carrying simplex for thecompetitive mapping. Several concrete discrete-timecompetition models are further analyzed, which doadmit heteroclinic cycles. The criteria on the stabil-ity of the heteroclinic cycle for each model are alsogiven.

Monotone Semiflows with Respectto High-Rank Cones on a BanachSpace

Jianhong WuYork, CanadaLirui Feng, Yi Wang

We consider semiflows in general Banach spaces moti-vated by monotone cyclic feedback systems or di↵er-ential equations with integer-valued Lyapunov func-tionals. These semiflows enjoy strong monotonicityproperties with respect to cones of high ranks, whichimply order-related structures on the !-limit sets ofprecompact semi-orbits. We show that for a pseudo-ordered precompact semi-orbit the !-limit set ⌦ is ei-ther ordered, or is contained in the set of equilibria,

or possesses a certain ordered homoclinic property.In particular, we show that if ⌦ contains no equilib-rium, then ⌦ itself is ordered and hence the dynamicsof the semiflow on ⌦ is topologically conjugate to acompact flow on Rk with k being the rank. We alsoestablish a Poincare-Bendixson type Theorem in thecase where k = 2. All our results are establishedwithout the smoothness condition on the semiflow,allowing applications to such cellular or physiologicalfeedback systems with piecewise linear vector fieldsand to such infinite dimensional systems where theC1-Closing Lemma or smooth manifold theory hasnot been developed.

Asymptotic Behaviour, SpreadingSpeeds and Travelling Waves ofSome Dynamical Systems

Xingfu ZouUniversity of Western Ontario, CanadaTaishan Yi

I will report some recent results on asymptotic behav-ior, spreading speeds and existence/non-existence oftravelling waves of some dynamical systems in formof discrete-time dynamical system. As a byproduct,we obtain some results on the global attractivityofnontrivial constant fixed point and on the existence ofnon-constant fixed point. We then apply the main re-sults to three model systems: (i) a spatially nonlocalintegro-di↵erence equation; (ii) a reaction-di↵usionequation with spatial nonlocality and time delay inthe reaction term; (iii) an equation with nonlocal dif-fusion and delayed non-monotone nonlinearity in thereaction term. The obtained results for these threeequations improve some existing ones by removingthe symmetry of the kernel functions and relaxingthe conditions on the nonlinear terms.


Special Session 16: Dissipative Systems and Applications

Georg Hetzer, Auburn University, USAWenxian Shen, Auburn University, USA

Lourdes Tello Del Castillo, Universidad Politecnica de Madrid, Spain

Dissipative systems arise in many applications. The special session will feature talks from infinite dynamicalsystems theory and random dynamical systems to evolutionary partial di↵erential equations and numericalsimulation. The scope of applications covers reaction-di↵usion systems with local and nonlocal dispersal,ecology, and climate modeling.

Analyticity in Time and Space fora System of Semilinear EvolutionEquations with Variable Coe�cients

Falko BaustianUniversity of Rostock, Germany

We consider the classical Cauchy problem for astrongly parabolic system of M semilinear partial dif-ferential equations of order 2m with analytic coe�-cients

8>>>>>>>>><

>>>>>>>>>:

@u@t

+P�x, t, 1

i@@x

�= f (0)

✓x, t;

⇣@|�|utialx�

⌘

|�|m

◆+

+

8<

:

0 if m = 1;NP

j=1

@@x

j

f (j)✓x, t;

⇣@|�|u@x�

⌘

|�|m

◆if m � 2;

for (x, t) 2 RN ⇥ (0, T );u(x, 0) = u

0

(x) for x 2 RN ,

but with the initial data u0

only in the real interpo-lation space Bs;p,p(RN ).We show analyticity in the space and time variablesof the unique strict solutions u. For the analyt-icity in time we investigate an abstract nonlinearCauchy problem and operators with the maximal Lp-regularity property. For the analyticity in space weapproximate the initial value with analytic functionsand use suitable estimates of the corresponding solu-tions on a complex domain.The results are generalisation of the linear case inP. Takac: Space-Time analyticity of weak solutionsto linear parabolic systems with variable coe�cients,Journal of Functional Analysis 236, 50–88, 2012.

Positive Solutions in Logistic Prob-lems with Nonlinear Mixed Bound-ary Conditions a Spatial Hetero-geneities

Santiago Cano-CasanovaComillas Pontifical University, Spain

In this talk will be analyzed the existence, uniquenessand stability of the positive solutions of a very gen-eral class of logistic problems containing spatial het-erogeneities in the PDE and on the boundary condi-tions and with a nonlinear flux on the boundary, withpositive, negative or changing sign along it. The re-sults will be given in terms of the nodal behaviourand sign of the potencials appearing in the PDE andon the boundary conditions. The boundary condi-

tions considered are Dirichlet on a component of theboundary and nonlinear Robin or Neumann bound-ary conditions on the other. The main techniquesused to achieve the results are bifurcation, continua-tion, monotonicity and blow up methods.

Random Versus Stochastic LatticesDynamical Systems

Tomas CaraballoUniversidad de Sevilla, Spain

The objective of this talk is to report on recent ad-vances in the topic of random dynamical systems gen-erated by stochastic lattice di↵erential systems. Wewill focus on problems containing additive and mul-tiplicative noise and will emphasize the di↵erenceswhen considering a finite number of noisy terms ateach node (essentially the same noise in each node)or a di↵erent noisy perturbation at each one. We willshow how these systems generate a random dynami-cal system possessing a random attractor.

Controlled Explosions: DynamicsAfter Blow-Up Time for SemilinearProblems with a Dynamic Bound-ary Condition

Alfonso CasalTechnical University of Madrid, SpainJesus Ildefonso Diaz, Gregorio Diaz, JoseManuel Vegas

It is well know that in many nonlinear dynamicalproblems the maximal existence time T

maximal

of so-lutions is defined trough the blow-up time of someof the norm of the solution as, e.g., ku(T1)kL1 =+1. Nevertheless, in the case of some ordinarydi↵erential equations it is possible to control theblow-up in such a way that the solution of thecontrolled equation let well defined after the blow-up time T1 (see, A.C. Casal, J.I. Dıaz, J.M. Ve-gas, Complete recuperation after the blow up timefor semilinear problems.Discrete and Continuous Dy-namical Systems. Dynamical Systems, Di↵eren-tial Equations and Applications, AIMS Proceedings,doi:10.3934/proc.2015.0223 2015 pp. 223-229).


The main goal of this work is to extend such controlprocess to some semilinear boundary value problemof the type⇢

��y(r, t) + |y(r, t)|m�1 y(r, t) = 0 in BR(0)⇥ (0, Tmaximal

),@y@t(R, t) + @y

@n(R, t) = |y(r, t)|p�1 y(r, t) + u(t) on @BR(0)⇥ (0, T

maximal

),

where m, p > 0, BR(0) is the ball of RN of radiumR and n is the normal unit exterior vector. In a firstpart we assume that no control is applied (u(t) ⌘ 0)and show that if p > (m+ 1)/2 > 1 then T1

Spatial Population Models withFitness Based Dispersal

Chris CosnerUniversity of Miami, USA

Traditional continuous time models in spatial ecologytypically describe movement in terms of linear dif-fusion and advection, which combine with nonlinearpopulation dynamics to produce semilinear equationsand systems. However, if organisms are assumed tomove up gradients of their reproductive fitness, andfitness is density dependent (for example logistic), theresulting models are quasilinear and may have othernovel features. This talk will describe some modelsinvolving fitness dependent dispersal and some re-sults and challenges in the analysis of such models.

Global and Nonglobal Existence ofSolutions of Source Types of De-generate Parabolic Equations witha Singular Absorption: completequenching phenomenon.

Anh Dao NguyenTon Duc Thang University, VietnamJesus Ildefonso Diaz

We consider nonnegative solutions of source type ofdegenerate parabolic equations with a singular ab-sorption term:

@tu� (|ux|p�2ux)x + u��{u>0} = f(u, x, t), in I ⇥ (0, T ),

with the homogeneous zero boundary condition, inthat p > 2, � 2 (0, 1), and I = (x

1

, x2

) is an open in-terval in R. To show the existence result, we prove apointwise estimate for |ux|, the so called gradient es-timate in N -dimension. We also consider the globaland non-global existence of solutions of the aboveequation. Moreover, we prove that any solution mustvanish identically after a finite time if either the ini-tial data, or the source term is small enough.

Inverse Problems for ParabolicEquations Arising in Ther-mochronology

Dmitry GlotovAuburn University, USAWillis E. Hames, A. J. Meir, Sedar Ngoma

The reconstruction of the temperature history ofminerals allows geologists to better understand var-ious processes within Earth’s crust. One datingmethod is based on the decay of radiogenic potassiumto stable argon. The concentration of argon, which issubject to thermally activated di↵usion, is governedby a parabolic equation with a time-dependent di↵u-sion coe�cient. We study the inverse coe�cient andrelated inverse source problems for this equation. Toensure uniqueness we introduce an integral constraintmotivated by the type of data reported in geologicalliterature. For the constrained problem, we establishexistence and uniqueness. We propose a scheme forsolving the equations numerically and report on theobserved rates of convergence.

Stability Analysis for Positive So-lutions for Classes of SemilinearElliptic Boundary Value Problemswith Nonlinear Boundary condi-tions

Jerome Goddard IIAuburn University Montgomery, USAR. Shivaji

In this talk, we will investigate the stability prop-erties of nontrivial positive steady state solutions ofsemilinear initial-boundary value problems with non-linear boundary conditions. In particular, we will em-ploy a Principle of Linearized Stability for this classof problems to prove su�cient conditions for stabil-ity and instability of positive steady state solutions.These results shed some light on the combined e↵ectsof the reaction term and the boundary nonlinearityon stability properties. If time permits, we will alsodiscuss existence results and provide complete bifur-cation curves in the case of dimension one.

On the Well Posedness of a Dissi-pative Nonlinear System for Li-IonBatteries

David Gomez-CastroUniversidad Complutense de Madrid, SpainJ.I. Dıaz, A.M. Ramos

We study a dissipative model developed in the frame-work of Li-ion batteries, which is a leading candidatefor the generation of automomotive and aerospaceapplications. After presenting some details on themodel, we prove the existence and uniqueness of itssolutions. We consider a pseudo-two dimensionalmodel coupled to a lumped thermal model, that fol-lows the macrohomogeneous approach presented byNewman in 1973. So far, only some numerical studies


of special cases of the nonlinear system or some anal-ysis of the linearized system have been made. Herewe prove, for the first time in the literature, the ex-istence and uniqueness of solutions of the completenonlinear model. The model can be considered asstrongly nonlinear since it consists of several couplednonlinear elliptic and degenerate parabolic equationsin di↵erent spatial dimensions and also involves somenonlinear boundary conditions of Robin type.

Numerical Approximation of Re-gions of Attraction of EquilibriumSolutions in a Climate EBM

Arturo HidalgoUniversidad Politecnica de Madrid, Spain

We study an energy balance model arising in Cli-matology. The model is sensible to the fluctuationsof the Solar constant. The number of equilibriumstates depends on the Solar constant and the shapeof the planetary coalbedo (one of the nonlinearity ofthe model). In this work, we study the regions ofattraction of the stationary states by numerical ap-proximation based upon the finite volume methodwith Weighted Essentially Non-Oscillatory (WENO)reconstruction. This results are in collaboration withL. Tello (UPM).

On Limiting Behavior of StationaryMeasures for Stochastic EvolutionSystems with Small Noise Intensity

Jifa JiangShanghai Normal University, Peoples Rep of ChinaChen Lifeng, Dong Zhao, Zhai Jianliang

This talk presents the limit behavior of a familyof stationary measures for various stochastic evolu-tion systems, which include stochastic ordinary dif-ferential equations, stochastic functional di↵erentialequations and stochastic partial di↵erential equationsdriven by either Brown motion or Levy process withsmall noise intensity as well as stochastic approxi-mation with constant step. Under mild regular con-ditions, it is proved that as noise intensity tends tozero, any limit measure of a tight family for station-ary measures in sense of weak convergence is an in-variant measure of the corresponding systems with-out noise, whose support is contained in the Birkho↵center. This result plays a role both in understandingconcentrating location for stationary measures withsmall noise and in stability for deterministic systemsundergoing noise perturbation. The proof is uniform,and is divided into three steps: probability conver-gence, invariance for limiting measure and the sup-port for limiting measure. Several examples are con-structed to illustrate peculiar property: the supportsfor limiting measures are saddles, however, almostevery trajectory for corresponding deterministic sys-tems without noise is asymptotic to cycles connectingsaddles.

Attractors for the GeneralShigesada-Kawasaki-TeramotoModels on Planar Domains

Dung LeUniversity of Texas at San Antonio, USA

The existence of global and exponential atrractorsis established for generalized Shigesada-Kawasaki-Teramoto models on planar domains. The cross dif-fusion and reaction can have polynomial growth ofany order. If time permits we will discuss the casewhen the self di↵usion is large and show that the dy-namics of the cross di↵usion system will be describedby that of the corresponding ODE system.

A Small-Gain Theorem for Nonlin-ear Stochastic Systems with Inputsand Outputs

Xiang LvShanghai Normal University, Peoples Rep of ChinaJifa Jiang

This paper studies a small gain theorem for nonlinearstochastic equations driven by additive white noise inboth trajectories and stationary distribution. Moti-vated by the most recent work of Freitas and Son-tag, we firstly define the ”input-to-state characteris-tic operator” K(u) of the system in a suitably cho-sen input space via backward Ito integral, and thenfor a given output function h, define the ”gain oper-ator” as the composition of output function h andthe input-to-state characteristic operator K(u) onthe input space. Suppose that the output functionis either order-preserving or anti-order-preserving inthe usual vector order and the global Lipschitz con-stant of the output function is less than the absoluteof the negative principal eigenvalue of linear matrix.Then we prove the so-called ”small gain theorem”:the gain operator has a unique fixed point, the im-age for input-to-state characteristic operator at thefixed point is a globally attracting stochastic equilib-rium for the random dynamical system generated bythe stochastic system. Under the same assumptionfor the relation between the Lipschitz constant of theoutput function and maximal real part of stable lin-ear matrix, we prove that the stochastic system has aunique stationary distribution, which is regarded as astationary distribution version of small gain theorem.These results can be applied to stochastic coopera-tive, competitive and predator-prey systems, or evenothers.


A Robust and E�cient AdaptiveMultigrid Solver for the OptimalControl of Phase Field Formulationsof Geometric Evolution laws

Anotida MadzvamuseUniversity of Sussex, EnglandF. Yang, C. Venkataraman, V. Styles

In this talk, I will present a novel solution strategy toe�ciently and accurately compute approximate solu-tions to semilinear optimal control problems, focus-ing on the optimal control of phase field formulationsof geometric evolution laws. The optimal control ofgeometric evolution laws arises in a number of ap-plications in fields including material science, imageprocessing, tumour growth and cell motility. Despitethis, many open problems remain in the analysis andapproximation of such problems.In the current work we focus on a phase field formula-tion of the optimal control problem, hence exploitingthe well developed mathematical theory for the opti-mal control of semilinear parabolic partial di↵erentialequations. Approximation of the resulting optimalcontrol problem is computationally challenging, re-quiring massive amounts of computational time andmemory storage. The main focus of this work is topropose, derive, implement and test an e�cient solu-tion method for such problems.The solver for the discretised partial di↵erential equa-tions is based upon a geometric multigrid methodincorporating advanced techniques to deal with thenonlinearities in the problem and utilising adap-tive mesh refinement. An in-house two-grid solutionstrategy for the forward and adjoint problems, thatsignificantly reduces memory requirements and CPUtime, is proposed and investigated computationally.Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employedto further improve e�ciency. Along with a detaileddescription of our proposed solution method togetherwith its implementation we present a number of com-putational results that demonstrate and evaluate ouralgorithms with respect to accuracy and e�ciency. Ahighlight of the present work is simulation results onthe optimal control of phase field formulations of geo-metric evolution laws in 3-D which would be compu-tationally in-feasible without the solution strategiesproposed in the present work.

On Some Nonlinear Problems inPoroelasticity

Amnon MeirSouthern Methodist University, USA

Poromechanics is the science of energy, motion, andforces, and their e↵ect on porous material and in par-ticular the swelling and shrinking of fluid-saturatedporous media. Modeling and predicting the mechan-ical behavior of fluid-infiltrated porous media is sig-nificant since many natural substances, for exam-ple, rocks, soils, clays, shales, biological tissues, andbones, as well as man-made materials, such as, foams,

gels, concrete, water-solute drug carriers, and ceram-ics are all elastic porous materials (hence poroelas-tic). In this talk I will describe some nonlinear prob-lems in poroelasticity and their mathematical analy-sis. I will also describe finite element based numericalmethods for approximating solutions of (nonlinear)model problems in poroelasticity, and the availablea-priori error estimates.

On Doubly Nonlinear EvolutionEquations with Nonpotential OrDynamic Relation Between theState Variables

Jochen MerkerHTWK Leipzig University of Applied Sciences,GermanyAles Matas

This talk is about doubly nonlinear evolution equa-tions of the form d

dtBu + Au = f , where A,B are

nonlinear operators and B does not admit a poten-tial. A particular case are systems of doubly nonlin-ear reaction-di↵usion equations

@v@t

� div (a(ru)) = f ,

where u is vector-valued and the operator Au =�div (a(ru)) may be degenerate or singular. After areview of non-potential static relations v = b(u) thecase of an additional dynamic equations for u deter-mining the relation between u and v is discussed, likee.g. @u

@t� �u = 1

✏(v � b(u)) with a relaxation time

✏ > 0.

On the Uniqueness of Weak Solu-tions for a Tra�c Flow Model inTwo Lanes in One Direction

Juan Francisco PadialUniversidad Politecnica de Madrid, Spain

We study a the tra�c flow on long highways with towlines in same direction. Assuming that the driverscontinuously adjust their speed towards what theyconsider to be ideal value under the local tra�c con-ditions; we propose a new mathematical model interms of a class of nonlinear parabolic systems withzero–order coupling. Let ⇢

1

(t, x) be the density inthe slower lane, ⇢

2

(t, x) be the density in the fasterlane at any time t and position x. If we assume thatit is possible for the cars to change lanes, we havethe following doubly nonlinear degenerate parabolicsystem


⇢@@t⇢1

� @@x

�1

@@x⇢1

� V1⇤ (⇢1)

�= F (t, x,⇢

1

, ⇢2

),@@t⇢2

� @@x

�2

@@x⇢2

� V2⇤ (⇢2)

�= �F (t, x,⇢

1

, ⇢2

),

where i is a positive quantity which tells how muchthe change in the number of cars is considered by thedriver and Vi⇤ (⇢i) = ⇢iWi⇤ (⇢i) with Wi⇤ be the cor-responding flow speeds for i = 1, 2 . The net e↵ect oftime changes in density and space changes in flow ata point x and a time t are equal to the exchange be-tween the two lanes and it is described by function F(see, e. g., Gazis, D.C., Tra�c Science. John Wiley& Son, New York 1974; Haberman R., MathematicalModels: Mechanical Vibrations, Population Dynam-ics and Tra�c Flow. Prentice Hall, 1977). Severaltypes of functions F can be chose: a model in whichlane changes were proportional to the square of thedensity in the lane that was the source of depart-ing vehicles and the first power of the emptiness (un-used space) of the lane–admitting vehicles, can be ex-pressed by F (⇢

1

, ⇢2

) := �↵1

⇢21

[�2

�⇢2

]+↵2

⇢22

[�1

�⇢1

]where �i are the maximum allowable densities in therespective lanes.We develop a method based in doubling of the timevariable to prove the uniqueness of weak solution.This technique is inspired in a method introduced byS.N. Kruzhkov in order to prove a L1–contractionproperty for entropy solutions of hyperbolic prob-lems.(*) Partially Supported by the Project MTM2014–57113 of DGISPI (Spain)

Quantitative Concentration of Sta-tionary Measures

Zhongwei ShenUniversity of Alberta, CanadaMin Ji, Yingfei Yi

This talk concerns stationary measures of Fokker-Planck equations associated with a system of ordi-nary di↵erential equations perturbed by “small” mul-tiplicative white noises. I will present some quantita-tive results on the concentration of stationary mea-sures as noises vanish. In particular, the stability orinstability of local attractors and local repellers willbe presented. This directly relates to the stabilityor instability of the ODE system under small whitenoise perturbations.

Global Extinction of Populationwith Allee E↵ect in Advective En-vironment

Junping ShiCollege of William and Mary, USA

We show that in a spatial population model withstrong Allee e↵ect, the population becomes extinctno matter how large the initial population is, if theadvection is strong enough. Hence the extinction

equilibrium is globally asymptotically stable, which isquite di↵erent from the case of small or no advectionthat the dynamics is bistable. We will show resultsin both ODE patch model and reaction-di↵usion-advection model.

On Compact Support Solutions toParabolic Problems with the P-Laplacian

Peter TakacUniversity of Rostock, GermanyJiri Benedikt, Petr Girg, Lukas Kotrla

The validity of the weak and strong comparison prin-ciples for degenerate parabolic equations with the p-Laplacian operator will be discussed for p > 2 and1 < p < 2.This problem is reduced to the comparison of thetrivial solution (⌘ 0) with a nonnegative solutionu(x, t). The problem is closely related also to thequestion of uniqueness of a nonnegative solution viathe weak comparison principle. In this presentation,for p > 2 realistic counterexamples to the unique-ness of a nonnegative solution, the weak compari-son principle, and the strong maximum principle areconstructed with a nonsmooth reaction function thatsatisfies neither a Lipschitz nor an Osgood standarduniqueness condition. Nonnegative multi-bump solu-tions with spatially disconnected compact supportsand zero initial data are constructed between sub-and super-solutions that have supports of the sametype.

On a Green Roof MathematicalModel

Lourdes TelloUniversidad Politecnica de Madrid, Spain

We study a problem of parabolic type based on anenergy balance for buildings with vegetation cover(green roof). The model represents the evolution ofthe temperature in two layers of a green roof: the veg-etation layer and the soil layer. The model is basedon an energy balance and it includes the interactionsbetween these two layers. The model includes dif-ferent shapes of green roof. The spatial domain is asurface and the unknowns are the temperature in avegetal layer and the temperature in the substrate.We present a model including roof’s shape and themathematical treatment of it. Part of this work is incollaboration with M.L. Vilar (UPM).

On a Parabolic-ODE System withChemotactic Terms

J. Ignacio TelloUniversidad Politecnica de Madrid, Spain

We consider a competitive system of di↵erentialequations describing the behavior of two biologicalspecies. The system is weakly coupled and one ofthe species has the capacity of di↵use and moves to-


ward the higher concentration of the second speciesfollowing its gradient, the density function satisfiesa second order parabolic equation with chemotacticterms. The second species does not have motility ca-pacity and satisfies an ordinary di↵erential equation.We prove that the solutions are uniformly boundedand exist globally in time. The asymptotic behaviorof solutions is also studied for a range of parametersand initial data.

Dynamics of Almost-PeriodicForced Parabolic Equations onthe Circle

Yi WangUniversity of Sci&Tech of China, Peoples Rep ofChinaWenxian Shen, Dun Zhou

We consider the skew-product semiflow which is gen-erated by an almost-periodic forced scalar reaction-di↵usion equation on the circle.The structure of theminimal set M is thoroughly investigated under theassumption that the center space associated with Mis no more than 2-dimensional. Such situation nat-urally occurs while, for instance, M is hyperbolic oruniquely ergodic. It is shown in this paper that M isa 1-cover of the hull H(f) provided that M is hyper-bolic. If dimV c(M) = 1 (resp. dimV c(M) = 2 withdimV u(M) being odd), then either M is an almost1-cover of H(f) and topologically conjugate to a min-imal flow in R⇥H(f); or M can be (resp. residually)embedded into an almost periodically (resp. almostautomorphically) forced circle-flow S1 ⇥H(f).

Persistence Criteria for the Nonlo-cal Model and Applications

Xiaoxia XieIllinois Institute of Technology, USAWenxian Shen

Long range dispersal is a common phenomenon in bi-ology and ecology. To have a better understanding ofthe evolution of biodiversity in some ecosystem, thereis a need to understand the influence of nonlocal dis-persal on the survival/persistence of a population.In this talk, I will report on a recent study concern-ing persistence criteria in some nonlocal models ontemporal and spatial heterogeneous environment.I will first present some spectral theory of the as-sociated eigenvalue problem, such as the existenceof the principal eigenvalue. Secondly, I will showsome results of the eigenvalue problem with indefiniteweight functions, which have practical importance inthe context of reserve design or pest control.

Transition Fronts of Fisher-KPPEquations in Locally Spatially Inho-mogeneous Patchy Environments

Aijun ZhangDrexel University, USAErik Van Vleck

The current paper is devoted to the study of spatialpropagation dynamics of species in locally spatiallyinhomogeneous patchy environments or media. For alattice di↵erential equation with monostable nonlin-earity in a discrete periodic media, it is well-knownthat there exists a minimal wave speed such that atraveling wave exists if and only if the wave speed isabove this minimal wave speed. We shall show thatstrongly localized spatial inhomogeneous patchy en-vironments may prevent the existence of transitionfronts. Transition fronts may exist in weaklier local-ized spatial inhomogeneous patchy environments butonly in a finite range of speeds, which implies that itis plausible to obtain a maximal wave speed of exis-tence of transition fronts.


Special Session 17: Quantitative Geometric and Functional Inequalitiesand New Trends in Nonlinear PDEs

Alessio Figalli, UT Austin, USAEmanuel Indrei, CMU, USA

Enrico Valdinoci, WIAS, Germany

The aim of this special session is two-fold: first, to discuss quantitative versions of various geometric andfunctional inequalities and their applications; second, to identify new trends in nonlinear partialdi↵erential equations, including the study of free boundary problems and geometric PDEs.

Systems of Partial Di↵erentialEquations Arising from PopulationDynamics: Free Boundary Prob-lems As a Result of segregation.

Veronica Rita Antunes de Soares QuitaloPurdue University, USALuis Ca↵arelli, Stefania Patrizi, Monica Tor-res

In this talk we will present our latest results for twophase free boundary problems arising from popula-tion dynamics. We will focus on fully nonlinear sys-tems with local interaction and linear systems witha (non local) long range interaction.

Mean Value Theorems for and Ob-stacle Problems on RiemannianManifolds

Brian BensonKansas State University, USAIvan Blank, Jeremy LeCrone

In Euclidean space, mean value theorems for theLaplacian are used to prove many fundamental re-sults in PDE such as maximum principles, Harnackinequalities, interior estimates, and semicontinuityof weakly superharmonic functions. In 1998, Caf-farelli discussed a proof of the mean value theorem forthe Laplacian which di↵ers from the usual approach.Specifically, this di↵erent approach could be gener-alized to a divergence-form elliptic operator when aGreen’s function for the operator and a solution to arelated obstacle problem are known. Generalizing re-sults of many authors, a mean value theorem for theLaplacian on a strongly non-parabolic Riemannianmanifold is given by Ni. Motivated by the work of Niand the usefulness of mean value theorems, we willdiscuss adapting Ca↵arelli’s approach to the settingof Riemanian manifolds to prove mean value theo-rems for elliptic operators.

Classes of Improved Moser-Trudinger Inequalities and TheirGeometric Applications

Alessandro CarlottoETH, Switzerland

Moser-Trudinger inequalities concern, roughly speak-ing, the exponential integrability of functions belong-ing to suitable Sobolev spaces and naturally appearin a variety of geometric contexts, ranging from theconformal to the Kaehler setting. After a generalintroduction, I will describe some recent variationson these themes motivated by the curvature pre-scription problem in the category of Riemann sur-faces with conical singularities. More specifically, Iwill briefly present an important class of improved,scaling-invariant inequalities which lie at the heart ofthe variational/Morse-theoretic approach of the au-thor and A. Malchiodi to the singular uniformizationproblem.

Further Time Regularity forParabolic Equations

Hector Chang-LaraColumbia University, USADennis Kriventsov

In contrast with second order parabolic equations,a particular phenomena of nonlocal models is theloss of time regularity. Even for the fractional heatequation a sudden change in time of the boundarydata (over the complement of the domain) gets im-mediately noticed by the nonlocal term and usuallycreates a discontinuity for the time derivative of thesolution. Together with D. Kriventsov, we recentlyestablished a priori Schauder estimates for nonlocal,fully nonlinear parabolic equations, addressing theregularity of the time derivative of the solution undermild assumption on the boundary data. Our resultextend to new estimates for second order equations.


On a Fractional Monge-AmpereOperator

Fernando CharroUniversitat Politecnica de Catalunya, SpainLuis Ca↵arelli

In this talk we consider a fractional analogue of theMonge-Ampere operator. Our operator is a concaveenvelope of fractional linear operators that are a�netransformations of determinant one of a given mul-tiple of the fractional Laplacian. We set up a rela-tively simple framework of global solutions prescrib-ing data at infinity and global barriers. In our keyestimate, we show that the operator remains strictlyelliptic, which allows to apply known regularity re-sults for uniformly elliptic operators and deduce thatsolutions are classical.

A-Free Rigidity and Applications tothe Compressible Euler System

Elisabetta ChiodaroliEPFL, Lausanne, SwitzerlandE. Feireisl, O. Kreml, E. Wiedemann

Szekelyhidi and Wiedemann showed that anymeasure-valued solution to the incompressible Eulerequations in several space dimensions can be gen-erated by a sequence of exact solutions. This meansthat measure-valued solutions and weak solutions aresubstantially the same for incompressible Euler, thusleading to a very large set of weak solutions. Inthis talk we address the corresponding problem forthe compressible Euler system: can every measure-valued solution to the compressible Euler equationsbe approximated by a sequence of weak solutions?We show that the answer is negative: generalizing awell-known rigidity result of Ball and James, we givean explicit example of a measure-valued solution forthe compressible Euler equations which can not begenerated by a sequence of distributional solutions.We also give an abstract necessary condition in theform of a Jensen-type inequality for measure-valuedsolutions to be generated by weak solutions. Thedichotomy between weak and measure-valued solu-tions in the compressible case is in contrast with theincompressible situation.

Deficit Estimates for the GaussianLogarithmic Sobolev Inequality

Max FathiUC Berkeley, USAMichel Ledoux, Emanuel Indrei

I will present a few results on estimating the deficitin the Gaussian logarithmic Sobolev inequality, aswell as in Talagrand’s transport-entropy inequality.These estimates have some applications in the studyof long-time behavior of di↵usion processes.

Global Well-Posedness of the HighDimensional Maxwell-Dirac Equa-tion for Small Critical Data

Cristian GavrusUC Berkeley, USASung-Jin Oh

We discuss the global well-posedness of the masslessMaxwell-Dirac equation in Coulomb gauge on R1+d

(d � 4) for data with small scale-critical Sobolevnorm, as well as modified scattering of the solutions.Main components of our proof are A) uncovering nullstructure of Maxwell-Dirac in the Coulomb gauge,and B) construction of a microlocal-parametrix forthe underlying paradi↵erential magnetic Dirac equa-tion. A key step for achieving both is to exploit andjustify a deep analogy between Maxwell-Dirac andMaxwell-Klein-Gordon (for which an analogous re-sult was proved earlier by Krieger-Sterbenz-Tataru).This is joint work with Sung-Jin Oh.

Approximation Schemes for Op-timal Constants and ExtremalFunctions in Sobolev Inequalities

Ryan HyndUniversity of Pennsylvania, USA

We employ discrete and continuous time flows toapproximate optimal constants and functions forwhich equality holds in various inequalities such asPoincare’s and Morrey’s inequality. The discretetime scheme is based on the inverse iteration methodfor square matrices and the continuous time flow is aparticular type of doubly nonlinear evolution.

Convergence of ThresholdingSchemes for Mean-Curvature Flow

Tim LauxMPI MIS Leipzig, GermanyFelix Otto, Drew Swartz

The thresholding scheme, a time discretization formean-curvature flow was introduced by Meriman,Bence and Osher in 1992. In the talk we presentnew convergence results for several variants of thisscheme, in particular in the multi-phase case with ar-bitrary surface tensions. The proofs are based on theinterpretation of the thresholding scheme as a mini-mizing movements scheme by Esedoglu and Otto in2014. This interpretation means that the threshold-ing scheme preserves the structure of (multi-phase)mean-curvature flow as a gradient flow w. r. t. thetotal interfacial energy. More precisely, the thresh-olding scheme is a minimizing movements scheme foran energy that �-converges to the total interfacialenergy. In this sense, our proof is similar to theconvergence results of Almgren, Taylor and Wangin 1993 and Luckhaus and Sturzenhecker in 1995,which establish convergence of a more academic min-imizing movements scheme. Like the one of Luck-haus and Sturzenhecker, ours is a conditional con-


vergence result, which means that we have to assumethat the time-integrated energy of the approximationconverges to the time-integrated energy of the limit.This is a natural assumption, which is however notensured by the compactness coming from the basicestimates.

The Shape of Capillarity Dropletsin a Container

Mihalia CorneliaUniversity of Texas at Austin, USAFrancesco Maggi

The capillarity droplet problem concerns the mini-mization of the Gauss free energy of a set, where theGauss free energy is defined as the sum of the sur-face and potential energies of that set. In this talk,we consider minimizers of the Gauss free energy fora liquid droplet bounded in a container with C1,1

boundary. We discuss a quantitative description ofthe shape of global minimizers and their regularityproperties in the small volume regime. This work wasperformed in collaboration with Francesco Maggi.

The Sharp Quantitative EuclideanConcentration Inequality

Connor MooneyUT Austin, USAAlessio Figalli, Francesco Maggi

A special case of the Brunn-Minkowski inequalitystates that, among sets with fixed volume in Rn,balls have r-neighborhoods of smallest volume foreach r > 0. We will discuss a sharp quantitativeversion of this result.

A Strong Form of the QuantitativeWul↵ Inequality

Robin NeumayerUniversity of Texas at Austin, USA

For a set E that almost minimizes perimeter amongsets of the same volume, quantitative isoperimetricinequalities measure how close E is to the uniqueperimeter minimizer. A recent paper of Fusco andJulin gives a quantitative isoperimetric inequality

where the deficit in the inequality controls the os-cillation of the boundary of E. In this talk, we willgeneralize this result to the anisotropic case, whereperimeter is weighted with respect to some fixed con-vex set K.

Di↵erential Calculus in Wasserstein-Orlicz Spaces

Levon NurbekyanKing Abdullah University of Science and Technology(KAUST), Saudi ArabiaAlessio Figalli

In this work, we study metric properties of theWasserstein-Orlicz spaces of probability measures.We generalize results by L. Ambrosio, N. Gigli andG. Savare on gradient flows in p-Wasserstein spacesto the case of Wasserstein-Orlicz spaces and applythese methods to study certain classes of parabolicPDEs.

A Min-Max Formula for LipschitzOperators That Satisfy the GlobalComparison Principle.

Russell SchwabMichigan State University, USANestor Guillen

We investigate Lipschitz maps, I, mapping C2(D) !C(D), where D is an appropriate domain. The globalcomparison principle (GCP) simply states that when-ever two functions are ordered in D and touch at apoint, i.e. u(x) v(x) for all x and u(z) = v(z) forsome z 2 D, then also the mapping I has the sameorder, i.e. I(u, z) I(v, z). It has been known sincethe 1960s, by Courrege, that if I is a linear mappingwith the GCP, then I must be represented as a lin-ear drift-jump-di↵usion operator that may have bothlocal and integro-di↵erential parts. It has also longbeen known and utilized that when I has the GCPand is both local and Lipschitz it will be a min-maxover linear and local drift-di↵usion operators, withzero nonlocal part. In this talk we discuss some re-cent work that bridges the gap between these situa-tions to cover the nonlinear and nonlocal setting forthe map, I. These results open up the possibility tostudy Dirichlet-to-Neumann mappings for fully non-linear equations as integro-di↵erential operators onthe boundary as well as have implications for exis-tence and uniqueness theory for weak solutions ofintegro-di↵erential equations. This is joint work withNestor Guillen


Special Session 18: Tra�c Flow Models and Their Application in Tra�cEngineering

Benedetto Piccoli, Rutgers University, USABenjamin Seibold, Temple University, USA

Tra�c engineering is undergoing a fundamental transformation: modern sensor technologies (e.g. smart-phones) have caused a surge in new types of tra�c data; and autonomous vehicles and wireless vehicle-vehiclecommunication give rise to completely new means of controlling tra�c flow. With this rise of high-qualitytra�c data, and high-fidelity means of tra�c control, sophisticated mathematical tra�c models becomeincreasingly important in actual engineering practice. This special session brings together tra�c modelingresearchers who are also interested in practical perspectives, with engineers who are using tra�c modelsas fundamental tools in their research and engineering solutions. Senior and junior faculty, postdoctoralresearchers, and students, will present and discuss recent research on tra�c flow theory, tra�c computation,state estimation and prediction, and the control of tra�c flow.

Multi-Jam Solutions and La-grangian Data Assimilation inTra�c-Flow Models

Paul CarterBrown University, USACourtney Cochrane, Peter Leth Christiansen,Joseph DeGuire, Gaoyang Fan, Yuri B. Gai-didei, Carlos Gorria, Emma Holmes, MelissaMcGuirl, Patrick Murphy, Jenna Palmer,Bjorn Sandstede, Laura Slivinski, Mads PeterSorensen, Jens Starke, Chao Xia

The aims of this talk are twofold. Firstly, a micro-scopic optimal velocity model of tra�c flow is pre-sented which allows for the spontaneous formationof tra�c jam solutions. Connections with the fun-damental diagram of tra�c flow are also discussed.Secondly, a study of data assimilation for tra�c flowmodels is presented; the method allows for the as-similation of both Eulerian (sensor) and Lagrangian(GPS) data and works well in di↵erent tra�c scenar-ios using either ensemble Kalman or particle filters.The algorithm is also demonstrated to be capableof estimating parameters and assimilating real traf-fic data as well as data obtained from the aforemen-tioned microscopic model.

A Fast Algorithm for ComputingSolutions to the LWR Tra�c FlowModel with Internal Conditions

Christian ClaudelUT Austin, USAMichele Simoni

In this article, we focus on the problem of computingthe solutions to the LWR tra�c flow model with in-ternal conditions. The main issue arising in this typeof computation is the need for one to know the pa-rameters (actual velocity, passing rate) of the movingboundary condition at all times. These parametersboth impact and are impacted by the solution aroundthe internal condition, which makes this problem dif-ficult to solve. To date, the incorporation of internalconditions in the LWR model is done on a ad-hocbasis, in situations where the parameters of these in-ternal conditions are easy to determine, but no al-

gorithm exist to incorporate an arbitrary number ofinternal conditions into the LWR model. Using anequivalent formulation of the problem based on anHamilton Jacobi equation, we show that the movingboundary conditions can be modeled as a hybrid sys-tem with three states, and that the computation ofthe parameters of the internal conditions can be donee�ciently by using a semi-analytical decompositionof the solutions, which we outline. We then presenta few possible applications of this framework, fromthe optimization of the trajectories of slow vehiclesto optimal signal timing.

High-Order Methods Devoid of Ve-locity Overshoots for Second-OrderTra�c Models

Shumo CuiTemple University, USABenjamin Seibold

Traditional finite volume methods, applied to theAw-Rascle-Zhang (ARZ) model or generic second or-der models (GSOM), may lead to spurious overshootsin the velocity. The fundamental reason is that nei-ther velocity nor momentum is conservative variablesof the model. If left unattended, numerical defectswould first emerge as a velocity overshoot at con-tact waves and later contaminate the solution in agradually enlarging vicinity. In this talk, we presentan e�cient and flexible velocity correction approach,which can be added to most numerical schemes as ablack box substep. Moreover, the proposed correc-tion method is only triggered near the contact dis-continuities and preserves the formal order of the un-derlying scheme. The performance and robustness ofthe proposed method are demonstrated in a series ofnumerical examples.


Tra�c Flow Modeling and Simula-tions with Moving Bottlenecks

Maria Laura delle MonacheRutgers University, USA

We consider a strongly coupled PDE-ODE systemthat describes the influence of a slow and large vehicleon road tra�c. The model consists of a scalar conser-vation law accounting for the main tra�c evolution,while the trajectory of the slower vehicle is given byan ODE depending on the downstream tra�c density.The moving constraint is expressed by an inequalityon the flux, which models the bottleneck created inthe road by the presence of the slower vehicle. Weprove the existence of solutions to the Cauchy prob-lem for initial data of bounded variation. Moreover,we introduce two numerical methods for the trackingof the slower vehicle trajectory on a road. The firstalgorithm is a finite volume scheme that uses a locallynon uniform moving mesh that tracks the slower ve-hicle. The second method is a conservative schemethat uses a reconstruction technique. We performnumerical tests and compute numerically the orderof convergence.

A Nonstandard Second-Order For-mulation of the LWR Model

Wenlong JinUC Irvine, USA

It is well known that the LWR model can be consid-ered a limit of many second-order continuum modelswith a zero relaxation time. In this study, we presenta nonstandard second-order formulation of the LWRmodel, which can be considered as a limit of a numer-ical scheme. The di↵erence between the new formu-lation and the traditional formulation is discussed.Then by converting the second-order model into theLagrangian coordinates, we demonstrate that its dis-crete version is the same as that of the LWR model,which was derived based on the Hopf-Lax formula.In addition, we introduce a new criterion, absolutelycollision-free condition, for the well-definedness ofthe discrete version. With numerical examples, wedemonstrate that the new model admits smooth solu-tions but has shock waves as the limit case. Potentialapplications of the new formulation are discussed.

Tra�c Regulation Via ControlledSpeed Limit

Benedetto PiccoliRutgers University, USAMaria Laura Delle Monache, Francesco Rossi

We present an optimal control problem for tra�cregulation via variable speed limit. The tra�c flowdynamics is described with the Lighthill-Whitham-Richards (LWR) model with Newell-Daganzo fluxfunction. We aim at minimizing the L2 quadraticerror to a desired outflow, given an inflow on a sin-gle road. We first provide existence of a minimizer

and compute analytically the cost functional varia-tions due to needle-like variation in the control pol-icy. Then, we compare three strategies: instanta-neous policy; random exploration of control space;steepest descent using numerical expression of gra-dient. We show that the gradient technique is ableto achieve a cost within 10% of random explorationminimum with better computational performances.

Mathematical Models for CapacityDrop Phenomena

Benjamin SeiboldTemple University, USA

Most contemporary tra�c models are based on a flowvs. density relationship (“fundamental diagram’)that is continuous. In contrast, many tra�c flowcontrol heuristics (e.g., ramp metering) are based ona capacity drop argument: for a certain density (ordensity range), tra�c flow may exist in a high flow(“free flow’) or in a low flow (“congested’) state; thus,when transitioning from free flow to congestion, thecapacity of the flow drops. We demonstrate how sucha drop in the fundamental diagram, or even hysteresisbehavior, can be included in continuum tra�c mod-els. Specifically, we first show how not to model thephenomenon, and then present two suitable way howto actually do it: one describing a straight road, andanother one a bottleneck.

Robust Control of AutonomousVehicle Trajectories

Jonathan SprinkleUniversity of Arizona, USARahul Bhadani, Shumo Cui, Benjamin Seibold

In this paper we describe a robust treatment oftracking trajectories with an autonomous vehicle.In employing autonomous behaviors for tra�c con-trol there will inevitably be disturbances introducedthrough model error, non-planar surfaces, sensornoise, and delay in both sensing and actuation. Wedescribe how we address these issues through robustcontrol techniques. The trajectories we follow includeposition and orientation as part of their specifica-tion: but the most interesting aspect of these tra-jectories is the time-varying description of the state.This is opposed to a traditional approach of follow-ing a trajectory at any speed (with expected errorin all dimensions of the state vector), as long as thespeed does not exceed a maximum value. However,for tra�c control to reduce tra�c waves, most of thedampening approaches are time-varying trajectories.With this in mind, it becomes necessary to considerthe delay of following the reference trajectory, andhow this may a↵ect drivers in the flow. We includesimulation data demonstrating the results, as well asdata from a full-sized robotic Ford Escape.


Control of Microscopic Tra�c FlowVia a Single Autonomous Vehicle

Raphael SternUniversity of Illinois at Urbana-Champaign, USA

This work focuses on tra�c control to locally mitigateinstabilities that adversely a↵ect fuel consumption(e.g., stop-and-go waves) via precise velocity controlof a single autonomous vehicle (AV) on the highway.The approach is to use the sensors onboard the AVto detect congestion events, and then close the loopby carefully following prescribed velocity controllersthat are demonstrated to stabilize the tra�c flow.Specifically, this work considers the use of a feedbackcontroller to prescribe the necessary acceleration pro-file to dampen tra�c waves. The main finding is thateven a single autonomous vehicle may substantiallyreduce undesirable tra�c waves in its vicinity whenproperly controlled.

On Some Constant-Speed andStop-And-Go Solutions of a Car-Following Model

Eugen StumpfUniversity of Hamburg, Germany

In this talk we consider a simple optimal velocity car-following model describing the dynamics of infinitelymany cars moving, one after another, along a singlelane road. Using a traveling wave ansatz and apply-ing a time transformation, we deduce a parameter-

dependent delay di↵erential equation whose solutionsform wave front solutions for the tra�c model. In thisway, we will not only be able to prove the existenceand analyze the stability of constant-speed solutions,but we will also be able to address the appearance ofstop-and-go tra�c. The obtained results are in accor-dance with earlier studies of a similar car-followingmodel on a circular road where the existence of stop-and-go tra�c can be evidently observed in experi-ments.

E�cient Tra�c State Estimationand Incident Detection

Dan WorkUniversity of Illinois at Urbana-Champaign, USARen Wang

This talk links the tra�c state estimation problemwith the tra�c incident detection problem, resultingin a unified framework to solve both problems simul-taneously. The joint problem is posed as a hybridstate estimation problem, where a continuous vari-able denotes the tra�c state and a discrete modelvariable identifies the location and severity of an in-cident. The hybrid state is estimated using a multiplemodel particle filter to accommodate the nonlinearityand switching dynamics of the tra�c incident model,and is validated in simulation and with field data.The joint framework can improve both incident de-tection capabilities and post incident tra�c state es-timation. Compared to our earlier work, a new vari-ation of a multiple model particle filter is proposedwhich reduces the computation time by three ordersof magnitude.


Special Session 19: Modern Applications of Mathematical andComputational Sciences

Katie Newhall, University of North Carolina at Chapel Hill, USARafail Abramov, University of Illinois at Chicago, USAGregor Kovacic, Rensselaer Polytechnic Institute, USA

This special session encompasses a broad scope of modern applications of applied and computational math-ematics, as well as mathematical science. Covered topics include applications of dynamical systems inneuroscience and population dynamics, applications in fluid dynamics, microbiology, electromagnetism, tur-bulence and other related areas.

Di↵usive Boltzmann Equation, ItsFluid Dynamics, Couette Flow andKnudsen Layers

Rafail AbramovUniversity of Illinois at Chicago, USA

In the current work we propose a di↵usive modifica-tion of the Boltzmann equation. This naturally leadsto the corresponding di↵usive fluid dynamics equa-tions, which we numerically investigate in a simpleCouette flow setting. This di↵usive modification isbased on the assumption of the “imperfect“ modelcollision term, which is unable to track all collisionsin the corresponding real gas particle system. Thee↵ect of missed collisions is then modeled by an ap-propriately scaled long-term homogenization processof the particle dynamics. The corresponding di↵u-sive fluid dynamics equations are produced in a stan-dard way by closing the hierarchy of the momentequations using either the Euler or the Grad clo-sure. In the numerical experiments with the Cou-ette flow, we discover that the di↵usive Euler equa-tions behave similarly to the conventional Navier-Stokes equations, while the di↵usive Grad equationsadditionally exhibit Knudsen-like velocity boundarylayers. We compare the simulations with the cor-responding Direct Simulation Monte Carlo (DSMC)results. Argon and the air are studied as examples.

Instability of Steep Ocean Wavesand Whitecapping.

Sergey DyachenkoUniversity of Illinois, USAAlan C. Newell

Wavebreaking in deep oceans is a challenge thatstill defies complete scientific understanding. Sailorsknow that at wind speeds of approximately 5 m/sec,the random looking windblown surface begins todevelop patches of white foam (‘whitecaps‘) nearsharply angled wave crests. We idealize such a sealocally by a family of close to maximum amplitudeStokes waves and show, using highly accurate simu-lation algorithms based on a conformal map repre-sentation, that perturbed Stokes waves develop theuniversal feature of an overturning plunging jet. We

analyze both the cases when surface tension is ab-sent and present. In the latter case, we show theplunging jet is regularized by capillary waves whichrapidly become nonlinear Crapper waves in whosetrough pockets whitecaps may be spawned.

On-Site and O↵-Site BoundStates of the Discrete NonlinearSchroedinger Equation and ThePeierls-Nabarro Barrier

Michael JenkinsonRensselaer Polytechnic Institute, USAMichael I. Weinstein

We construct several families of symmetric localizedstanding waves (breathers) to the one-, two-, andthree-dimensional discrete nonlinear Schroedingerequation (DNLS) with cubic nonlinearity using bi-furcation methods about the continuum limit. Suchwaves and their energy di↵erences play a role in thepropagation of localized states of DNLS across thelattice. The energy di↵erences, which we prove to ex-ponentially small in a natural parameter, are relatedto the Peierls-Nabarro Barrier in discrete systems,first investigated by M. Peyrard and M.D. Kruskal(1984). These results may be generalized to di↵erentlattice geometries and inter-site coupling parameters.Finally, we discuss the local stability properties ofsuch bound states. This is joint work with MichaelI. Weinstein.

Synchronizing Cortical DynamicsVia Gap Junctions Between Excita-tory Neurons

Jennifer KileRensselaer Polytechnic Institute, USAGregor Kovacic, David Cai

Brain networks are known to give rise to global os-cillations that are linked to synchronized neuronalactivity, which has been shown to contribute to cog-nitive processes such as motor performance, learningand memory. Electric coupling through gap junctionsmay facilitate the emergence of synchronized oscilla-tions in the cortex, and influence their properties.Electrical synapses, or gap junctions, connect the cy-tosolic contents of coupled neurons, allowing the di-rect transfer of electrical signals between the cells.While such synapses between interneurons in the cor-tex have been studied, electric coupling between ex-


citatory, pyramidal neurons have only recently beendiscovered. In order to further study these hypothe-ses, we have developed a detailed, comprehensivemodel with both synaptic and electric coupling forboth excitatory and inhibitory neurons using a mod-ified version of the Hodgkin-Huxley equations. Thenetwork incorporates local synaptic connections be-tween pairs of neurons using known models from theliterature to describe the kinetics of the synapses.This model also includes gap junctions between in-hibitory neurons, as well as pairs of excitatory neu-rons. Through this model, we examine the resultingdynamical regimes from the inclusion of both electricand synaptic connections, with a specific interest inthe emergence and properties of synchrony.

Hydrodynamic Limits of GrainCoarsening in Two Dimensions

Joe KlobusickyRPI, USAGovind Menon, Robert Pego

In two dimensions, areas of individual grains infoams and metals evolve at constant rates based ontheir topologies. When a grain vanishes, neighbor-ing grains may gain or lose sides to satisfy the cellnetwork’s trivalency requirement. Various curvaturebased and mean field models handle the problem ofside redistribution di↵erently. In this talk, we com-pare these models, and also their corresponding non-linear transport equations which describe evolutionfor a grain’s area and topology. We also introducea particle system, through piecewise deterministicMarkov processes, which captures both deterministicdrift of grain areas and random mean field side re-assignment. We test this model against others withstochastic simulations on a large number of grains.

Modeling of Gas Flow Networkswith Stochastic Load: Theory andNumerical Experiment

Alexander KorotkevichUniversity of New Mexico, USAS.A. Dyachenko, M. Chertkov, V.V. Lebedev,S.N. Backhause

We propose a method for gas transportation net-work modelling based on operator splitting tech-nique. This approach is stable for cases with loopsin the graph of the network. For the fluctuatingconsumption in the network, which is a standardsituation for heterogeneous power generation net-works, where gas powered generators work as a powersources when solar or wind power station is not gener-ating due to weather change, we demonstrate growthof pressure fluctuations with time. Theoretical com-putations are supported by numerical simulations us-ing di↵erent physical models of the network.

Is Our Sensing Compressed?

Gregor KovacicRensselaer Polytechnic Institute, USAVictor Barranca, Douglas Zhou, David Cai

Considering many natural stimuli are sparse, can asensory system evolve to take advantage of sparsity?We show significant downstream reductions in thenumbers of neurons transmitting stimuli in early sen-sory pathways might be a consequence of sparsity.Our work points to a potential mechanism for trans-mitting stimuli related to compressed-sensing (CS)data acquisition. Through simulation of integrate-and-fire point-neuron models and reconstruction viafiring-rate models, we examine the characteristics ofnetworks that optimally encode sparsity and the roleof receptive fields in stimulus sampling.

Metastable Transitions in Sleep-Wake Networks

Peter KramerRensselaer Polytechnic Institute, USAAnthony Trubiano, Fatih Olmez, Jung-EunKim, Janet Best

We have developed a computational neuronal net-work model for sleep-wake transitions which exhibitsboth a power law region and an exponential tail inthe probability density function for the time betweensleep/wake switches. Such power-law and exponen-tial features are seen in experimental recordings ofyoung mammals. We analyze the nature of the tran-sitions contributing to both the power law and expo-nential region, and what distinguishes them.

Symmetrization of Rare Event Sam-plers for Stochastic Di↵erentialEquations

Andrew LeachUniversity of Arizona, USAKevin Lin, Matthias Morzfeld

Many interesting behaviors in stochastic di↵erentialequations (SDE) occur infrequently and are di�cultto observe through direct simulation. Recently, opti-mal control based sampling methods have been pro-posed for e�cient simulation of rare events in SDE.We analyze the performance of these techniques whenthe noise parameter is small, and show that the rel-ative variance of such methods is order one in thenoise parameter. Moreover, we show that this ordercan be improved by a symmetrization procedure akinto antithetic variates. We illustrate our small noiseanalysis with numerical examples, and compare thecontrol based samplers to other methods.


Wave Patterns in an Excitable Neu-ronal Network

Christina LeeOxford College of Emory University, USAGregor Kovacic

This talk describes a study of spiral- and target-like waves traveling in a two-dimensional network ofintegrate-and-fire neurons with close-neighbor cou-pling. The individual neurons are driven by Poissontrains of incoming spikes. Each wave nucleates as aresult of a fluctuation in the drive. It begins as a tar-get or a spiral, and eventually evolves into a straight“zebra“ -like grating. Some of the waves contain de-fects arising from collisions with other waves. Thewavelength and wave speed of the patterns were in-vestigated, as were the temporal power spectra of theoscillations experienced by the individual neurons aswaves were passing through them.

The E↵ective Dispersion Relation ofthe Nonlinear Schrodinger Equation

Katelyn LeismanRensselaer Polytechnic Institute, USAGregor Kovacic, David Cai

The linear part of the Nonlinear Schrodinger Equa-tion (NLS) (iqt = qxx) has dispersion relation ! = k2.We don‘t necessarily expect solutions to the NLS tobehave nicely or have any kind of e↵ective disper-sion relation, since we expect nonlinear waves to bestrongly coupled and not sinusoidal in time. How-ever, I have seen that solutions to the NLS are actu-ally weakly coupled and are often nearly sinusoidal intime with a dominant frequency, often behaving sim-ilarly to modulated plane waves. In fact, when I lookat long-time average of either a solution with manysoliton like pulses or with many unstable modes, thepower spectral density does indicate a quadratic dis-persion relation that has been shifted by a constantproportional to the amplitude of the initial condition:

! = k2 � 2A where A = kq(k,0)k22⇡

.

Nonlinear Combining of MultipleLaser Beams in Multimode OpticalFiber

Pavel LushnikovUniversity of New Mexico, USANatalia Vladimirova

We simulate combining of 127 laser beams intodi↵raction-limited beam by beam self-focusing (col-lapse) in multimode optical fiber with fiber diameterabout 1mm. Beams with total power above criti-cal are combined in near field at the entrance. Beamquality of the combined beam is not a↵ected by mod-erate fluctuations of the combining beams phases.

First-Passage Time to Clear theWay for Receptor-Ligand Bindingin a Crowded Environment

Jay NewbyUniversity of North Carolina, Chapel Hill, USA

I will present theoretical support for a hypothe-sis about cell-cell contact, which plays a criticalrole in immune function. A fundamental questionfor all cell-cell interfaces is how receptors and lig-ands come into contact, despite being separated bylarge molecules, the extracellular fluid, and otherstructures in the glycocalyx. The cell membraneis a crowded domain filled with large glycoproteinsthat impair interactions between smaller pairs ofmolecules, such as the T cell receptor and its lig-and, which is a key step in immunological informationprocessing and decision-making. A first passage timeproblem allows us to gauge whether a reaction zonecan be cleared of large molecules through passivedi↵usion on biologically relevant timescales. I com-bine numerical and asymptotic approaches to obtaina complete picture of the first passage time, whichshows that passive di↵usion alone would take far toolong to account for experimentally observed cell-cellcontact formation times. The result suggests thatcell-cell contact formation may involve previously un-known active mechanical processes.

The Causes of Metastability andTheir E↵ects on Transition Times

Katie NewhallUNC Chapel Hill, USA

Many experimental systems can spend extended pe-riods of time relative to their natural time scale in lo-calized regions of phase space, transiting infrequentlybetween them. This display of metastability canarise in stochastically driven systems due to the pres-ence of large energy barriers, or in deterministic sys-tems due to the presence of narrow passages in phasespace. To investigate metastability in these di↵erentcases, we take the Langevin equation and determinethe e↵ects of small damping, small noise, and dimen-sionality on the dynamics and mean transition time.

Modeling the Evolving Oscilla-tory Dynamics of the Rat LocusCoeruleus Through Early Infancy

Mainak PatelCollege of William and Mary, USABadal joshi

The mammalian locus coeruleus (LC) is a brain-stem structure that displays extensive interconnec-tions with numerous brain regions, and in particularplays a prominent role in the regulation of sleep andarousal. Postnatal LC development is known to dras-tically alter sleep-wake switching behavior throughearly infancy, and, in rats, exerts its most signifi-


cant influence from about postnatal day 8 to post-natal day 21 (P8-P21). Physiologically, several dra-matic changes are seen in LC functionality throughthis time period. Prior to P8, LC neurons are exten-sively coupled via electrical gap junctions and chem-ical synapses, and the entire LC network exhibitssynchronized ⇠0.3 Hz subthreshold oscillations andspiking. From P8 to P21, the network oscillation fre-quency rises up to ⇠3 Hz (at P21) while the ampli-tude of the network oscillation decreases. BeyondP21, synchronized network oscillations vanish andgap junction coupling is sparse or nonexistent. Inthis work, we develop a large-scale, biophysically re-alistic model of the rat LC and we use this model toexamine the changing physiology of the LC throughthe pivotal P8-P21 developmental period. We findthat progressive gap junction pruning is su�cient toaccount for all of the physiological changes observedfrom P8-P21. Furthermore, we discuss the relevanceof this work to the physiology of sleep-wake cyclingbehavior and the applications of this work to sleep-wake neurophysiology that will be explored in ourfuture modeling e↵orts.

Pattern and Defect Formation byIon Bombardment of Binary Sub-strates

Matt PennybackerUniversity of New Mexico, USAPatrick Shipman, R. Mark Bradley

When a solid surface is bombarded with a broad ionbeam, a wide variety of self-assembled nanoscale pat-terns can emerge, including nanodots arranged inhexagonal arrays of remarkable regularity. We dis-cuss a theory that explains the formation of the strik-ingly regular hexagonal arrays of nanodots that canoccur when binary materials are bombarded. In thistheory, the coupling between a surface layer of alteredchemical composition and the topography of the sur-face is the key to the observed pattern formation. Weanalyze how a soft mode related to the mean sputteryield can lead to defect formation and give rise to lessordered patterns.

Idealized Models of Insect Olfaction

Pamela PyzzaOhio Wesleyan University, USAGregor Kovacic, David Cai

The functionality of olfaction is known to be sharedamong a range of phyla from insects to mammals, andthe locust is an important animal model for study-ing olfaction. Experiments suggest that odors, de-

tected through receptors on its antennas, are relayedto sensory neurons in the antennal lobe. There, theytrigger a series of synchronous oscillations, followedby slow dynamical modulation of their firing rates,which slowly subside after the stimulus has been re-moved. We model the e↵ects of a white-smell-typeodor using an integrate-and-fire network and firing-rate model, with both fast excitatory and inhibitoryand slow inhibitory conductance responses. The fastinhibitory conductance response, together with theexcitation, creates initial oscillations, and the slowcomponent then damps the oscillations and aids inthe creation of the slow firing rate patterns that fol-low yet later. We propose a possible mechanism forgenerating this dynamical sequence to be a slow pas-sage through a saddle-node-on-a-circle bifurcation.

Waveaction Spectra for Fully Non-linear MMT Model

Michael SchwarzRensselaer Polytechnic Institute, USADavid Cai, Gregor Kovacic, Peter Kramer

We investigate a version of the Majda-McLaughlin-Tabak model of dispersive wave turbulence where thelinear term in the time derivative is removed. Weconsider driven-damped and undriven, undampedcases of the model. Our theoretical predictions forthe waveaction spectrum, which are made using sta-tistical mechanical methods as well as argumentsreminiscent of Kolmogorov’s theory of turbulence,are found to agree with time dynamics simulations.

Transverse Instability of ElectronPlasma Waves Study Via Direct2+2D Vlasov Simulations

Denis SilantyevUniversity of New Mexico, USAPavel Lushnikov, Harvey Rose

Transverse instability could be viewed as initial stageof electron plasma waves (EPWs) filamentation. Weperformed direct 2+2D Vlasov-Poisson simulationsof collisionless plasma to systematically study thegrowth rates of oblique modes of finite-amplitudeEPW depending on its amplitude, wavenumber, an-gle of the oblique mode wavevector relative to theEPW’s wavevector and the configuration of thetrapped electrons in the EPW. Simulation results arecompared to the theoretical predictions of simplifiedmodels.


Special Session 20: Models for Treatment of Prostate Cancer

Fabio Milner, Arizona State University, USAYang Kuang, Arizona State University, USA

The session will be focused on the description, analysis, and calibration of various models of treatment ofprostate cancer in di↵erent stages. Most prostate cancers depend on the hormone testosterone to grow.About one-third of prostate cancer patients require hormone therapy when their cancer spreads beyond theprostate or recurs after treatment. Some models will be concerned with intermittent ablation therapy, otherswith continuous ablation therapy when resistance develops and a new line of treatment is added to suppressintramural testosterone production. Models are calibrated using clinical data from di↵erent sources thattypically track the Prostate-Specific Antigen (PSA) concentration in patient blood.

Predicting the Outcome of ProstateCancer Patients Undergoing In-termittent Androgen DeprivationTherapy

Rebecca EverettNorth Carolina State University, USAA. M. Packer, Y. Kuang

Prostate cancer is often treated by intermittent an-drogen deprivation therapy since prostate cells de-pend on androgens for proliferation and survival. Wepresent a mathematical model that uses the Droopequation to apply the idea of androgen as a limit-ing nutrient. Using clinical data, we test the model’spredictive accuracy and predict whether or not a pa-tient can undergo another o↵-treatment cycle, thusimproving their quality of life. Ideas for future workwill be discussed.

The Potential for Immunotherapyin Combination with Androgen Ab-lation for the Treatment of ProstateCancer

Harsh JainFlorida State University, USA

Androgen ablation is the standard treatment formetastatic prostate cancer; but the onset of castra-tion resistant disease has necessitated the develop-ment of alternative treatments like immunotherapy,wherein the hosts immune system is trained to tar-get the tumor. New evidence points to a potentialfor synergy with androgen ablation. We will inves-tigate the potential for such a combination using adetailed mathematical model that includes temporaldynamics of key cell players and chemokines togetherwith detailed intracellular pathways relevant to drugaction.We will also discuss the limited success of anti-cancer vaccines like Provenge due to immune modu-lation by the prostate cells and therefore the need fora third kind of drug, namely those that target thiscross talk.

Data Assimilation in Mathemati-cal Models of Cancer Growth andTreatment

Eric KostelichArizona State University, USAJavier Baez, Yang Kuang

This talk will consider some applications of data as-similation on models of tumor growth and treatment.The ensemble Kalman filter and its variants providea computationally e�cient way to estimate initialconditions and parameters and their associated un-certainties in dynamical systems. I will also discusssome results regarding identifiability and bias in themodel variables. These concepts will be illustrated byassimilating clinical trial data into a di↵erential equa-tion models of prostate cancer growth and treatment.

Parameter Distribution in ProstateCancer Models

Fabio MilnerArizona State University, USAMaria Peters

Mathematical models of prostate cancer—and ofmany other biological systems—often involve a largenumber of unknown parameters, posing the problemof their identification. Since very often there is noway of knowing their exact values, di↵erent tech-niques are used to fit the models to individual casesin an e↵ort to predict the evolution of the particularsystem. Reliable estimates for the parameters usingthese techniques require a significant amount of col-lected data. In contrast, a database of many individ-ual cases containing each a relatively small numberof data points, amounts to a dataset large enough forreliably identifying population parameter distibutionsrather than single values for each parameter. Al-though estimated parameter distributions cannot beused to predict cancer progression in individual cases,they can be a useful tool in identifying general fea-tures or behavior of the system. To track progressionof prostate cancer it is common to use blood concen-tration of prostate specific antigen (PSA). The Hiratamodel predicts PSA levels based on 10 free parame-ters and the initial values of three di↵erent prostatecancer cell populations. We use MCMC (MarkovChain Monte Carlo) methods to identify parameterdistributions for those 10 parameters.


Evolutionary Stable Strategies inNutrient-Dependent Prostate Can-cer Growth

John NagyArizona State University, USAKirsten Karr, David Ung, Karl Lundin

Cancer cells require nutrient concentrations to bewithin a fixed range to maintain viability. When con-centrations become too low, cells face nutrient defi-ciency, or starvation; when too high, cells can su↵ermetabolic costs associated with toxicity, secretion ordetoxification. However, the relationship between tu-mor growth and nutrient concentrations is complexand di�cult to quantify. We have developed a math-ematical model which explores this relationship, fo-cusing on androgen as the nutrient in question. Thismodel extends the Portz et al. prostate cancer modelto include costs of androgen dependence and inde-pendence. We consider two iterations of this model:one that features a constant per-capita death rate,and one that assumes that cell morality is a unimodalfunction of intracellular nutrient concentration. Herewe show, using the theory of adaptive dynamics, thatthe interplay between costs and benefits of castra-tion resistance drive natural selection toward specificEvolutionary Stable Strategies (ESS). These resultssuggest targets for potential improvement of clinicaloutcomes by manipulating the selective environmentwithin advanced prostate malignancies.

Global Dynamics of a Model of JointHormone Treatment with DendriticCell Vaccine for Prostate Cancer

Erica RutterArizona State University, USAYang Kuang

Androgen deprivation therapy (ADT) is often usedto treat advanced prostate cancer. This type ofhormone therapy is e↵ective initially, but eventuallycastration-resistant prostate cancer emerges, whichcannot be treated with ADT. Intermittent androgendeprivation therapy is suggested as an alternativewhich may lessen side e↵ects, lower therapy costs,and potentially increase time to treatment resistance.Immunotherapy is often used once patients developresistance to hormone therapy, in the form of den-dritic cell vaccines. We model hormone therapy (in-termittent and continual ADT) in conjunction withdendritic cell vaccine therapy. We numerically inves-tigate the e↵ect scheduling of dendritic cell vaccineshas on patient longevity. From basic local analy-sis, we determine a personalized dendritic cell vac-cine dosage for disease-free equilibrium stability. Weexamine a quasi-steady state system and classify theglobal dynamics.


Special Session 21: Bifurcations and Asymptotic Analysis of Solutions ofNonlinear Models

Jann-Long Chern, Central University, TaiwanYoshio Yamada, Waseda University, JapanShoji Yotsutani, Ryukoku University, Japan

The aim of this special session is to exchange recent results, ideas and techniques on nonlinear ellipticand parabolic PDEs, including reaction-di↵usion systems and free boundary problems, from mathematicalphysics, chemical reactions, mathematical biology, medical science and some other fields. In particular, weare interested in the global bifurcation structure for such models. Combinations of numerical simulations andtheoretical approaches with asymptotic analysis will be very useful to understand the nonlinear phenomenatogether with underlying structure of solutions. We will give opportunities to both established and juniorresearchers working in the related area to present their recent results.

On the Uniqueness and Structureof Solutions to the System Arisingfrom Maxwell-Chern-Simons O(3)Sgma Model

Zhi-You ChenNational Changhua University of Education, TaiwanJann-Long Chern

In this talk, we prove the uniqueness of topologi-cal multivortex solutions for the self-dual Maxwell-Chern-Simons O(3) sigma model with Chern-Simonscoupling parameter su�ciently large and the chargeof electron either su�ciently small or large. Besides,we also establish the sharp region of flux-pairs for thenon-topological solutions and provide the classifica-tion of radial solutions of all types for single vortex-point case.

Stability of Equilibria for Epidemi-ological Models with TemporaryAcquired Immunity

Yoichi EnatsuTokyo University of Science, Japan

To theoretically understand the disease transmissionin the population, the asymptotic behavior of the so-lutions of epidemiological models have been widelystudied. In particular, incorporating immunity to adisease of recovered individuals has now been consid-ered to be an important concept of the modelling. Inthis talk, we introduce the recent works on the sta-bility of equilibria of the model governed by a classof nonlinear delay di↵erential equations. We also dis-cuss the e↵ect of the waning immunity on the stabil-ity of an endemic equilibrium of the model with twoconstant delays that represent latency time and in-fectious period, via the linearization of the model atthe equilibrium.

Bifurcation from Equilibrium Pointsfor a Modified Swift-HohenbergEquation

Jongmin HanKyung Hee University, KoreaYuncherl Choi, Taeyoung Ha, Doo Seok Lee

In this talk, we consider the dynamical bifurcationof a modified Swift-Hohenberg equation(MSHE). Asthe control parameter crosses a critical value, it isshown that the MSHE bifurcates from a trivial solu-tion to an attractor which determines the long timedynamics of the system. Using the center manifoldanalysis, we describe the bifurcated attractor in de-tail.

On the Stability of Time PeriodicSolutions of the Primitive Equations

Chun-Hsiung HsiaNational Taiwan University, TaiwanMing-Cheng Shiue

In this joint work with Ming-Cheng Shiue, we con-sider the stability of time periodic solutions for theprimitive equations with periodic forcing terms. Weprovide a global stability analysis for the small forcecase.

The Domain Geometry and TheBubbling Phenomenon of RankTwo Gauge Theory

Hsin-Yuan HuangNational Sun Yat-sen University, TaiwanLei Zhang

Let G be the Green’s function on a flat torus. One in-triguing mystery of G is how the number of its criticalpoints is related to blowup solutions of certain PDEs.In this talk, I will show that if fully bubbling solu-tions of Liouville type to Chern-Simons Model withtwo Higgs particles exist, the G has exactly threecritical points. In addition, the necessary and suf-ficient conditions for the existence of fully bubblingsolutions with multiple bubbles will be presented.


Shadow System Approach to aPlankton Model Generating Harm-ful Algal Bloom

Hideo IkedaUniversity of Toyama, JapanMasayasu Mimura, Tommaso Scotti

Spatially localized blooms of toxic plankton specieshave negative impacts on other organisms via theproduction of toxins, mechanical damage, or byother means. Such blooms are nowadays a world-wide spread environmental issue. To understandthe mechanism behind this phenomenon, a two-prey(toxic and nontoxic phytoplankton)-one-predator(zooplankton) Lotka-Volterra system with di↵usionhas been considered in a previous paper. Numer-ical results suggest the occurrence of stable non-constant equilibrium solutions, that is, spatially lo-calized blooms of the toxic prey. Such blooms appearfor intermediate values of the rate of toxicity µ whenthe ratio D of the di↵usion rates of the predator andthe two prey is rather large. In this paper, we con-sider a one-dimensional limiting system (we call it ashadow system) in (0, L) as D ! 1 and discuss theexistence and stability of non-constant equilibriumsolutions with large amplitude when µ is globally var-ied. We also show that the structure of non-constantequilibrium solutions sensitively depends on L as wellas µ.

Singular Solutions to the Equationof the Scalar-Field Type on the UnitSphere

Yoshitsugu KabeyaOsaka Prefecture Unviersity, JapanSoohyun Bae, Jann-Long Chern, Shoji Yot-sutani

We consider the nonlinear elliptic equation of theform⇤ u��u+up = 0 on the unit sphere Sn ⇢ Rn+1

with n � 2 and with � being a real parameter, where⇤ stands for the Laplace-Beltrami operator on theunit sphere. We discuss the existence of positive sin-gular solutions and the structure of solutions to theequation.

Generation of Singularity and LargeTime Behaviors of Solutions fora Free Boundary Problem of aReaction-Di↵usion equation

Yuki KanekoWaseda University, JapanYoshio Yamada

We discuss a free boundary problem for a reaction-di↵usion equation in multi-dimensions. This prob-lem may be used to model the spreading of biolog-ical species, where unknown functions are popula-tion density and spreading front of the species. Ifan initial function is defined in an annulus, the inner

boundary possibly shrinks until it touches the originand disappears (called Singularity). Hence it is ap-propriate to introduce a weak solution to study theproblem after the disappearance of the inner bound-ary, and we study large-time behaviors. Putting amonostable nonlinearity as a reaction term and as-suming that the outer boundary is su�ciently farfrom the inner boundary, we will show that Singu-larity appears in a finite time. Moreover we will de-rive some information on spreading and vanishing forlarge time behaviors.

Secondary Bifurcation for a Nonlo-cal Allen-Cahn Equation

Kousuke KutoUniversity of Electro-Communications, JapanTatsuki Mori, Tohru Tsujikawa, Shoji Yotsu-tani

This talk is concerned with the Neumann problem ofthe 1D stationary Allen-Cahn equation with a nonlo-cal term. Since this nonlocal term is given as the av-erage of the unknown function, each odd-symmetricsolution counteracts the nonlocal e↵ect, and thereby,becomes a stationary solution of the Allen-Cahnequation without nonlocal term. Namely, the setof odd-symmetric solutions of the nonlocal prob-lem forms a bifurcation branch of the Chafee-Intanteproblem which emanates from a pitchfork bifurca-tion point on the branch of the trivial solution. Ourmain result reveals that the nonlocal term induces asymmetry breaking bifurcation point on the branchof odd-symmetric solutions. This talk also mentionsthe uniqueness of such secondary bifurcation pointsand the global behavior of the bifurcation branch ofasymmetric solutions.

Some Well-Posedness Problem forthe Compressible Navier-StokesEquations

Ying-Chieh LinNational University of Kaohsiung, Taiwan

In this talk we concern with the isentropic compress-ible Navier-Stokes equations with density dependentviscosity on the n-dimensional torus Tn (n = 2 or 3)and with initial density vanishing somewhere insideTn. Let ⌦ (t) denote the domain where the fluid den-sity is positive at time t. We assume that the bound-ary of ⌦ moves along with the fluid velocity. We willpresent our strategy to get the local well-posednessof the problem.


The Fisher-KPP Equation witha Free Boundary and A MovingBoundary

Hiroshi MatsuzawaNational Institute of Technology, Numazu College,Japan

In this talk, we concern with free boundary problemof Fisher-KPP equation

ut = uxx + u(1� u), t > 0, ct < x < h(t),

where c > 0 is a given constant, h(t) is a free bound-ary which is determined by the Stefan-like condition.This model may be used to describe the spreadingof a new or invasive species with population den-sity u(t, x) over a one dimensional habitat. The freeboundary x = h(t) represents the spreading front. Inthis model, we impose zero Dirichlet boundary condi-tion at left boundary x = ct. This means that the leftboundary of the habitat is a very hostile environmentfor the species and that the habitat is eroded awayby the left moving boundary at a constant speed c.In this talk we will give a trichotomy result, thatis, for any initial data, one of the three situations(vanishing, spreading and transition) happens. Thisresult is related to the result of Fisher-KPP equationwith a shifting-environment, obtained by Du, Weiand Zhou.

Convergence of the Hydrodynami-cal Limit for Generalized CarlemanModels

Hironari MiyoshiWaseda University, JapanMasayoshi Tsutsumi

We consider the initial-boundary value problem for a2-speed system of first order semilinear hyperbolicequations with the homogeneous boundary condi-tions. We establish the existence of global weaksolutions in L1 by the theory of nonlinear contrac-tion semigroups. Using the monotone method andthe div-curl lemma, we investigate the hydrodynamiclimit of the sum of the solutions of the hyperbolicsystem and show that the limit verifies the doublynonlinear parabolic equations.

Global Structure of Stationary So-lutions to a Cell Polarization Model

Tatsuki MoriRyukoku University, JapanKousuke Kuto, Tohru Tsujikawa, Shoji Yot-sutani

Various cell polarization models are proposed byS.Ishihara, et al. (Phys. Rev. E 75 015203(R), 2007)and M.Otsuji, et al. (PLoS Compt. Biol. 3: e108,2007). We investigate global structure of station-ary solutions to a cell polarization model proposed

by Y.Mori, A.Jilkine and L.Edelstein-Keshet (SIAMJ.Appl Math, 2011), which is closely related with re-sults by them. We study both infinite and finite di↵u-sion coe�cient case. We show several mathematicaland numerical results.

Spectral Comparison in a General-ized Phase-Field Type System

Yoshihisa MoritaRyukoku University, Japan

We are concerned with the system ↵ut � �vt =d�u+ f(u) + v, �ut + �vt = �v, in a bounded do-main with the Neumann boundary conditions, where↵,�, �,� and d are positive parameters and f(u) issmooth. The system allows a Lyapunov function andthe stationary problem is reduced to a scalar equa-tion with a nonlocal term. We study the linearizedeigenvalue problem of an equilibrium solution, andby a spectral comparison method we show that theunstable dimension coincides with that of the simplereigenvalue problem of the scalar equation. This talkis based on a joint work with S.Jimbo and a jointwork with E.Latos and T.Suzuki.

On the Equilibrium States of anInextensible Elastic Ring Under theUniform Pressure

Minoru MuraiOsaka city University Advanced MathematicalInstitute, JapanWaichiro Matsumoto, Shoji Yotsutani

We consider a variational problem proposed byTadjbakhsh-Odeh to study the equilibrium states ofan elastic inextensible ring under a uniform pres-sure. Takagi-Watanabe mathematically investigatethe structure of the solutions of the Euler-Lagrangeequation for this problem. They gave very interestingresults for it.However, there are some parts to be clarified. In thistalk, we will give a complete answer to the globalstructure of the solutions.

Chaotic Traveling Pulses in SomeReaction-Di↵usion System

Masaharu NagayamaHokkaido University, JapanKei-Ichi Ueda, Masaaki Yadome

We consider the extended Gray-Scott model as one ofthe reaction di↵usion system which have a excitableproperty with a mon-stable equilibrium point. Thismodel has an oscillatory traveling pulse which bifur-cate form a traveling pulse. Using a global bifurca-tion calculation, we find that the oscillatory travel-ing pulse is destabilized by peirod-doubling bifurca-tion. As a result, we discover the oscillatory travelingpulse to make a chaotic motion, and we make clearthat the solution is chaos by calculating the maximalLyapunov exponent.


Asymptotic Behaviour of Equilib-rium States of Reaction-Di↵usionSystems with Mass Conservation

Tien-Tsan ShiehNational Taiwan University, TaiwanJann-Long Chern, Yoshihisa Morita

We study stationary solutions for a 1-D cross-di↵usion system, a conceptual model for cell polarity,

ut = d�u� g(u+ v) + v,

vt = �v + g(u+ v)� v

where g(w) = w/(w + 1)2. In particular, we are in-terested in asymptotic behaviours of the solution forthe system as d ! 0. We show that the solution witha minimizing energy exhibits the phenomena of massconcentration. Using a blow-up technique, we alsofind the limiting profile for the solution.

Multiple-Angle Formulas of Gener-alized Trigonometric Functions withTwo Parameters

Shingo TakeuchiShibaura Institute of Technology, Japan

Generalized trigonometric functions with two param-eters were introduced by Drabek and Manasevich tostudy an inhomogeneous eigenvalue problem of the p-Laplacian. Concerning these functions, no multiple-angle formula has been known except for the classi-cal cases and a special case discovered by Edmunds-Gurka-Lang, not to mention addition theorems. Inthis talk, we will present new multiple-angle formu-las which are established between two kinds of thegeneralized trigonometric functions, and apply theformulas to generalize classical topics related to thetrigonometric functions.

Classification of Standing WaveSolutions to a Coupled SchrodingerSystem

Yong-Li TangFeng Chia University, TaiwanZhi-You Chen

In this talk, a Schrodinger-type system is considered.We establish the existence and uniqueness of solu-tions for the Dirichlet boundary value problem of thesystem. Besides, the nonexistence of ground statesolutions under certain conditions on nonlinearities,and a complete structure of various types of solutionsare also provided.

On the Maximizing Problem Asso-ciated with Trudinger-Moser TypeInequalities

Hidemitsu WadadeKanazawa University / Institute of Science andEngineering, JapanMichinori Ishiwata

In this talk, we consider the existence and non-existence of maximizers associated with Trudinger-Moser type inequalities. Recently, A Trudinger-Moser inequality was derived by B. Ruf, JFA, 219(2005) for the two spacial dimension and extendedto the higher spacial dimensions by Y. Li-B. Ruf,Indiana UMJ, 57 (2008), which are in-homogeneoustype inequalities in the whole space. In the papersLi-B. Ruf, Indiana UMJ, 57 (2008) and M. Ishi-wata, Math. Ann. 351 (2011), the authors consid-ered the variational problems associated with theseTrudinger-Moser type inequalities and proved thatthe existence and non-existence results depending onthe exponents appearing in the exponential type in-tegrals. We revisit the existence and non-existenceproblems for the above inequalities and clarify thee↵ects of the norm-normalization to the structure ofthese variational problems.

Limiting Classification on Lin-earized Eigenvalue Problems for1-Dimensional Scalar Field Equa-tion

Tohru WakasaKyushu Institute of Technology, Japan

We are interested in classical linearized eigenvalueproblems for one dimensional reaction-di↵usion equa-tions with a small di↵usion parameter. In the pre-vious work by Wakasa and Yotsutani, the case ofbistable nonlinearities has been investigated, andasymptotic formulas of eigenvalues and eigenfunc-tions as the parameter tends to zero are obtained. Itcan be observed from the asymptotic formulas thatall of eigenpairs are classified into the finite number ofgroup, which is characterized by the associated limitproblem. In this talk we will focus on the scalar fieldequations, and will investigate the limiting classifica-tions on the eigenvalue and eigenfunctions associatedwith the spike solutions.


Nonlinear M-Accretive OperatorTheoretic Approach to Parabolic-Parabolic Keller-Segel Systems

Noriaki YoshinoTokyo University of Science, Japan

In this talk, we deal with parabolic-parabolic chemo-taxis systems by using the theory for nonlinear m-accretive operators. In the case of parabolic-elliptictype, an approach by nonlinear m-accretive opera-

tors to the systems was developed and the solvabilityof the systems was established by Marinoschi (2013),and Yokota and Yoshino (2014, 2015). However thecase of parabolic-parabolic type was left unsolved.We develop the theory to establish existence of solu-tions to the parabolic-parabolic chemotaxis systems.


Special Session 22: Dynamics and Games

Alberto Adrego Pinto, University of Porto, PortugalMichel Benaim, Institut de Mathematiques, Universite de Neuchatel, Switzerland

The session aims to bring together world top researchers and practitioners from the fields of DynamicalSystems, Game Theory and applications to such areas as Biology, Economics, Engineering, Energy, NaturalResources and Social Sciences. This session is organized by the founders and editors-in-chief of the Journalof Dynamics and Games (JDG), published by the American Institute of Mathematical Sciences (AIMS).

Anosov Di↵eomorphisms and Self-Renormalizable Sequences

Joao AlmeidaLIAAD - INESC TEC and Polytechnic Institute ofBraganca, PortugalA.A. Pinto

We consider a hyperbolic toral automorphism A :T ! T induced by the matrix

✓a bc d

◆2 GL(2,Z),

where T = R2/Z2.We use Adler, Tresser and Worfolk decomposition ofA to give an explicit construction of the stable andunstable C1+ self-renormalizable sequences.

Bayesian-Nash Equilibria in Theoryof Planned Behavior

Joao AlmeidaLIAAD - INESC TEC and Polytechnic Institute ofBraganca, PortugalL. Almeida, J. Cruz, H. Ferreira, J.P.Almeida, B. Oliveira, Alberto A. Pinto

We construct a model, using Game Theory, for theTheory of Planned Behavior and we propose theBayesian-Nash Equilibria as one of many possiblemechanisms to transform human intentions into be-havior decisions. We show that saturation, boredomand frustration can lead to the adoption of a vari-ety of di↵erent behavior decisions, as opposed to nosaturation, which leads to the adoption of a singleconsistent behavior decision.

Coupled Drift-Di↵usion StochasticProcesses As a Model for DecisionMaking

Reggie CaginalpUniversity of Pittsburgh, USABrent Doiron

We consider N drift-di↵usion escape processes, andallow them to interact via a kick at the escape time.The influence of the interaction on the number ofprocesses that escape through a certain gate is inves-tigated. A two-body case is first discussed, and itis investigated with both Monte-Carlo simulations aswell as solving the Fokker-Planck equation. Monte-Carlo simulations for a large number of these coupledprocesses are then considered.

Externality E↵ects in the Formationof Societies

Abdelrahim MousaBirzeit University, IsraelR. Soeiro, A. Pinto

We study a finite decision model where the util-ity function is an additive combination of a per-sonal valuation component and an interaction com-ponent. Individuals are characterized according tothese two components (their valuation type and ex-ternality type), and also according to their crowdingtype (how they influence others). We study how posi-tive externalities lead to type symmetries in the set ofNash equilibria, while negative externalities allow theexistence of equilibria that are not type-symmetric.In particular, we show that positive externalities leadto equilibria having a unique partition into a mini-mum number of societies (similar individuals usingthe same strategy, see [27]); and negative external-ities lead to equilibria with multiple societal parti-tions, some with the maximum number of societies.

Cournot Competition with Uncer-tainty in the Production Costs

Bruno OliveiraFCNA Universidade do Porto and INESC TEC,PortugalJoana Becker Paulo, Allberto A. Pinto

In an economy with a single sector, under Cournotcompetition with complete information, firms choosethe optimal quantities that maximize their profits.This maximum is a unique perfect Nash equilibriumand depends on the values of the parameters of thefirms, in particular, their production costs. We studya static game where uncertainty is on the productioncosts, that can be either high or low. Before pro-duction starts, firms are uncertain of the productioncost of the other firm and are certain of the value oftheir production cost. There are six distinct possi-ble cases. For each case, we have characterized theNash Equilibria and have obtained explicit formulasfor the output quantities of each firm and their re-spective profits.


Prices in Random Exchange Mar-kets and Cobb-Douglas Utility

Bruno OliveiraFCNA Universidade do Porto and INESC TEC,PortugalA. Yusuf, B. Finkenstadt, A. N. Yannacopou-los, A. A. Pinto

We study a random matching economy, where pairsof participants trade two goods and follow Cobb-Douglas utility functions. Under the appropriatesymmetry conditions, depending on the initial dis-tribution of endowments and the agents preferences,we show that the sequence of bilateral prices con-verges to the Walrasian price for this economy. Ad-ditionally, we study the e↵ect of an asymmetry inthe preferences on the di↵erence between the bilat-eral price and the Walrasian price for this economy.Moreover, we associate a selfishness factor to eachparticipant in this market. This brings up a gamealike the prisoner’s dilemma, where trade may occurat a price di↵erent from the bilateral, with advantageto the more selfish participant, or trade may not evenbe allowed. We discuss how the selfishness a↵ects thesequence of prices and the increase in utility.

Impact of Nash and Social Equilib-ria in an International Trade Model

Alberto PintoUniversity of Porto, PortugalF. Martins, M. Choubdar, J. Zubelli

We study an international trade model consisting of astrategic game in the tari↵s of the governments. Weconsider a two-stage game where, at the first stage,governments of each country choose their tari↵s com-petitively or socially for certain utilities that are rel-evant economic quantities, such as total output pro-duced by the home firm, total quantity in the homemarket, inverse demand, consumer’s savings, profitsof the firms, custom revenue of the country and wel-fare of the country. In the second stage, firms choosecompetitively (Nash) their home and export quanti-ties. We compare the competitive (Nash) tari↵s with

the social tari↵s and classify the game according tothe coincidence or not of these equilibria for eachutility. The lack of coincidence of these equilibria isa main di�culty in international trade that can bepartially dealt with the use of trade agreements.

Local Market Structure in aHotelling Town

Alberto PintoUniversity of Porto, PortugalTelmo Parreira, J. P. Almeida

We develop a theoretical framework to study thelocation-price competition in a Hotelling-type net-work game, extending the Hotelling model, with lin-ear transportation costs, from a line (city) to a net-work (town). We show the existence of a pure Nashequilibrium price if, and only if, some explicit con-ditions on the production costs and on the networkstructure hold. Furthermore, we prove that the localoptimal localization of the firms are at the cross-roadsof the town.

A Mathematical Model of Radical-ization

Manuele SantopreteWilfrid Laurier University, CanadaConnell McCluskey

Radicalization is the process by which people cometo adopt increasingly extreme political or religiousideologies. In recent years radicalization has becomea major concern for national security because it canlead to violent extremism. Governments and secu-rity services are making a substantial e↵ort to betterunderstand the radicalization process and to identifythe psychological, social, economic, and political cir-cumstances that lead to violent extremism. It is inthis context that this talk attempts to describe radi-calization mathematically by modelling the spread ofextremist ideology as the spread of an infectious dis-ease. This is done by using a compartmental epidemi-ological model. We try to use this model to evaluatethe e↵ectiveness of some strategies to counter violentextremism.


Special Session 23: Numerical Methods for Phase-Field Models

Xiaoming Wang, Florida State University, USASteven Wise, University of Tennessee Knoxville, USA

Phase field models are becoming ever more important in the study of many multi-phase physical, chemicalor biological processes. There has been a recent surge in the development of fast and accurate numericalmethods for various phase field models that are of importance in applications. The purpose of this mini-symposium is to provide a platform for experts to report the state of the art progress and discuss futuredirections in numerical methods relevant to phase field models.

New Epitaxial Thin Film Modelsand Numerical Approximation

Wenbin ChenFudan University, Peoples Rep of ChinaZhenhua Chen, Jin Cheng, Yanqiu Wang

This paper concerns new continuum phenomeno-logical model for epitaxial thin film growth withthree di↵erent forms of the Ehrlich-Schwoebel cur-rent. Two of these forms were first proposed by Politiand Villain and then studied by Evans, Thiel andBartelt. The other one is completely new. Followingthe techniques used in Li and Liu, we present rig-orous analysis of the well-posedness, regularity andtime stability for the new model. We also studiedboth the global and the local behavior of the surfaceroughness in the growth process. The new modeldi↵ers from other known models in that it features alinear convex part and a nonlinear concave part, andthus by using a convex-concave time splitting scheme,one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicitparts, which is much easier to implement than con-ventional models requiring nonlinear implicit parts.Despite this fundamental di↵erence in the model, nu-merical experiments show that the nonlinear morpho-logical instability of the new model agrees well withresults of other models , which indicates that the newmodel correctly captures the essential morphologicalstates in the thin film growth process.

A Second Order in Time FiniteElement Scheme for the Cahn-Hilliard-Navier-Stokes Equation

Amanda DiegelLouisiana State University, USASteven Wise, Cheng Wang, Xiaoming Wang

In this talk, we present a second order in time mixedfinite element method for the Cahn-Hilliard equationcoupled with a Navier-Stokes flow that models phaseseparation and coupled fluid flow in immiscible bi-nary fluids in two and three dimensions. We willdiscuss the main results of the numerical scheme in-cluding the following. We show that our scheme isunconditionally energy stable with respect to a spa-tially discrete analogue of the continuous free en-ergy of the system. Additionally, we show that thediscrete phase variable is bounded in L1 (0, T ;L1)and the discrete chemical potential is bounded in

L1 �0, T ;L2

�, for any time and space step sizes, in

two and three dimensions, and for any finite finaltime T . We subsequently prove that these variablesconverge with optimal rates in the appropriate energynorms in both two and three dimensions.

Thermodynamically ConsistentModeling and Computations forTwo-Phase Flows with VariableDensity

Zhenlin GuoUniversity of California Irvine, USAPing Lin, John Lowengrub, Steven Wise

In this talk, we will present a phase-field model forbinary incompressible fluid with thermocapillary ef-fects, which allows for the di↵erent properties (den-sities, viscosities and heat conductivities) of eachcomponent while maintaining thermodynamic con-sistency. The governing equations of the modelincluding the Navier-Stokes equations with addi-tional stress term, Cahn-Hilliard equations and en-ergy balance equation are derived within a thermody-namic framework based on entropy generation, whichguarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that theinterfacial conditions of the classical sharp-interfacemodels can be recovered from our phase-field model.Some numerical examples for the multiphase flowswith and without thermocapillary e↵ects will be pre-sented. The results are compared to the correspond-ing analytical solutions and existing numerical resultsas validations for our model.

Decoupled Unconditioanlly StableSchemes for Cahn-Hilliard-DarcyType Equations

Daozhi HanIndiana University Bloomington, USAXiaoming Wang

In this talk, we will present first-order and second-order, decoupled, unconditionally energy stable nu-merical schemes for solving the Cahn-Hilliard-Darcytype equations. The Darcy equations treated hereinclude the classical Darcy equation and a variationof it with the time derivative retained. Numericalexamples will be presented as well.


Continuum Models of Network For-mation in Ionomer Membranes

Keith PromislowMichigan State University, USA

Ionomer membranes, in particular Nafion, are well-known to be hysteric materials which display longtime-scales associated with transient behavior, time-scales that are far outside the reach of even the mostcoarse-grained particle based simulations. An un-derstanding of the mechanisms behind the transientsstates of Nafion is most readily obtained by the de-velopment of relatively simple models, based upon adissipation of a free energy, which resolves the com-petition between various morphological states. TheFunctionalized Cahn-Hilliard free energy incorpo-rates solvation energy of pendant ionic groups againstinterfacial bending energy and various contributionsto the solvent-phase pressure, to develop an di↵use-interface expression for the interfacial free energywithin functionalized-polymer/solvent mixtures. Wepresent this free energy, a multiscale analysis of thesolvent phase morphology, discuss challenges in thenumerical resolution of the model, as well as identi-fying several possible scaling regimes for these mostcomplex materials.

Characterizing the Stabilization Sizefor Semi-Implicit Fourier-SpectralMethod to Phase Field Equations

Zhonghua QiaoThe Hong Kong Polytechnic University, Hong KongDong Li, Tao Tang

Recent results in the literature provide computa-tional evidence that stabilized semi-implicit time-stepping method can e�ciently simulate phase fieldproblems involving fourth-order nonlinear di↵usion,with typical examples like the Cahn-Hilliard equationand the thin film type equation. The up-to-date the-oretical explanation of the numerical stability relieson the assumption that the derivative of the nonlin-ear potential function satisfies a Lipschitz type con-dition, which in a rigorous sense, implies the bound-edness of the numerical solution. In this work weremove the Lipschitz assumption on the nonlinear-ity and prove unconditional energy stability for thestabilized semi-implicit time-stepping methods. It isshown that the size of stabilization term depends onthe initial energy and the perturbation parameter butis independent of the time step. The correspondingerror analysis is also established under minimal non-linearity and regularity assumptions.

A Di↵use Interface Model for Two-Phase Ferrofluid Flows

Abner SalgadoUniversity of Tennessee, USARicardo H. Nochetto, Ignacio Tomas

A ferrofluid is a liquid which becomes strongly mag-netized in the presence of applied magnetic fields.It is a colloid made of nanoscale monodomain ferro-magnetic particles suspended in a carrier fluid. Theseparticles are suspended by Brownian motion and willnot precipitate nor clump under normal conditions.Ferrofluids are dielectric and paramagnetic.There are two well established PDE models used asa mathematical description for the behavior of fer-rofluids: the Rosensweig and Shliomis models. Thesedeal with one-phase flows, which is the case of manytechnological applications. However, some applica-tions arise naturally in the form of a two-phase flow:one of the phases has magnetic properties while theother one does not (magnetic manipulation of mi-crochannel flows, microvalves, magnetically guidedtransport, etc.).We develop a model describing the behavior of two-phase ferrofluid flows using phase field techniques andpresent an energy-stable numerical scheme for it. Fora simplified version of this model and the correspond-ing numerical scheme we prove, in addition to stabil-ity, convergence and, as a consequence, existence ofsolutions. With a series of numerical experiments weillustrate the potential of these simple models andtheir ability to capture basic phenomenological fea-tures of ferrofluids such as the Rosensweig instability.

Decoupled, Linear and Energy Sta-ble Schemes for Phase-Field Models

Jie ShenPurdue University, USA

I shall present unconditionally energy stable, decou-pled numerical schemes which only require solving asequence of linear elliptic equations at each time stepfor solving this coupled nonlinear system, and showample numerical results which not only demonstratethe e↵ectiveness of the numerical schemes, but alsovalidate the flexibility and robustness of the phase-field model.


Long-Time Stability of a Regular-ized Family of Models for Homo-geneous Incompressible Two-PhaseFlows

Florentina ToneUniversity of West Florida, USAT. Tachim Medjo, C. Tone

In this talk we present results on the stability of thefully implicit Euler scheme for a regularized family ofmodels for an incompressible two-phase flow model.More precisely, we consider the time discretisationscheme and with the aid of the discrete Gronwalllemma and of the discrete uniform Gronwall lemmawe prove that the numerical scheme is stable.

A Second Order Accurate and Ef-ficient Numerical Scheme for theCahn-Hilliard-Darcy System

Xiaoming WangFlorida State University, USADaozhi Han

We propose a novel second order in time, decoupledand unconditionally stable numerical scheme for solv-ing the Cahn-Hilliard-Darcy (CHD) system whichmodels two-phase flow in porous medium or in a Hele-Shaw cell. The scheme is based on the ideas of secondorder convexsplitting for the Cahn-Hilliard equationand pressure-correction for the Darcy equation. Weshow that the scheme is uniquely solvable, uncondi-tionally energy stable and mass-conservative. Amplenumerical results are presented to gauge the e�ciencyand robustness of our scheme.

An Arbitrary-Lagrangian-Eulerian-Phase-Field Method for Contact-Line Dynamics on Moving Particles

Pengtao YueVirginia Tech, USA

In this talk, we will present a hybrid Arbitrary-Lagrangian-Eulerian(ALE)-Phase-Field method forthe direct numerical simulation of multiphase flowswhere fluid interfaces, moving rigid particles, and

moving contact lines coexist. Practical applicationsinclude Pickering emulsions, froth flotation, and bi-olocomotion at fluid interface. An ALE algorithmbased on the finite element method and an adaptivemoving mesh is used to track the moving boundariesof rigid particles. A phase-field method based on thesame moving mesh is used to capture the fluid in-terfaces; meanwhile, the Cahn-Hilliard di↵usion au-tomatically takes care of the stress singularity at themoving contact line when a fluid interface intersectsa solid surface. To fully resolve the di↵use interface,mesh is locally refined at the fluid interface. All thegoverning equations, i.e., equations for fluids, inter-faces, and particles, are solved implicitly in a uni-fied variational framework. As a result, the hydrody-namic forces and moments on particles do not appearexplicitly in the formulation and an energy law holdsfor the whole system. In the end, we will presentsome results on the water entry problem and cap-illary interaction between floating particles, with afocus on the e↵ect of contact-line dynamics.

On Energy-Stable Schemes forComplex-Fluid Models

Jia ZhaoUniversity of South Carolina, USAQi Wang, Xiaofeng Yang

Complex fluids are fluids whose micro-structure haveimpact on the fluid macroscopic properties, which in-clude complex fluid mixtures of di↵erent types. Inthis talk, I will first present a systematic develop-ment of a general hydrodynamic model for complexfluid system using the generalized Onsager relation.Then, a semi-discrete scheme to solve this generalmodel, which satisfies the discrete energy dissipationlaw, will be presented. Specific tricks on linearizingand decoupling the schemes will be presented for par-ticular reduced models. In the end, several 3D simu-lations will be shown to illustrate the e↵ectiveness ofour schemes.


Special Session 24: SPDEs/SDEs and Stochastic Systems withControl/Optimization and Applications

Wanyang Dai, Nanjing University, Peoples Rep of China

We will discuss theory, methods, and numerical schemes of stochastic partial di↵erential equations (SPDEs),stochastic ordinary di↵erential equations (SDEs), and general stochastic dynamical systems. Furthermore,we will also talk about their interactions with and applications in stochastic optimal controls and stochasticdi↵erential games, queueing networks and discrete event systems, statistical mechanics and quantum physics,service and financial systems, computer and communication networks, etc.

Particle Representations forStochastic Partial Di↵erential Equa-tions with Boundary Conditions

Dan CrisanImperial College London, EnglandC. Janjigian, T. G. Kurtz

I discuss a weighted particle representation for aclass of stochastic partial di↵erential equations withDirichlet boundary conditions. The locations andweights of the particles satisfy an infinite system ofstochastic di↵erential equations (SDEs). The evolu-tion of the particles is modelled by an infinite sys-tem of stochastic di↵erential equations with reflect-ing boundary condition and driven by independentfinite dimensional Brownian motions. The weights ofthe particles evolve according to an infinite system ofstochastic di↵erential equations driven by a commoncylindrical noise W and interact through V , the as-sociated weighted empirical measure. When the par-ticles hit the boundary their corresponding weightsare assigned a pre-specified value. We show the exis-tence and uniqueness of a solution of the infinite di-mensional system of stochastic di↵erential equationsmodeling the location and the weights of the par-ticles. We also prove that the associated weightedempirical measure V is the unique solution of a non-linear stochastic partial di↵erential equation drivenby W with Dirichlet boundary condition. The workis motivated by and applied to the stochastic Allen-Cahn equation. This joint work with C. Janjigianand T. G. Kurtz.

A Unified System of SPDEs withLevy Jumps Vs. Stochastic Di↵er-ential Games

Wanyang DaiNanjing University, Peoples Rep of China

We study a unified system of stochastic partial dif-ferential equations (SPDEs) with Levy jumps ina forward-backward coupling manner. The par-tial di↵erential operators in its drift, di↵usion, andjump coe�cients are in time-variable and position-parameters over a domain (e.g., a hyperbox or amanifold). A solution to the system is defined by a4-tuple random vector-field process evolving in time.Since our unified system is a general-dimensional vec-tor one with general nonlinearity and general high-order, the popular computation (e.g., integration byparts) based proof method can not be applied. Thus,

we develop an approach to prove the well-posednessof an adapted 4-tuple strong solution to the systemin a topological space and under a sequence of gen-eralized local linear growth and Lipschitz conditions.As the further investigation of our system, we for-mulate a non-zero-sum stochastic di↵erential game(SDG) problem with general number of players. Bya 4-tuple solution to the system, we get a Paretooptimal Nash equilibrium policy to the SDG. In ad-dition, illustrative examples from quantum physics,statistical mechanics, and queueing networks are alsopresented.

Stochastic Systems with Memoryand Jumps

Giulia di NunnoUniversity of Oslo, NorwayD.R. Banos, F. Cordoni, L. Di Persio, E. Rose

Stochastic systems with memory naturally appearin life science, economy, and finance. We take themodeling point of view of stochastic functional delayequations and we study these structures when thedriving noises admit jumps. Our results concern ex-istence and uniqueness of strong solutions, estimatesfor the moments and the fundamental tools of calcu-lus, such as the Ito formula. We study the robustnessof the solution to the change of noises. Specifically,we consider the noises with infinite activity jumpsversus an adequately corrected Gaussian noise. Ourtechniques include tools of infinite dimensional cal-culus and the stochastic calculus via regularization.

Optimal Life-Insurance Selectionand Purchase Within a Market ofSeveral Life-Insurance Providers

Abdelrahim MousaBirzeit University, IsraelD. Pinheiro, A. A. Pinto

We consider the problem faced by a wage-earnerwith an uncertain lifetime having to reach decisionsconcerning consumption and life-insurance purchase,while investing his savings in a financial market com-prised of one risk-free security and an arbitrary num-ber of risky securities whose prices are determined bydi↵usive linear stochastic di↵erential equations. Weassume that life-insurance is continuously availablefor the wage-earner to buy from a market composedby a fixed number of life-insurance companies o↵ering


pairwise distinct life-insurance contracts. We charac-terize the optimal consumption, investment and life-insurance selection and purchase strategies for thewage-earner with an uncertain lifetime and whosegoal is to maximize the expected utility obtainedfrom his family consumption, from the size of theestate in the event of premature death, and from thesize of the estate at the time of retirement. We usedynamic programming techniques to obtain an ex-plicit solution in the case of discounted constant rel-ative risk aversion (CRRA) utility functions.

Di↵erentiability of Stochastic Flowsand Sensitivity Analysis of ReflectedDi↵usions in Convex Polyhedral Do-mains

Kavita RamananBrown University, IndiaDavid Lipshutz

Di↵erentiability of flows and sensitivity analysis areclassical topics in dynamical systems. However, theanalysis of these properties for constrained processes,which arise in a variety of applications, is challeng-ing due to the discontinuities in the dynamics atthe boundary of the domain, and is further compli-cated when the boundary is non-smooth. We showthat the study of both flows and sensitivities of con-strained processes in convex polyhedral domains canbe largely reduced to the study of directional deriva-tives of an associated map, called the extended Sko-rokhod map, and we introduce an axiomatic frame-work to characterize these directional derivatives. Inaddition, we establish pathwise di↵erentiability of alarge class of reflected di↵usions in convex polyhe-dral domains and show that they can be describedin terms of certain constrained stochastic di↵erentialequations with time-varying domains and directionsof reflection. This is joint work with David Lipshutz.

Backward Uniqueness for a Class ofSPDE

Michael RocknerBielefeld University, GermanyViorel Barbu

We present recent results on backward uniqueness ofsolutions to stochastic semilinear parabolic equationsand also for the tamed 3D Navier-Stokes equationsdriven by linear multiplicative Gaussian noises. Inthe first case we use a rescaling transformation toreduce the SPDE to a random PDE. Applicationsto approximate controllability of nonlinear stochasticparabolic equations with initial controllers are given.The method of proof relies on the logarithmic con-vexity property known to hold for solutions to lin-ear evolution equations in Hilbert spaces with self-adjoint principal part.

Stochastic Control of Path-Dependent Systems, Applicationto the Principal-Agent Problem

Nizar TouziEcole Polytechnique, FranceDylan Possamai, Jaksa Cvitanic

We consider a general formulation of the Principal-Agent problem. Our approach is the following: wefirst find the contract that is optimal among thosefor which the Agent’s value function allows for thedynamic programmin approach. We then show theoptimization over this restricted family represents nomoss of generality. Hence, this reduces the non-zero sum stochastic di↵erential game to a standardstochastic control problem which can then be ana-lyzed by standard tools of control theory.Our proofs rely on the Backward Stochastic Di↵er-ential Equations approach to Non-Markovian stoc-chastic control, and more specifically, on the recentextensions to the second order case.

Backward Stochastic Dynamicswith a Subdi↵erential Operator andNon-Local Parabolic VariationalInequalities

Phillip YamChinese University of Hong Kong, Hong KongAlain Bensoussan, Yiqun Li

In this talk, we introduce the first systematic studyon the unique existence of the solution of back-ward stochastic dynamical variational inequalities(BSDVI) on a general complete filtered probabilityspace. On the one hand, penalized method is usedwith a novel application of the backward-Gronwallinequality to construct the a-priori estimates of theunknown martingale, including that of its indeter-minate quadratic variation, adapted to the generalfiltration. On the other hand, the unique existenceof the weak solution of the parabolic variational in-equality is also established by making an associationwith a suitable BSDVI. It should be emphasized thatthe concept of viscosity solution could not be adoptedsince the potential function involved is non-local.

Robust Dynkin Game

Song YaoUniversity of Pittsburgh, USAErhan Bayraktar

We analyze a robust version of the Dynkin gameover a set P of mutually singular probabilities. Wefirst prove that conservative player’s lower and uppervalue coincide (Let us denote the value by V). Such aresult connects the robust Dynkin game with second-order doubly reflected backward stochastic di↵eren-tial equations. Also, we show that the value process


V is a submartingale under an appropriately definednonlinear expectations up to the first time ⌧⇤ when Vmeets the lower payo↵ process. If the probability setP is weakly compact, one can even find an optimaltriple (P⇤, ⌧⇤, �⇤) for the value V

0

.

Exponential Convergence for 3DStochastic Primitive Equations ofthe Large Scale Ocean

Dong ZhaoAcademy of Mathematics and Systems Science CAS,Peoples Rep of China

In this paper, we consider the ergodicity for thethree-dimensional stochastic primitive equations ofthe large scale oceanic motion. We proved that if thenoise is su�ciently smooth and non-degenerate, theweak solutions converge exponentially fast to equilib-rium. Moreover, the uniqueness of invariant measureis stated.

Method of Evolving Junctions(MEJ) for Optimal Control withConstraints

Haomin ZhouGeorgia Tech, USAShui-Nee Chow, Magnus Egerstedt, WuchenLi, and Jun Lu

We design a new stochastic di↵erential equation(SDE) based algorithm to e�ciently compute thesolutions of a class of infinite dimensional optimalcontrol problems with constraints on both state and

control variables. The main ideas include two parts.1) Use junctions to separate paths into segments onwhich no constraint changes from active to in-active,or vice versa. In this way, we transfer the originalinfinite dimensional optimal control problems into fi-nite dimensional optimizations. 2) Employ the inter-mittent di↵usion (ID), a SDE based global optimiza-tion strategy, to compute the solutions e�ciently. Itcan find the global optimal solution in our numer-ical experiments. We illustrate the performance ofthis algorithm by several shortest path problems, thefrogger problem and generalized Nash equilibrium ex-amples.

On Feller and Strong Feller Prop-erties of Regime-Switching JumpDi↵usions

Chao ZhuUniversity of Wisconsin-Milwaukee, USAFubao Xi

This work considers the martingale problem for aclass of weakly coupled Levy type operators. It isshown that under some mild conditions, the martin-gale problem is well-posed and uniquely determinesa strong Markov process (X,⇤). The process (X,⇤),called a regime-switching jump di↵usion with Levytype jumps, is further shown to posses Feller andstrong Feller properties via the coupling method.


Special Session 25: Applied Analysis and Dynamics in Engineering andSciences

Thomas Hagen, University of Memphis, USAFlorian Rupp, German University of Technology, Oman

The goal of this session is to bring together mathematicians who work in di↵erent areas of applied mathe-matics and might thus not meet and exchange ideas and points of view. Consequently, the session programaddresses a cross section of theoretical and computational developments and their applications to fluid dy-namics, solid mechanics and life sciences. Areas of interest include the theory of di↵erential equations,in particular evolution equations and stochastic di↵erential equations, stability and asymptotics, controltheoretic issues, numerical results and computational methods, and related aspects.

Rational Decays for Fluid-StructurePDE Models

George AvalosUniversity of Nebraska-Lincoln, USARoberto Triggiani

In this paper, we consider a fluid-structure PDEmodel of longstanding interest within the mathemat-ical and biological sciences. Here, a Stokes systemand vector-valued wave equation comprise the cou-pled PDE system under study; these respective PDEcomponents come into contact via a boundary inter-face. For this fluid structure system, our main re-sult is as follows: Under an appropriate geometricassumption which precludes imaginary point spec-trum for the associated semigroup generator, thenfor smooth initial data - i.e., data in the domain ofsaid generator - the corresponding solutions decay ata certain polynomial rate.

On the Stationary Solutions ofNon-Isothermal Film Casting withUnknown Frost Point

Shaun CeciLe Moyne College, USAThomas Hagen

In this talk, we will examine a one-dimensional modelfor film casting, an industrial process used to man-ufacture thin sheets and films from a highly vis-cous polymer melt. The flow is assumed to be non-isothermal and dominated by viscous forces withtemperature-dependent viscosity and an a priori un-known frost point. In particular, we will focus ourdiscussion on the existence, uniqueness, and lin-ear stability of stationary solutions to the governingequations.

Computation of Lyapunov Func-tions by Convex Optimization

Sigurdur HafsteinReykjavik University, Iceland

A Lyapunov function for a dynamical system deliv-ers valuable information on the system’s qualitativebehavior. The algorithmic computation of Lyapunovfunctions for nonlinear systems has received consid-

erable attention in the last two decades and there hasbeen major progress in the methods developed. Wegive an overview of some of the methods developedand precent some recent developments in the compu-tation of control- and ISS (input-to-state stability)Lyapunov functions for nonlinear systems.

Surface Tension Driven Flow inNetworks

Thomas HagenUniversity of Memphis, USA

In this presentation we discuss the dynamics andstability of surface tension driven fluid flow in net-works of channels. The mechanism behind the dy-namics of such flows is volume scavenging of capil-lary drops due to changes in pressure. We study theresulting droplet coarsening for Newtonian and non-Newtonian fluid models in the underlying gradient-like flow equations. An important aspect of the studyis the occurrence of heteroclinic orbits connecting sta-tionary solutions.

Basins of Attraction in Stochasti-cally Excited Systems

Florian RuppGerman University of Technology in Oman, Oman

We will discuss approximation techniques for basinsof attractions of stochastically asymptotically sta-ble equilibrium points in dynamical systems gener-ated by Stochastic and Random Ordinary Di↵eren-tial Equations. Hereby special attention will be givento an extension of the deterministic sums of squaremethod to compute suitable level sets of stochasticLyapunov functions were we give error bounds for theapproximation and validate the quality of the approx-imation. Our examples will in particular cover firstexamples from earthquake and o↵shore engineering.


Approximate Fourier-Series Solu-tions of Stochastic Sine-GordonEquations

Henri SchurzSIU, USA

An analysis of approximate Fourier solutions of mod-ified stochastic Sine-Gordon equations is presented.We prove existence and uniqueness of Fourier se-ries solutions under Dirichlet-type boundary condi-tions. For this purpose, make use of Lyapunov-functionals (energy-type based methods), truncateand control the dynamics of the original infinite-dimensional stochastic system by finite-dimensionalsystems of stochastic di↵erential equations. We ar-rive at uniform energy-type estimates for its seriessolutions. If time permits, we shall also discuss somestability results. The advantage is seen by a directpreparation for the use and qualitative control of ad-equate numerical methods.

Semigroup Well-Posedness forthe Total Linearization of a Free-Boundary Hydro-Elastic Interaction

Daniel ToundykovUniversity of Nebraska-Lincoln, USALorena Bociu, Jean-Paul Zolesio

We investigate wellposedness of a linearizationaround a steady regime of a free-boundary fluid-structure interaction model. The hydro-elastic equa-tions and the free boundary were linearized togetherwhich results in a system rather di↵erent from theclassical coupling of the Stokes flow and linear elas-

todynamics. New terms emerge on the common in-terface, some of them involving boundary curvatures.We proceed to establish that the associated evolutionoperator generates a strongly continuous semigroup.

Numerical Solutions of a Class ofSingular Neutral Functional Di↵er-ential Equations on Graded Meshes

Janos TuriUTD, USAPedro Perez-Nagera

In this talk we present case studies to illustrate thedependence of the rate of convergence of numericalschemes for singular neutral equations (SNFDEs) onthe particular discretization employed in the compu-tation. Based on our numerical experiments we ob-serve that the ideal mesh (i.e., resulting in the highestachievable rate of convergence) for the SNFDE un-der consideration is a discretization corresponding toequal integrals of its kernel function.

Existence of Periodic and MultipleSpike Standing Waves in CoupledReaction-Di↵usion Systems

Fu ZhangCheyney University of Pennsylvania, USAStuart P. Hastings

In this talk we present our proof of the existence ofperiodic patterns in a coupled homogeneous reaction-di↵usion system employing topological shooting ar-guments and the existence of multiple spike standingwaves in a coupled non-homogeneous system by aperturbation method.


Special Session 26: Hamiltonian Systems and the Planetary Problem

Gabriella Pinzari, University of Naples “Federico II”, Italy

The study of Hamiltonian systems is of relevant interest for Physics. These may consist of classical systems,namely with a finite number of degrees of freedom, like the N-body problem, the planetary problem, therigid body, billiards, the spin-orbit system, or, more generally, of extended systems, with a infinite numberof degrees of freedom, like, for example, the Schrodinger equation, the wave equation, the Euler equations ofhydrodynamics. Since the early 50s, many robust techniques have been applied firstly to classical and nextalso to extended systems, like the theorem of Kolmogorov, Moser and Arnold, the theorem of Nekhorossev,Arnold’s instability for systems with more than two degrees of freedom, splitting of separatrices, variationaltechniques, Mather theory. In this special session, we aim to gather specialist in this field, to outline thestatus of the art and perspectives.

Di↵usion Along Chains of NormallyHyperbolic Cylinders

Marian GideaYeshiva University, USAJean-Pierre Marco

We consider chains of 3-dimensional normally hyper-bolic invariant cylinders with boundary. We assumethat the unstable manifolds of each cylinder inter-sects both the stable manifold of the same cylin-der, and the stable manifold of the next cylinder inthe chain. We make some further assumptions onthese intersections, that amount to the existence of acertain family of locally defined scattering maps oneach cylinder, and on the dynamics restricted to eachcylinder. Under these assumptions we prove the theexistence of di↵usion orbits that drift along the cylin-der chains. Our approach is geometric, extending amethod introduced by Moeckel on constructing con-necting orbits inside a zone of instability for a twistmap on the annuls. The motivation of our work re-sides with the a priori stable case of the Arnold dif-fusion problem.

The Scattering Map in a Piezoelec-tric Energy Harvester

Albert GranadosTechnical University of Denmark, DenmarkTere Seara

In this talk we consider an energy harvesting sys-tem based on two piezoelectric oscillators modeled bydu�ng equations. When forced to oscillate, for in-stance when driven by a small periodic vibration, theoscillators create an electrical current which chargesan accumulator (a capacitor or a battery). The elec-trical circuit also couples the oscillators adding anextra dimension to the system. We aim to iden-tify trajectories that benefit the absorption of en-ergy from the source and somehow optimize the en-ergy harvester. To this end, we use techniques inwhich is based a common approach for the study ofArnold di↵usion, typically associated with celestialdynamics. In the absence of dissipation (given by thedamping in the du�ng equations), coupling and forc-ing, the system possesses a 3-dimensional NormallyHyperbolic Manifold. We use a modified Melnikovmethod to study the existence of 4-dimensional ho-

moclinic intersections in the presence of coupling andthe small periodic forcing. We then study the scatter-ing map associated with homoclinic excursions allow-ing us to identify trajectories injecting energy fromthe source to one of the oscillators.

Arnold Di↵usion of Charged Parti-cles in ABC Magnetic Fields

Alejandro LuqueInstituto de Ciencias Matem‘aticas, SpainDaniel Peralta-Salas

In this talk we prove the existence of di↵using so-lutions in the motion of a charged particle in thepresence of an ABC magnetic field. The equations ofmotion are modeled by a 3DOF Hamiltonian systemdepending on two parameters. For small values ofthese parameters, we obtain a normally hyperbolicinvariant manifold and we apply the so-called geo-metric methods for a priori unstable systems devel-oped by A. Delshams, R. de la Llave, and T.M. Seara.We characterize explicitly su�cient conditions for theexistence of a transition chain of invariant tori havingheteroclinic connections, thus obtaining global insta-bility (Arnold di↵usion). We also check the obtainedconditions in a computer assisted proof. This is ajoint work with Daniel Peralta-Salas.

On the Lax-Oleinik Semigroup ofSome Gravitational Problems

Ezequiel MadernaUniversidad de la Republica, Uruguay

The Lax-Oleinik semigroup associated to a TonelliLagrangian on a compact manifold gives a very fruit-ful link between the dynamics of the Euler-Lagrangeflow and the viscosity solutions of the Hamilton-Jacobi equation. More precisely, the invariant setscoming from the Aubry-Mather theory can be char-acterized in terms of the fixed points of the Lax-Oleinik semigroup, or weak KAM solutions. I willshow in this talk that this method also works formore general Lagrangian systems with singularities,like several gravitational problems.


Oscillatory Orbits in the RestrictedElliptic Planar Three Body Problem

Pau MartinUniversitat Politecnica de Catalunya, SpainMarcel Guardia, Tere M Seara

The restricted planar elliptic three body problemmodels the motion of a massless body under theNewtonian gravitational force of two other bodies,the primaries, which evolve in Keplerian ellipses.A trajectory is called oscillatory if it leaves everybounded region but returns infinitely often to somefixed bounded region. We prove the existence of suchtype of trajectories for any values for the masses ofthe primaries provided the eccentricity of the Keple-rian ellipses is small.

From Moser’s Normal Form toDissipative KAM Theory. an Appli-cation to the Spin-Orbit Problem.

Jessica Elisa MassettiUniversite Paris-Dauphine and IMCCE, Observa-toire de Paris, France

In 1967 J. Moser established a powerful normal formtheorem for real analytic perturbations of vectorfields possessing an invariant reducible quasi-periodictorus of Diophantine frequencies. From this normalform, in some particular cases issued from Hamil-tonian Mechanics and its dissipative versions issuedfrom Celestial Mechanics, we show the existence ofparticular remarkable normal forms: a la Hermanand a la Russmann. Through these normal forms,it’s possible to deduce KAM-type results if the sys-tem depends in an opportune way on a su�cientnumber of free parameters - internal or external toit. The persistence result is hence obtained througha technique of elimination of parameters, set up byRussmann, Herman and other authors in the 80s-90s. In this geometric frame the dissipative spin-orbit problem of Celestial Mechanics (recently pre-sented by Celletti-Chierchia and Locatelli-Stefanelli),can more easily be handled: deducing the existenceof quasi-periodic attractors becomes a particular caseof small dimension. Moreover, the process of elimi-nation of parameters highlights relations among dis-sipation, frequency and perturbation proper to thissystem and brings out a better understanding of theirrole, opening the way to a global study in the param-eters‘ space on the persistence of di↵erent kinds ofmotions under perturbation.

Beatings for the NLS Equation

Michela ProcesiUniversita di Roma tre, ItalyEmanuele Haus

We prove the existence and stability of a class of sim-ple quasi-periodic solutions for the NLS equation ontori which exhibit energy transfer phoenomena.

The Problem of Global Regularityfor Water Waves

Fabio PusateriPrinceton University, USA

We will discuss some recent works on the problem ofglobal regularity for the water waves equations, fo-cusing in particular on the role played by resonancesand normal forms in the understanding of the long-time behavior.

Numerical Study of the 3:1 Reso-nance with Application to Di↵usionin the RTBP

Pablo RoldanITAM, Mexico

Astronomical observations show that the Main As-teroid Belt has some gaps corresponding to those as-teroids in mean-motion resonance with Jupiter. Thisphysical phenomenon can be explained in terms ofArnold di↵usion. We consider the classical (planar,circular) Restricted Three Body Problem, modelingthe Sun-Jupiter-Asteroid system, and study the ge-ometric structure of the 3 : 1 resonance numerically.Namely, we compute the Normally Hyperbolic Invari-ant Manifold of resonant periodic orbits, its associ-ated (un)stable manifolds, two di↵erent homoclinicmanifolds, and their corresponding splitting func-tions.

An Approximation Theorem inClassical Mechanics

Cristina StoicaWilfrid Laurier University, Waterloo, Canada

A theorem by K. Meyer and D. Schmidt says that“The reduced three-body problem in two or three di-mensions with one small mass is approximately theproduct of the restricted problem and a harmonicoscillator (Transactions AMS, 352, 2000). This theo-rem was used to prove dynamical continuation resultsfrom the classical restricted circular three-body prob-lem to the three-body problem with one small mass.We examine the analogue statement in a broaderclass mechanical systems and state a definition ofrestricted problems. We state and prove a similartheorem applicable to a larger class of mechanicalsystems. We present applications to the sphericaldouble pendulum with a small mass at the free end,the spatial (N+1)-body systems with one small mass,and gravitationally coupled systems formed by a rigidbody and a small point mass.


Positive Lyapunov Exponents forSome Randomly Perturbed 2DConservative Maps

Jinxin XueUniversity of Chicago, USAAlex Blumenthal, Lai-Sang Young

Positive Lyapunov exponent is an important char-acterization of the exponential instability and chaosin dynamical systems. However, it is a well-knownhard problem to prove positive Lyapunov exponentsin conservative concrete systems. On the other hand,conservative maps appears naturally by taking thePoincare return map in Hamiltonian systems, andthe Chirikov standard map can be considered as a

model of the Poincare map near separatrix in Hamil-tonian systems with two degrees of freedom. In thistalk, we show the positive Lyapunov exponent of aclass of two dimensional maps with the help of a tinyrandom perturbation.

Normally Hyperbolic Laminationsin a Priori Unstable Systems

Ke ZhangDepartment of Mathematics, University of Toronto,CanadaVadim Kaloshin, Jianlu Zhang

We construct normally hyperbolic laminations for ana priori unstable Hamiltonian system, using the sep-aratrix maps of Treschev. Normally hyperbolic lam-inations provide a model of stochasticity in Arnolddi↵usion.


Special Session 27: Advances in the Mathematical Modeling of FailurePhenomena and Interfaces in Materials

Marco Morandotti, SISSA - International School for Advanced Studies, ItalyMarco Barchiesi, Universita di Napoli “Federico II”, Italy

Jose Matias, Instituto Superior Tecnico, Universidade de Lisboa, Portugal

In recent years, there has been an ever-increasing interest in the development of mathematical techniquescapable to describe the interplay between microscopic and macroscopic theories in material science. Bridgingdi↵erent (length and time) scales is crucial to grasp the fine structural behavior of the equilibrium configu-rations. The challenge is two-fold: capturing the emergence of microstructure from meso- and macroscopicmodels, and deriving e↵ective macroscopic models from microscopic ones. Tackling these issues is interestingboth for the mathematical and for the engineering communities: for the former, this involves dealing with theminimization of non-convex and non-local energies; for the latter, it provides formalization and validation ofexperimental models. This special session will focus on current research topics including dislocation theory,pattern formation, fracture mechanics, plasticity, and their numerical implementation.

Ground States for a Ternary Systemwith Coulomb Interaction

Marco BonaciniUniversity of Bonn, GermanyHans Knuepfer

We study a variational model where two phases- interacting via attractive and repulsive Coulombforces - are embedded in a third homogeneousphase, describing for instance systems of copolymer-homopolymer blends or of surfactants in water solu-tions. The energy of the system is the sum of a localinterfacial contribution and a nonlocal interaction ofCoulomb type. We establish existence and regular-ity properties of global minimizers, together with afull characterization of minimizers in the small massregime. Furthermore, we prove uniform bounds onthe potential of minimizing configurations, which inturn imply some qualitative estimates about the ge-ometry of minimizers in the large mass regime.

Cohesive Fracture Evolutions: Exis-tence Results and Applications

Filippo CagnettiUniversity of Sussex, England

I will start by recalling an abstract existence theo-rem for the time evolution of cohesive fractures. Theabove result has some interesting features, but itis not easily implementable by a computer. I willthen discuss a recent work, in collaboration withMarco Artina, Massimo Fornasier, and FrancescoSolombrino (from Technical University of Munich),in which the model is modified, in such a way thatnumerical simulations can be done.

Periodic Critical Points of the Ohta-Kawasaki Functional

Riccardo CristoferiCarnegie Mellon University, USA

In this talk we present some new observations aboutperiodic critical points and local minimizers of a non-local isoperimetric problem arising in the modeling ofdiblock copolymers. In particular, by using a purelyvariational procedure, we show that it is possible toconstruct (locally minimizing) periodic critical pointswhose shape resemble that of any given strictly sta-ble constant mean curvature (periodic) hypersurface.

Existence and Uniqueness of Dy-namic Evolutions for a Peeling Testin Dimension One

Gianni dal MasoSISSA, ItalyGiuliano Lazzaroni, Lorenzo Nardini

In this paper we present a one-dimensional model ofa dynamic peeling test for a thin film, where the waveequation is coupled with a Gri�th criterion for thepropagation of the debonding front. Our main resultsprovide existence and uniqueness for the solution tothis coupled problem under di↵erent assumptions onthe data.

Wul↵ Shape Emergence inGraphene

Elisa DavoliUniversity of Vienna, AustriaPaolo Piovano, Ulisse Stefanelli

Graphene samples are identified as minimizers of con-figurational energies featuring both two- and three-body atomic-interaction terms. This variationalviewpoint allows for a detailed description of ground-state geometries as connected subsets of a regularhexagonal lattice. We investigate here how these ge-ometries evolve as the number n of carbon atomsin the graphene sample increases. By means of an


equivalent characterization of minimality via a dis-crete isoperimetric inequality, we prove that groundstates converge to the ideal hexagonal Wul↵ shapeas n ! +1. Precisely, we show that ground statesdeviate from such hexagonal Wul↵ shape by at mostKn3/4 + o(n3/4) atoms.

Ground States of a Two PhaseModel with Cross and Self Attrac-tive Interactions.

Lucia de LucaTechnical University of Munich, GermanyMarco Cicalese, Matteo Novaga, MarcelloPonsiglione

We consider a variational model for two interact-ing species (or phases), subject to cross and self at-tractive interactions. We show existence and severalqualitative properties of minimizers. Depending onthe strengths of the attractive forces, minimizers canexhibit di↵erent behaviors: phase mixing or phaseseparation with nested or disjoint phases. For thespecial case of Coulomb interaction forces, we fullycharacterize the ground state configurations, by giv-ing its explicit shape.

Second Order Gamma-Convergencefor the Modica Mortola Functional

Irene FonsecaCarnegie Mellon University, USAGianni Dal Maso, Gurgen Hayrapetyan,Giovanni Leoni, Matteo Rinaldi, BarbaraZwicknagl

The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirich-let boundary condition is studied when the parame-ter that determines the width of the transition layerstends to zero. The first order term in the asymptoticdevelopment by Gamma-convergence is well-known,and is related to a suitable anisotropic perimeterof the interface. Here it is shown that, dependingon symmetry and growth hypotheses on the doublewell potential, the second order term in the Gamma-convergence expansion is zero. Slow motion is ad-dressed, and related estimates of the rate of conver-gence of solutions of the associated Allen-Cahn equa-tion to the minimum value are discussed.

Linear and Nonlinear Stability ofBilayers Under the FunctionalizedCahn-Hilliard Gradient Flow

Gurgen HayrapetyanOhio University, USAK. Promislow

Functionalized energies, such as the Functional-ized Cahn-Hilliard, model phase separation in am-phiphilic systems, in which interface production isenergetically favorable, but is limited by competitionfor surfactant phase, which wets the interface. This

is in contrast to classical phase-separating energies,such as the Cahn-Hilliard, in which interfacial areais energetically penalized. We discuss the linear andnonlinear stability of bilayer interfaces under an asso-ciated mass-preserving gradient flow. In particular,for su�ciently small perturbations of radial bilayers,we show that there is a unique decomposition intoa non-radial bilayer and a decaying remainder, andthat as the remainder decays the perturbed inter-face relaxes back radial symmetry through a sharp-interface motion governed to leading order by a lin-earized Willmore flow.

Integral Representation for Func-tionals Defined on SBDp in Dimen-sion Two

Flaviana IurlanoIAM, University of Bonn, GermanySergio Conti, Matteo Focardi

We present an integral representation result for func-tionals with growth conditions which give coercivityon the space SBDp(⌦), for⌦ ⇢ R2. The space SBDp

of functions whose distributional strain is the sum ofan Lp part and a bounded measure supported on aset of finite H1-dimensional measure appears natu-rally in the study of fracture and damage models.Our result is based on the construction of a local ap-proximation by W 1,p functions. We also obtain ageneralization of Korn’s inequality in the SBDp set-ting.

Rigidity of Discrete Energies withSurface Scaling: Interactions Be-yond Nearest Neighbours VersusNon-Interpenetration

Giuliano LazzaroniSISSA, Trieste, ItalyRoberto Alicandro, Mariapia Palombaro

We present some discrete models for crystals withsurface scaling of the interaction energy. We assumethat at least nearest and next-to-nearest neighbourinteractions are taken into account. Our purpose isto show that interactions beyond nearest neighbourshave the role of penalising changes of orientationand, to some extent, they may replace the positive-determinant constraint that is usually required whenonly nearest neighbours are accounted for.

A Model for Dislocations in Epitax-ially Strained Elastic Films

Giovanni LeoniCarnegie Mellon University, USANicola Fusco, Irene Fonseca, MassimilianoMorini

We present a variational model for nucleation of dis-locations in epitaxially strained films. This is jointwork with Nicola Fusco, Irene Fonseca, and Massim-iliano Morini.


Optimal Location of Dislocations ina Crystal with Prescribed ExternalStrain

Ilaria LucardesiSISSA, Trieste, ItalyM. Morandotti, R. Scala, D. Zucco

Dislocations are point defects appearing in crystalsand explain the observation of plastic deformations.In this talk I propose a location problem for screwdislocations, in presence of a Dirichlet boundary con-dition, which prevents dislocations to migrate to theboundary and leave the domain. The study is car-ried out with techniques of Gamma-convergence andPDEs, in the framework of the so-called “core-radiusapproach“.

Variational Models for Dislocationsat Semi-Coherent Interfaces

Marcello PonsiglioneSapienza University of Rome, ItalySilvio Fanzon, Mariapia Palombaro

We discuss some simple variational models for dislo-cations at semi-coherent interfaces. The energy func-tional describes the competition between two terms:a surface energy induced by dislocations and a far

field elastic energy, spent to decrease the amount ofneeded dislocations. We prove that the former scaleslike the surface area of the interface, the latter likeits diameter. The proposed continuum model is de-duced from the semi-discrete theory of dislocations.Even if we deal with finite elasticity, linearized elas-ticity naturally emerges in our analysis since the farfield strain contribution vanishes as the interface sizeincreases.

A Bridging Mechanism in the Ho-mogenization of Brittle Compositeswith Soft Inclusions

Caterina Ida ZeppieriUniversity of Muenster, GermanyMarco Barchiesi, Giuliano Lazzaroni

We study the limit behavior of the energy associ-ated with a purely brittle composite whose micro-structure is characterized by soft inclusions period-ically embedded in a sti↵er matrix. We exhibit anelementary micro-geometry for the composite whichgives rise, in the limit, to a cohesive-zone energy.


Special Session 28: Recent Developments Related to Conservation Lawsand Hamilton-Jacobi Equations

Laura Caravenna, Universita di Padova, ItalyAnnalisa Cesaroni, University of Padova, Italy

Hung Vinh Tran, University of Wisconsin Madison, USA

The session focuses on some recent developments of first order nonlinear Partial Di↵erential Equations, andin particular conservation laws, Hamilton-Jacobi equations and related topics such as dynamical propertiesand homogenization. Recently the joint analysis of conservation laws and Hamilton-Jacobi equations onheterogeneous structures has received an increasing attention. This includes problems on networks and theirapplications to modeling of tra�c flows, homogenization in periodic and random media, dynamical propertiesof solutions, etc. One of the main motivation for problems on networks is the application to dynamic modelsof tra�c flow, e.g. the flowing of cars on a highway or of gas along pipelines or of packages of data ontelecommunication networks. The established mathematical framework of these models consists of singleconservation laws, systems of conservation or balance laws running on a network modeled as a topologicalgraph. These models are widely used also in engineering: it is an area where fundamental studies areinteresting but which is also directly related to applications. More recently, a complementary analysis of thenetwork dynamics based on Hamilton-Jacobi equations has also been developed. The current trend consistsof proposing new models/problems, studying their well-posednesses, dynamical properties (optimal controlformulas, large time behaviors), and related homogenization problems. Numerical approaches are as wellof great interest. Homogenization problems are about finding averaged (e↵ective) properties of solutionsto inhomogeneous equations depending on small parameters and set in self averaging media. The areais moving fast in various perspectives: qualitative and quantitative properties of the e↵ective equations,rates of convergences, stochastic homogenization of front propagations, and non-convex Hamilton-Jacobiequations, etc. Moreover, there have been a lot of developments connecting homogenization and problemson networks such as homogenization on junction framework, Hamilton-Jacobi equations on a network asa limit of a singularly perturbed problem in optimal control defined on thin strips around the network.The aim of this session is to bring together mathematicians working on conservation laws and Hamilton-Jacobi from di↵erent backgrounds and perspectives, including homogenization. It will be a great occasionto present the new advances and directions of di↵erent research groups, in order to interact and to improvethe understanding of these exciting topics that have been considered intensively in the last few years.

Global Existence of Prandtl-MeyerReflection and Optimal RegularityResults

Myoungjean BaePOSTECH, KoreaGui-Qiang G. Chen, Mikhail Feldman

Prandtl (1936) first employed the shock polar analy-sis to show that, when a steady supersonic flow im-pinges a solid wedge whose angle is less than a criticalangle (i.e., the detachment angle), there are two pos-sible configurations: the weak shock solution and thestrong shock solution, and conjectured that the weakshock solution is physically admissible. The funda-mental issue of whether one or both of the strong andthe weak shocks are physically admissible has beenvigorously debated over several decades and has notyet been settled in a definite manner. In this talk,I address this longstanding open issue and presentrecent analysis to establish the stability theorem forsteady weak shock solutions as the long-time asymp-totics of unsteady flows for all the physical param-eters up to the detachment angle for potential flow.

An HJB Equation and RegularityResults for a Time-Optimal ControlProblem in the Space of ProbabilityMeasures

Giulia CavagnariUniversity of Trento, ItalyAntonio Marigonda

We present some results related to the study of atime-optimal control problem in the space of proba-bility measures endowed with the topology inducedby the Wasserstein metric. Such a formulation seemsto be quite natural to model situations in which theknowledge of the initial state is only probabilistic asit happens when measurements are a↵ected by noises,or to describe the behaviour of multi-agent systems.Potential applications are problems arising in pedes-trian dynamics where a possible objective consists insteering a mass of people outside a room in the min-imum amount of time. The dynamics is given bya controlled continuity equation, which can be seenas a superposition of admissible curves of an under-lying optimal control problem, while the target setis defined by duality. Through suitable definitionsof sub/super-di↵erentials we formulate an Hamilton-Jacobi-Bellman equation solved, in this suitable vis-cosity sense, by the generalized minimum time func-tion T (·), provided its continuity. We discuss alsosome attainability results and we give further con-


ditions yielding Lipschitz-continuity regularity of T .The main open problem remains the formulation ofan HJB equation requiring less regularity on T , andto prove a comparison principle, granting uniquenessof the solution.

Semi-Geostrophic System withVariable Coriolis Parameter

Jingrui ChengUniversity of Wisconsin-Madison, USAMichael Cullen, Mikhail Feldman

The semi-geostrophic system (abbreviated as SG) is amodel of large-scale atmospheric/ocean flows. Previ-ous works about the SG system have been restrictedto the case of constant Coriolis force, where we writethe equation in dual coordinates and solve. Thismethod does not apply for variable Coriolis param-eter case. We develop a time-stepping procedure toovercome this di�culty and prove local existence anduniqueness of smooth solutions to SG system. This isjoint work with Michael Cullen and Mikhail Feldman.

Loss of Regularity for Linear Trans-port Equations

Gianluca CrippaUniversity of Basel, SwitzerlandGiovanni Alberti, Anna L. Mazzucato

For a linear transport equation

@tu+ b ·ru = 0

with a Lipschitz velocity field b, the classical Cauchy-Lipschitz theory ensures propagation in time of the(Lipschitz) regularity of the initial datum. Althoughfor less regular (Sobolev or BV , for instance) ve-locity fields a well-posedness theory for this equa-tion is by now available (based on seminal resultsby DiPerna-Lions and Ambrosio), it turns out thatthe issue of the propagation in time of the regularityis much more delicate. In this talk I will report ona joint work with Alberti and Mazzucato, in whichSobolev velocity fields and smooth initial data areconstructed, in such a way that any fractional reg-ularity of the solution is instantaneously destroyed.Connections to mixing phenomena in fluids will alsobe mentioned.

Stochastic Homogenisation forDegenerate Hamilton-Jacobi Equa-tions.

Federica DragoniCardi↵ University, WalesClaudio Marchi, Paola Mannucci

In the talk I Investigate the limit behaviour for afamily of Cauchy problems for Hamilton-Jacobi equa-tions describing a stochastic microscopic model. TheHamiltonian considered is not coercive in the to-tal gradient. The Hamiltonian depends on a lower

dimensional gradient variable which is associatedto a Carnot group structure. The rescaling isadapted to the Carnot group structure, therefore itis anisotropic. Under suitable stationary-ergodic as-sumptions on the Hamiltonian, the solutions of thestochastic microscopic models will converge to a func-tion independent of the random variable: the limitfunction can be characterised as the unique viscos-ity solution of a deterministic PDE. The key stepwill be to introduce suitable lower-dimensional con-strained variational problems.

Shock Reflection Problem: Exis-tence and Properties of Solutions

Mikhail FeldmanUniversity of Wisconsin-Madison, USAMyoungjean Bae, Gui-Qiang Chen, Wei Xiang

We discuss shock reflection problem for compressiblegas dynamics, and von Neumann conjectures on tran-sition between regular and Mach reflections. Thenwe will talk about existence and regularity of regu-lar reflection solutions for potential flow equation upto the detachment angle, and geometric propertiesof the free boundary (shock curve), including its con-vexity. Our approach is to reduce the shock reflectionproblem to a free boundary problem for a nonlinearequation of mixed elliptic-hyperbolic type. We willalso discuss known results and open questions regard-ing uniqueness and stability. The talk is based on thejoint works with Gui-Qiang Chen, Myoungjean Baeand Wei Xiang.

On Cell Problems for Hamilton-Jacobi Equations with Non-Coercive Hamiltonians and ItsApplication to HomogenizationProblems

Nao HamamukiHokkaido University, JapanAtsushi Nakayasu, Tokinaga Namba

We study a cell problem arising in homogenizationfor a Hamilton-Jacobi equation whose Hamiltonian isnot coercive. We introduce a generalized notion of ef-fective Hamiltonians by approximating the equationand characterize the solvability of the cell problem interms of the generalized e↵ective Hamiltonian. Un-der some su�cient conditions, the result is appliedto the associated homogenization problem. We alsoshow that homogenization for non-coercive equationsfails in general.


Homogenization of Hamilton-JacobiEquations in Dynamic Random En-vironments

Wenjia JingThe University of Chicago, USAPanagiotis E. Souganidis, Hung V. Tran

We consider stochastic homogenization of Hamilton-Jacobi equations in dynamic random environments,where the coe�cients of the equations, namely theHamiltonian and, for second order equations, the dif-fusion matrix, are highly oscillatory in space andtime. I will discuss how to generalize the metricapproach of stochastic homogenization developed forstatic random environment to the dynamic randomsetting, when uniform continuity (uniform with re-spect to the scale of oscillation and the random re-alization) of the minimal cost function is available.This talk is based on joint work with P.E. Souganidisand H.V. Tran.

Scalar Conservation Laws withMarkov Initial Data

David KasparBrown University, USAFraydoun Rezakhanlou

The inviscid Burgers‘ equation has the remarkableproperty that its dynamics preserve the class of spec-trally negative Levy initial data, as observed by Car-raro and Duchon (statistical solutions) and Bertoin(entropy solutions). Further, the evolution of theLevy measure admits a mean-field description, givenby the Smoluchowski coagulation equation with ad-ditive kernel. In this talk we discuss ongoing e↵ortsto generalize this result to scalar conservation laws,a special case where this is done, and a connectionwith integrable systems. Includes work with F. Reza-khanlou.

Numerical Approximation of En-semble Based Solutions to Incom-pressible Flow Equations

Filippo LeonardiETH Zurich, SwitzerlandS. Mishra, Ch. Schwab

We are interested in the behaviour of ensemble ofsolutions of incompressible Navier-Stokes and Eulerequations and in algorithms for the approximation ofthose solutions. In the context of viscous flows, fol-lowing the introduction of a proper notion of statisti-cal solution for the vorticity formulation, we presentan e�cient and convergent vorticity-based finite dif-ference algorithm for the approximation of such so-lutions, exploiting e�cient Multi-level Monte Carlotechniques. For this algorithm, we are able to proveconvergence rates under suitable assumptions. Forinviscid flows, we present a similar algorithm, based

on single-level Monte Carlo approximations, for thecomputation of approximations of admissible mea-sure valued solutions: a notion of solution arisingnaturally when considering a vanishing viscosity ap-proach to incompressible Euler equations.

Stochastic Homogenization ofReaction-Di↵usion Equations inIsotropic Media

Jessica LinUniversity of Wisconsin-Madison, USAAndrej Zlatos

We consider reaction-di↵usion equations with com-bustion nonlinearity in stationary ergodic andisotropic environments in dimensions d 3. Weprove existence of asymptotic, deterministic speedsof propagation for solutions with both spark-like andfront-like initial data. This leads to a general stochas-tic homogenization result which shows that on aver-age, the large-scale large-time behavior is governedby a deterministic Hamilton-Jacobi equation model-ing front propagation. Applications include predict-ing the evolution of forest fires in random heteroge-neous media. This talk is based on joint work withAndrej Zlatos.

Is a Nonlocal Di↵usion StrategyConvenient for Biological Popula-tions in Competition?

Annalisa MassaccesiUniversitat Zurich, SwitzerlandEnrico Valdinoci

We study the convenience of a nonlocal dispersalstrategy in a reaction-di↵usion system with a frac-tional Laplacian operator. We show that there arecircumstances - namely, a precise condition on thedistribution of the resource - under which a nonlo-cal dispersal behavior is favored. In particular, weconsider the linearization of a biological system thatmodels the interaction of two biological species, onewith local and one with nonlocal dispersal, that arecompeting for the same resource. We give a simple,concrete example of resources for which the equilib-rium with only the local population becomes linearlyunstable. In a sense, this example shows that non-local strategies can become successful even in an en-vironment in which purely local strategies are dom-inant at the beginning, provided that the resourceis su�ciently sparse. Indeed, the example consideredpresents a high variance of the distribution of the dis-persal, thus suggesting that the shortage of resourcesand their unbalanced supply may be some of the ba-sic ingredients that favor nonlocal strategies.


On Rate of the Convergence for theVanishing Discount Problem

Hiroyoshi MitakeHiroshima University, JapanKohei Soga

We will discuss on the vanishing discount problemfor Hamilton-Jacobi equations. Recently, it has beenproved that the whole family of solutions v� of thediscount problem with the factor � converges to asolution of the ergodic problem as � ! 0. In thistalk, we will consider some of specific case in 1D andparticularly discuss on a rate of the convergence.

Conservation Laws on Networks

Benedetto PiccoliRutgers University, USAMaria Laura Delle Monache

We present recent and less recent results on the the-ory of conservation laws on topological graphs, in-cluding existence and continuous dependence of so-lutions. Moreover, we will show applications to dif-ferent areas such as vehicular tra�c, supply chainsand water channels.

A Level Set Approach to the Crys-talline Mean Curvature Flow ofSurfaces

Norbert PozarKanazawa University, Japan

The crystalline mean curvature flow is a motion of asurface with normal velocity law V = f(⌫, �), wheref is a given function of the normal vector ⌫ and theso-called crystalline mean curvature �. This prob-lem appears in models of crystal growth formulatedas a gradient flow of the surface energy when thesurface energy density is not di↵erentiable. A char-acteristic feature of the evolution is the appearanceof flat parts of the surface, the facets of a crystal, onwhich the crystalline curvature is a nonlocal quan-tity. These facets are usually preserved by the flow,but they might also break or bend. Because of thisphenomenon, even a local-in-time well-posedness hadbeen open in dimensions higher than two except inspecial cases like convex initial data. We introduce anew notion of viscosity solutions for the level set for-mulation of the crystalline mean curvature flow. Inthree dimensions, we prove a comparison principle,stability and well-posedness of the initial value prob-lem for arbitrary bounded crystals. This talk is basedon joint work with Yoshikazu Giga from Universityof Tokyo.

A Counterexample Concerning Reg-ularity Properties for Systems ofConservation Laws

Laura SpinoloIMATI-CNR, Pavia, ItalyLaura Caravenna

In 1973 Schae↵er established a result that appliesto scalar conservation laws with convex fluxes andcan be loosely speaking formulated as follows: for ageneric smooth initial datum, the admissible solutionis smooth outside a locally finite number of curvesin the (t, x) plane. Here the term “generic” shouldbe interpreted in a suitable technical sense, relatedto the Baire Category Theorem. My talk will aimat discussing a recent explicit counterexample thatshows that Schae↵er’s Theorem does not extend tosystems of conservation laws.

Inverse Problems, Non-Roundnessand Flat Pieces Ofthe E↵ectiveBurning Velocity from an InviscidQuadratic Hamilton-Jacobi Model

Yifeng YuUC Irvine, USAWenjia Jing, Hung V. Tran

I will talk about some finer properties of the e↵ec-tive burning velocity from a combustion model intro-duced by Majda and Souganidis in 90s. We provedthat when the dimension is two and the flow of theambient fluid is either weak or very strong, the levelset of the e↵ective burning velocity has flat pieces.Implications on the e↵ective flame front and otherrelated inverse type problems will also be discussed.This is a joint work with Wenjia Jing and Hung Tran.

On the Camassa-Holm Type Equa-tions

Qingtian ZhangPenn State University, USAAlberto Bressan, Geng Chen, Mingjie Li

I will talk about the uniqueness of energy conser-vative weak solutions to Camassa-Holm and two-component Camassa-Holm equations, as well as thegeneric regularity of those solutions. Time permit-ting, I‘ll also mention the well-posedness of cubicCamassa-Holm equations.


Special Session 29: Advances in Theory & Application of ReactionDi↵usion Models

Jerome Goddard II, Auburn University Montgomery, USARatnasingham Shivaji, University of North Carolina Greensboro, USA

Application of reaction di↵usion models is seemingly endless with their use naturally arising in disciplinessuch as biology, ecology, chemistry, geology, physics, and engineering. Reaction di↵usion models have re-cently become even more useful in modeling physical and biological phenomena due to many importantdevelopments in the study of their dynamics. A key tool in understanding the dynamics of such modelsrequires detailed investigation of the structure solutions to the corresponding parabolic and elliptic partialdi↵erential equations. This investigation yields interesting nonlinear initial-boundary and boundary valueproblems of varied types. Even though the study of reaction di↵usion models has had a rich mathematicalhistory dating back to the 1960s, much is still not known about the structure of solutions to such problems.Several techniques have been developed and successfully used to solve these problems including, iterativemonotone methods, sub-super solutions, topological degree theory, and variational methods, among others.

A Nonlinear Model of Cancer Tu-mor Treatment with Cancer StemCells

Kristen AbernathyWinthrop University, USAHannah Horner, Alexander Middleton

We present a system of six nonlinear ordinary dif-ferential equations which model the interactions be-tween normal, cancer, endothelial, and cancer stemcell populations, as well as chemotherapy agent andanti-angiogenic agent concentrations. With analy-sis, it is shown that chemotherapy, with the co-administration of anti-angiogenic treatment, can pro-duce three states: persistence of cancer, cancer re-currence, and a cure state. Results are supportedby numerical simulations and bifurcation diagrams.We conclude with a discussion of the role of travelingwaves in cancer dynamics and possible extensions forfuture work.

A Mathematical Model of CancerStem Cell Driven Tumor Growthwith Radiation and ChemotherapyTreatment

Zach AbernathyWinthrop University, USASavannah Bates, Rebecca Santorella

In this talk, we build a tumor model that incorpo-rates the cancer stem cell hypothesis with chemother-apy and periodic radiation treatment using aspects ofcurrent models. We calculate conditions for the ex-istence and local stability of equilibria in the case ofno treatment as well as constant radiation with andwithout chemotherapy. Additionally, for periodic ra-diation treatment, su�cient conditions for the exis-tence of cancer persistence and cure state periodicsolutions are established. Conditions for global sta-bility of the periodic cure state are also derived usinga Lyapunov function. Numerical simulations demon-strate that treatments targeting cancer stem cells aremore e↵ective in eradicating cancer.

Analyticity in Time for an AbstractNonlinear Evolutionary Problem

Falko BaustianUniversity of Rostock, Germany

We consider an abstract nonlinear Cauchy problem

⇢dudt

�A(t, u(t))u(t) = f(t, u(t)) + g(t),u(0) = u

0

,

with the initial value in some real interpolation space.We investigate existence of a strict solution and an-alyticity in time for operators with the maximal Lp-regularity property. The results are applied to a gen-eral strongly parabolic system of semilinear partialdi↵erential equations. For this system we can showanalyticity in time and also in the space variables.To prove the analyticity in space we approximate theinitial value with analytic functions and use suitableestimates of the corresponding solutions on a com-plex domain. The results are a generalisation of thelinear case in P. Takac: Space-Time analyticity ofweak solutions to linear parabolic systems with vari-able coe�cients, Journal of Functional Analysis 236,50–88, 2012.

On the Solvability of Asymptoti-cally Linear Systems at Resonance

Maya ChhetriUNC Greensboro, USAPetr Girg

We employ the Lyapunov-Schmidt method to discussthe solvability of asymptotically linear system at res-onance at the simple eigenvalue of the correspond-ing linear eigenvalue problem. We consider nonlin-ear perturbations that are vanishing at infinity, un-bounded but sublinear at infinity and those satisfyingLandesman-Lazer type conditions for systems. Ourapproach allows us to treat systems that do not havevariational structure and are coupled in linear as wellas nonlinear part.


On Existence of Multiple PositiveSolutions for Elliptic Equations andSystems

David CostaUniversity of Nevada Las Vegas, USAAlfonso Castro, Ratnasingham Shivaji

After reviewing some of the literature on existence ofpositive solutions for elliptic equations, we present anew result on existence of multiple positive solutionsfor a class of elliptic systems in variational form.

Modeling the E↵ects of U-ShapedDensity Dependent Dispersal ViaReaction Di↵usion Equations

Jerome Goddard IIAuburn University Montgomery, USAR. Shivaji

Dispersal is broadly defined as movement from onehabitat patch to another and typically is consideredto encompass three stages: emigration, inter-patchmovement, and immigration. Dispersal can haveboth beneficial and detrimental e↵ects on the persis-tence of spatially structured systems. Recent empir-ical results indicate that certain organisms‘ emigra-tion from a patch is dependent on their own density–known as density dependent emigration. In fact, aU-shaped relationship between density and emigra-tion has been observed in several organisms in fieldstudies. To date, little is known about the patch-levelconsequences of such a dispersal strategy. In thistalk, we will discuss a population model built uponthe reaction di↵usion framework that is designed tomodel the patch-level e↵ects of U-shaped density de-pendent emigration. In particular, we will discuss theexistence and stability properties of positive steadystate solutions to this model for a one-dimensionalpatch. A brief discussion regarding ecological con-clusions of the model’s predictions will also be pre-sented.

Trajectory Attractor for a Reaction-Di↵usion Problem from ClimateModeling

Georg HetzerAuburn University, USA

Energy balance climate models lead to reaction-di↵usion problems with slow di↵usion and a set-valued reaction term on the 2-sphere. A hysteresisterm accounts for a frequent repetition of sudden andfast warming followed by much slower cooling as ob-served from paleoclimate proxy data. Existence of atrajectory attractor and the lack of attractor stabilitywill be discussed.

Bounded Solutions of NonlinearParabolic Equations in UnboundedDomains

Nsoki MavingaSwarthmore College, USA

We are concerned with the existence of full-boundedsolutions for nonlinear parabolic equations where thespace domain is unbounded with compact boundary(such as RN which has empty boundary, or an exte-rior space-domain) and the time domain is actuallythe entire real line. We give an example which showsthat the classical comparison principle does not holdin general on such domains. We establish L1 a-prioriestimates for solutions to linear boundary value prob-lems and derive a comparison/weak-maximum prin-ciple. We then define the notion of sub and super-solutions in this case, and by using comparison tech-niques, a-priori estimates and nonlinear approxima-tion methods, we derive the existence results.

Positive, Radial Solutions for a Su-perlinear Semipositone p-LaplacianProblem on the Exterior of a Ball

Quinn MorrisUniversity of North Carolina at Greensboro, USAInbo Sim, Ratnasingham Shivaji

We prove the existence of positive radial solutions toa class of semipositone p-Laplacian problems on theexterior of a ball subject to Dirichlet and nonlinearboundary conditions. Using variational methods weprove the existence of a solution, and then use a pri-ori estimates to prove the positivity of the solution.

Multiple Strong Solutions and Bi-furcation Structure for Di↵usionEquations with Nonlinear Bound-ary Flux

M. NkashamaUniversity of Alabama at Birmingham, USAN. Mavinga

We shall present multiplicity results for (strong) so-lutions of second order elliptic partialdi↵erential equations with nonlinear boundary con-ditions which include among others the so-calledSteklov type problems (i.e., harmonic-function so-lutions). We impose asymptotic conditions onthe boundary-nonlinearity and let the boundary-parameter vary. Our asymptotic conditions includethe so-called ‘very-strong-resonance‘ conditions aswell as oscillatory behavior. We set up the problemas a nonlinear ‘normal derivative trace‘ equation onthe boundary, and proceed to establish a priori esti-mates and prove multiplicity results (for large-normsolutions) when the parameter belongs to a (nontriv-ial) continuum of real numbers. The proofs are basedon degree theory, continuation methods and bifurca-tion from infinity techniques.


A Priori Bounds and Branches ofPositive Solutions for SubcriticalSemilinear Elliptic Systems

Rosa PardoUniversidad Complutense de Madrid, SpainNsoki Mavinga

We provide a-priori L1-bounds for classical positivesolutions of subcritical semilinear elliptic systems inbounded convex domains⌦ ⇢ RN . The criticalcase for a system ��u = f(v),��v = g(u), withDirichlet boundary conditions is the so called crit-ical hyperbola, that is f(t) = tp, g(s) = sq with1

p+1

+ 1

q+1

= N�2

N. We prove that all classical positive

solutions are a priori L1-bounded where the non-linearities are slightly below the critical hyperbola,

specifically f(v) =vp

[ln(e+ v)]↵, g(u) =

uq

[ln(e+ u)]�

with1

p+ 1+

1q + 1

=N � 2N

and ↵,�>2

N � 2.

Our analysis provides a new class of nonlinearitiesfor which classical positive solutions of Hamiltonianelliptic systems are a priori bounded.

In [1] the authors prove the existence of a-prioribounds for positive solutions of elliptic equations��u = f(u) with Dirichlet homogeneous bound-

ary conditions, when f(u) = uN+2N�2 / ln(e+ u)↵, with

↵ > 2/(N � 2), see [1, Corollary 2.2].

References

[1] A. Castro and R. Pardo, A priori bounds forPositive Solutions of Subcritical Elliptic Equa-tions, Rev. Mat. Complut. 28 (2015), 715-731.

[2] N. Mavinga and R. Pardo, A priori boundsand existence of positive solutions for subcrit-ical semilinear elliptic systems, Preprint.

Steklov-Robin Eigencurves

Stephen RobinsonWake Forest University, USADr. Mauricio Rivas

We investigate nontrivial solutions of the boundaryvalue problem

��u = µm2

u in⌦@u@⌫

+ bu = �m1

u on @⌦

where ⌦ is a smooth bounded domain in RN , (�, µ) 2R2, and the coe�cient function b and the weightsm

1

,m2

lie in appropriate Lp-spaces. In particular wecharacterize the sequence of eigencurves (�, µn(�))associated with the problem and then prove severalresults concerning the properties of those curves.

Spreading Speeds and Semi-WaveSolutions of Di↵usive KPP Equa-tions with a Free Boundary in TimeAlmost Periodic Environments

Wenxian ShenAuburn University, USAFang Li, Xing Liang

The current talk is concerned with spreading speedsand semi-wave solutions of di↵usive KPP equationswith a free boundary in time almost periodic environ-ments. It first discusses the criteria for the spread-ing to occur in such equations. It then provides acharacterization of the spreading speeds and showsthe existence of almost periodic semi-wave solutionsin the case that the spreading occurs. It should bepointed out that the spreading may not occur in dif-fusive KPP equations with a free boundary.

On Radial Solutions for SingularCombined Superlinear Elliptic Sys-tems on Annular Domains

Ratnasingham ShivajiUniversity of North Carolina at Greensboro, USAD. Hai

We prove the existence of a large positive solution tothe system

8<

:

�(rN�1�1

(u0))0 = �rN�1f1

(v), a < r < b,�(rN�1�

2

(v0))0 = �rN�1f2

(u), a < r < b,u(a) = 0 = u(b), v(a) = 0 = v(b),

where a > 0, � is a small positive param-eter, fi : (0,1) ! R are continuous and

limz!1��1

1

(f1

(c(��1

2

(f2

(z)))))

z= 1 for all c > 0.

Three Positive Solutions for One-Dimensional P-Laplacian Problemwith Sign-Changing Weight

Inbo SimUniversity of Ulsan, KoreaSatoshi Tanaka

We show that one-dimensional p-Laplacian with asign-changing weight which is subject to Dirichletboundary condition has three positive solutions sug-gesting suitable conditions on the weight functionand nonlinearity. Proofs are mainly based on thedirections of a bifurcation.


A Uniqueness Result for a Semi-positone p-Laplacian Problem onthe Exterior of a Ball

Byungjae SonUniversity of North Carolina at Greensboro, USARatnasingham Shivaji, Inbo Sim

We consider steady state reaction di↵usion equationson the exterior of a ball :

8<

:

��pu = �K(|x|)f(u) in ⌦E ,u = 0 on |x| = r

0

,u ! 0 when |x| ! 1,

where� pz := div(|rz|p�2rz), p 2 (1, n), � > 0,r0

> 0 and ⌦E := {x 2 Rn | |x| > r0

}. Herethe weight function K 2 C1([r

0

,1), (0,1)) satis-fies limr!1 K(r) = 0, and the reaction term f 2C1[0,1) is strictly increasing and satisfies f(0) is

negative, lims!1 f(s) = 1, lims!1f(s)

sp�1 = 0 andf(s)sq

is nonincreasing on [a,1) for some a > 0 andq 2 (0, p � 1). We establish the uniqueness of non-negative radial solution for �� 1.

On the Infinite Propagation Speedin Parabolic Problems with thep-Laplacian in a Domain for p < 2

Peter TakacUniversity of Rostock, GermanyJiri Benedikt, Petr Girg, Lukas Kotrla

The validity of the weak and strong comparisonprinciples for degenerate parabolic partial di↵eren-tial equations with the p-Laplace operator� p(u) =div(|ru|p�2ru) will be discussed for 1 < p < 2 (the“singular” case). This case is entirely di↵erent fromthe “degenerate” case p > 2 that allows for nonneg-ative ”multi-bump” solutions with a spatially com-pact support and zero initial data. For 1 < p < 2 weconsider the special case of comparing the trivial so-lution (⌘ 0, by hypothesis) with a nontrivial nonneg-ative solution u(x, t) that starts from nontrivial non-negative initial data u

0

(x). We will show that, evenfor a doubly nonlinear parabolic problem, there is a(su�ciently short) time-interval (0, T

0

) such that thesolution u(x, t) becomes positive immediately, i.e.,u(x, t) > 0 for all (x, t) 2 ⌦⇥T

0

. The spatial domain⌦ ⇢ RN is arbitrary, possibly unbounded. The previ-ous work was focused on the case ⌦= RN . This caseis much easier, as we are able to construct a subso-lution – a spherical wave – travelling arbitrarily fast.The case⌦ 6= RN requires the use of a rather so-phisticated local result to obtain u(x, t) > 0 for all(x, t) 2 ⌦ ⇥ T

0

.


Special Session 30: High Order Numerical Methods for PartialDi↵erential Equations

Wei Wang, Florida International University, USAZhongming Wang, Florida International University, USA

High order numerical methods have attracted considerable attention in scientific computation and engineer-ing community due to their accuracy. This minisymposium aims to present the most recent developmentsin the design, analysis, implementation and applications of high order methods. Topics may include discon-tinuous Galerkin (DG) method and weighted essentially non-oscillatory (WENO) method as well as theirapplications in fluid dynamics, kinetic theory, quantum mechanics, biophysics and electromagnetics.

Recovering Exponential Accuracy inSpectral Methods Involving Piece-wise Smooth Functions

Zheng ChenOak Ridge National Laboratory, USAChi-Wang Shu

Spectral methods achieve exponential accuracy bothon the approximation level and for solving partialdi↵erential equations, if the solution is analytic andperiodic. Lack of periodicity results in poor point-wise accuracy and O(1) errors near the discontinuity,even though the function is analytic. Such behavioris the so-called Gibbs Phenomenon. With Gegen-bauer reconstruction, the function with discontinu-ities can be recovered with exponential accuracy in-side each subinterval of analyticity. These techniqueshave been widely used as post-processing methods inmany fields. The methods highly rely on the an-alyticity of the function, and thus fail in the casethat the function has a singularity. The techniqueshave been extended via transformations to recoverthe exponential accuracy point-wisely for functionswhich have unbounded derivative singularities at endpoints, from the knowledge of point values on stan-dard collocation points or the first 2N + 1 Fouriercoe�cients. With this new reconstructions as post-processing methods, we are able to obtain exponen-tial accuracy of spectral methods applied to lineartransport equations involving such functions. Nu-merical results and applications will be shown in thistalk.

Local Discontinuous GalerkinMethod for Khokhlov-Zabolotskaya-Kuznetsov (KZK) Equation

Ching-Shan ChouOhio State University, USAWeizhou Sun, He Yang, Yulong Xing

Acoustic pulses with high intensity are widely usedin medical ultrasonics and sonar systems. Finite am-plitude e↵ects by high-intensity sound cannot be pre-dicted by linear acoustical theory, and considerationof the combined e↵ects of di↵raction, absorption andnonlinearity is require to study finite amplitude wavepropagation in thermoviscous fluid from an acoustic

source of finite size. Numerical simulations are chal-lenging due to the non-smoothness of the solutions.Here we present the LDG scheme for this type ofequations, and we will show the stability analysis andnumerical experiments.

A New Multiscale DiscontinuousGalerkin Method for the One-Dimensional Stationary SchrodingerEquation

Bo DongUniversity of Massachusetts Dartmouth, USAChi-Wang Shu, Wei Wang

We develop and analyze a new multiscale discon-tinuous Galerkin (DG) method for one-dimensionalstationary Schrodinger equations with open bound-ary conditions which have highly oscillating solu-tions. Our method uses a smaller finite element spacethan the WKB local DG method proposed in Wangand Shu (J Comput Phys 218:295–323, 2006) whileachieving the same order of accuracy with no res-onance errors. We prove that the DG approxima-tion converges optimally with respect to the meshsize h in L2-norm without the typical constraint thath has to be smaller than the wave length. Numer-ical experiments were carried out to verify the sec-ond order optimal convergence rate of the methodand to demonstrate its ability to capture oscillat-ing solutions on coarse meshes in the applications toSchrodinger equations.

A Hybrid Eulerian-LagrangianMethod for Free Boundary Prob-lems

David KellyFlorida International University, USA

In this paper the author considers the numerical so-lution of time-dependent free boundary problems.Free boundary problems are boundary-value prob-lems where part of the boundary is unknown andthus forms part of the solution. An e�cient, high-order, hybrid Eulerian-Lagrangian method for thetime evolution of the free boundary will be outlined.An application of the method to predict the evolu-tion of fully non-linear water waves over arbitrarybathymetry will be presented.


A Conservative Sweeping Methodfor Enforcing Maximum Principle

Yuan LiuMississippi State University, USA

In this talk, we will talk about a conservative bound-preserving sweeping procedure. The main advantageis the simplicity of implementation while maintainingthe high order of accuracy. Numerical examples areprovided to show the performance and e�ciency ofthe procedure.

High-Order Methods for Traveltimeand Amplitude with Application inGeometrical Optics

Songting LuoIowa State University, USA

We present e�cient methods to compute high or-der accurate traveltime and amplitude with applica-tions in geometrical optics. E�cient factorization ap-proaches are presented to resolve the upwind sourcesingularities such that high-order methods can be de-signed and applied e↵ectively to obtain high-orderaccurate traveltimes and amplitudes. With high ac-curate traveltimes and amplitudes, we present an ef-ficient method, namely the fast Huygens sweepingmethod, to solve the Helmholtz equation in the highfrequency regime. Numerical examples verify the ef-fectiveness of the methods.

Multi-Level Monte Carlo Methodfor Stochastic Optimal ControlProblems

Ju MingBeijing Computational Science Research Centre,Peoples Rep of ChinaQiang Du, Qi Sun

In this lecture, we consider the implementation ofmulti-level Monte Carlo (MLMC) method to an el-liptic optimal control problem with uncertain coe�-cients. Sample size formulas at each level of MLMCfrom the perspective of optimization, i.e., minimiz-ing the computational error/cost with given compu-tational cost/error, were derived. A gradient-basedoptimization algorithm using MLMC-based finite el-ement method was proposed and compared to the re-sults obtained by classical Monte Carlo method thatemploys many more degrees of freedom. These com-parisons show the e↵ectiveness and feasibility of theuse of MLMC for obtaining accurate optimal solu-tions, which are required to construct statistical mo-ments associated with the quantity of interest (QoI),of the stochastic control problem at low cost.

High-Order DG-FEM for Micro-Macro Partitioned Kinetic Models

James RossmanithIowa State University, USA

The dynamics of gases can be simulated using kineticor fluid models. Kinetic models are valid over mostof the spatial and temporal scales that are of physi-cal relevance in many application problems; however,they are computationally expensive due to the highdimensionality of phase space. Fluid models havea more limited range of validity, but are generallycomputationally more tractable than kinetic models.One critical aspect of fluid models is the question ofwhat assumptions to make in order to close the fluidmodel. In this work we develop a high-order discon-tinuous Galerkin finite element method (DG-FEM)for a so-called micro-macro decomposition approxi-mation of the kinetic equations. The micro-macrodecomposition approach allows us to obtain accuratesolutions of the fluid model, but instead of forcinga particular moment-closure approximation, whichwould typically only have a limited range of validity,this approach directly solves a version of the kineticequations and uses this solution to provide a closurefor the fluid equations. The proposed approach inthis work makes use of an e�cient semi-LagrangianDG method for solving the kinetic portion of the up-date. The resulting numerical method is validated onseveral standard test cases.

An Entropy Satisfying Discontinu-ous Galerkin Method for NonlinearFokker-Planck Equations

Zhongming WangFlorida International University, USAHailiang Liu

We propose a high order discontinuous Galerkin(DG) method for solving nonlinear Fokker-Planckequations with a gradient flow structure. For some ofthese models it is known that the transient solutionsconverge to steady-states when time tends to infin-ity. The scheme is shown to satisfy a discrete versionof the entropy dissipation law and preserve steady-states, therefore providing numerical solutions withsatisfying long-time behavior. The positivity of nu-merical solutions is enforced through a reconstruc-tion algorithm, based on positive cell averages. Forthe model with trivial potential, a parameter rangesu�cient for positivity preservation is rigorously es-tablished. For other cases, cell averages can be madepositive at each time step by tuning the numericalflux parameters. A selected set of numerical exam-ples is presented to confirm both the high-order ac-curacy and the e�ciency to capture the large-timeasymptotic.


Well-Balanced DiscontinuousGalerkin Methods for the EulerEquations Under GravitationalFields

Yulong XingUniversity of California Riverside, USA

Hydrodynamical evolution in a gravitational fieldarises in many astrophysical and atmospheric prob-lems. Improper treatment of the gravitationalforce can lead to a solution which oscillates aroundthe equilibrium. In this presentation, we pro-pose a recently developed well-balanced discontinu-ous Galerkin method for the Euler equations undergravitational fields, which can maintain the isother-mal and polytropic hydrostatic equilibrium states ex-actly. Some numerical tests are performed to verifythe well-balanced property, high-order accuracy, andgood resolution for smooth and discontinuous solu-tions.

Third Order Maximum-Principle-Satisfying Direct DiscontinuousGalerkin Methods for Time Depen-dent Convection Di↵usion equationson unstructured triangular meshes

Jue YanIowa State University, USA

We develop 3rd order maximum-principle-satisfyingdirect discontinuous Galerkin methods for convectiondi↵usion equations on unstructured triangular mesh.We carefully calculate the normal derivative numeri-cal flux across the cell boundary and prove that, withproper choice of parameter pair in the numerical fluxformula, the quadratic polynomial solution satisfiesthe strict maximum principle. The numerical solu-tion is bounded within the given range and thirdorder accuracy is maintained. There is no geomet-

ric restriction on the mesh and obtuse triangles areallowed in the partition. A sequence of numerical ex-amples are carried out to show the results includingincompressible flows are shown to demonstrate thetheoretical results.

Feedback Boundary Control of Hy-perbolic Systems with Sti↵ SourceTerm

Hui YuRWTH Aachen University, GermanyMichael Herty

We consider the feedback boundary control of hyper-bolic systems with sti↵ source term. Such equationsappear in many applications, such as gas dynamicsin pipes and water flow in canals. By combiningweighted Lyapunov functions, the structure is usedto derive new stabilization results. The result is il-lustrated with the numerical analysis on the decayrate of the Lyapunov function in terms of the sti↵parameter and an application to boundary stabiliza-tion of gas dynamics in pipes.

Compact WENO Limiters for Dis-continuous Galerkin Methods

Xinghui ZhongUniversity of Utah, USAChi-Wang Shu

We design compact limiters using WENO methodol-ogy for discontinuous Galerkin methods for solvinghyperbolic conservation laws, with the goal of ob-taining a robust and high order limiting procedure tosimultaneously achieve uniform high order accuracyand sharp, non-oscillatory shock transitions. Themain advantage of these compact limiters is theirsimplicity in implementation, especially on multi-dimensional unstructured meshes.


Special Session 31: Celestial Mechanics and Beyond

Zhifu Xie, Virginia State University, USAErnesto Perez-Chavela, Ernesto Perez-Chavela, Mexico

Alessandro Portaluri, Universita degli Studi di Torino, Italy

This special session will concentrate on the latest developments in the field of celestial mechanics which laidthe foundations for the birth of dynamical systems. The study of the N-body problem continues to attractresearchers in a wide range of fields including dynamical systems, topology, variational methods, algebraicgeometry, numerical methods and KAM theory. This special session provides a marketplace for ideas, andhelps identify trends and areas of new opportunity in the field. This session brings established researchersand recent Ph.D.s together, some of whom are women or from groups underrepresented in mathematics.Some specific topics to be covered include Hamiltonian system, Ergodic theory, variational methods, centralconfigurations, the N-body problem in spaces of constant curvature, discovery of new periodic solutions,regularization of collisions, stability of periodic solutions, spacecraft orbital design and applications of Morseindex and Maslov index to the N-body problem. If the schedule permits, we anticipate ending the sessionwith a discussion on open problems.

Lie Commutativity of PolynomialVector Fields

John Alexander ArredondoKonrad Lorenz University, ColombiaJesus Mucino Raymundo

Let X,Y be polynomial vector fields on R2 of degreeat most d. If [X,Y ] = 0 then X is integrable in thesense of Lie. Y is a symmetry of X, i.e., the local flowof Y sends local trajectories of X to local trajectoriesof X, preserving the parametrization along them. Inthis case X,Y have first integrals (real analytic andprobably multivalued) tf , tg on R2 � det(X,Y ) = 0.Our goal in this talk is the study of the implicationsof the number of common zeros of X and Y for theHamiltonian and non Hamiltonian case.

Singular Periodic Brake Orbits inthe Planar Pairwise SymmetricFour Body Problem

Lennard BakkerBrigham Young University, USAAmmon Lam

We investigate the existence and stability of symmet-ric singular periodic brake orbits in the equal mass,fully symmetric planar four body problem. Usingregularized coordinates, we remove the singularity ofbinary collision at the origin for each symmetric pair.We use topological and symmetry tools in our inves-tigation.

Exchange Orbits in the 2n+1 BodyProblem

Abimael BengocheaUniversidad Autonoma Metropolitana Iztapalapa,MexicoJorge Galan Vioque, Ernesto Perez Chavela

The horseshoe (exchange) orbits were first observedin the system conformed by Saturn, Janus andEpimetheus. In this talk we present the generaliza-tion of this kind of orbits to the 2n + 1 body prob-

lem. We discuss some properties of these orbits, forinstance, they are closely related with homographicsolutions. It is expected that for some exchange or-bits the variation of the moment of Inertia decreasesas n increases. We show some numerical results forthe case n = 2.

The Restricted Four Body Problemin the Solar System and Beyond.

Jaime Burgos-GarciaInstituto Tecnologico Autonomo de Mexico, Mexico

The restricted four body problem (r4bp) studies thedynamics of a massless particle under the gravita-tional influence of three point masses that lie in anequilateral configuration provided by the well-knownhomographic solution of the general three body prob-lem. This model has become relevant because ofthe observations of such configurations in our solarsystem, the most famous case is the configurationformed by the so called Trojan asteroids, Jupiterand the Sun although other Trojan asteroids havebeen detected for other planets and even between themoons of Saturn, however, recent studies show thatis very likely to find this kind of configuration in someobserved extrasolar systems. In this talk we will showhow the r4bp and some of its modifications could beimplemented to model the dynamics of small bodiesinteracting with these systems.

Convex Central Configurations ofthe Five-Body Problem

Kuo-Chang ChenNational Tsing Hua University, TaiwanJun-Shian Hsiao

In this talk we consider convex central configurationsof the five-body problem and prove some necessaryconditions for such configurations. In particular, fora planar central configuration with multiplier � andtotal mass M , we show that all exterior edges are lessthan r

0

= (M/�)1/3, at most two interior edges areless than or equal to r

0

, and its subsystem with fourmasses cannot be a central configuration. We also


obtain some other necessary conditions for strictlyconvex central configurations with five bodies, andshow examples with five bodies that have either oneor two interior edges less than or equal to r

0

. Wedevelop some formulae by W. L. Williams in 1938, inthe meanwhile we rectify some unsupported assump-tion in there.

A Property for Four-Body IsoscelesTrapezoid Central Configurations

Yiyang DengSiChuan University, Peoples Rep of ChinaBingyu Li, Shiqing Zhang

The four-body co-circural central configurations havetwo symmetric families,the kite and isosceles trape-zoid.Using mutual distances as coordinates,we provethat if the four-body central configurations is isosce-les trapezoid,the diagonals of the isosceles trapezoidcan‘t be vertical.

About the Existence of Symmetryin Trapezoid Central Configura-tions.

Luis Franco PerezUniversidad Autonoma Metropolitana Cuajimalpa,Mexico

In the four body problem, non-collinear convex cen-tral configurations have an axis of symmetry providedthat two bodies with equal mass are on the oppositevertices in the equilateral. In this case, the configu-ration must be of kite type. But if the pair of equalmasses are located at adjacent vertices of a trape-zoid, conforming a central cofiguration, the existenceof a symmetry axis is not clear, unless the remainingpair of masses were equal too. In this talk we givesome advances to show the existence of symmetry intrapezoid central configurations.

Melnikov Potential and HomoclinicIntersections in Higher DimensionalHamiltonian Systems

Marian GideaYeshiva University, USARafael de la Llave

One basic method to establish the existence of hy-perbolic horseshoes in Hamiltonian Systems (includ-ing models from Celestial Mechanics), is the Mel-nikov method. We present a version of the Melnikovmethod for higher dimensional Hamiltonian Systems,consisting of an unperturbed part, which can be de-scribed by a normally hyperbolic invariant manifoldwhose stable and unstable manifolds coincide, and atime-dependent perturbation. We define a Melnikovpotential given in terms of convergent integrals ofthe perturbation along homoclinic orbits of the un-perturbed system. We use the Melnikov potentialto measure the splitting of the stable and unstable

manifolds due to the perturbation. Unlike previousworks, the homoclinic orbits are not asymptotic toperiodic or quasi-periodic orbits, but to arbitrary or-bits. We also show that for generic perturbations weobtain horseshoe-type dynamics.

Invariant Manifolds of the ParabolicInfinity in the Restricted SpatialThree Body Problem

Pau MartinUniversitat Politecnica de Catalunya, SpainInmaculada Baldoma, Ernest Fontich

In the spatial restricted three body problem, infin-ity is foliated by parabolic periodic orbits. It isknown that this parabolic periodic orbits have twodimensional parabolic stable invariant manifolds. Weprove that these invariant manifolds have polynomialapproximation up to any order at the periodic or-bit. Such result is rather surprising, since, as wewill show, in general, invariant manifolds of parabolicpoints with dimension larger than one do not admitpolynomial approximations and, in fact, may haverather low regularity at the fixed point. Our resultfollows from a more general result on parabolic fixedpoints. Our prove provides an algorithm that canbe implemented to actually compute the approxima-tions.

N-Body Problem of Celestial Me-chanics and Numeric Simulation

Tiancheng OuyangBrigham Young University, USAZhifu Xie, Duokui Yan

In this talk, I will give a brief introduction of thisvariational methods of N-body Problem from 2001–2016 with emphasize the work of our collaborate re-search group. According to Newton’s Second Law,the motion of N point bodies with positive massesm

1

,m2

, . . . ,mN located at positions x1

, x2

, . . . , xN 2R3 is governed by the system of second-order nonlin-ear vector di↵erential equations

mixi =NX

j=1,j 6=i

Gmimj(xj � xi)

kxi � xjk3,

where the derivative is with respect to the time vari-able t, and G is the universal gravitational constant.By using variational methods and numerical compu-tation, some new discovered periodic orbits in 2D andand 3D will be present.


Stability of Relative Equilibria inCurved Spaces

Ernesto Perez-ChavelaITAM-Mexico, MexicoJuan M. Sanchez-Cerritos

We consider N–point positive masses moving on atwo dimensional space of constant curvature K. Us-ing the cotangent potential as a generalization of theNewtonian one on these spaces, we describe some es-pecial kind of periodic orbits where the mutual dis-tances among the particles remain constant for alltime. We classify these orbits called relative equilib-ria for the case N = 2, 3 and study the linear stabilityof them.

N-Body Problems on Surfaces ofRevolution

Cristina StoicaWilfrid Laurier University, Waterloo, Canada

In this talk I will present some dynamical aspects ofthe N-body problems on a two dimensional surface ofrevolution. I will discuss symmetries, relative equilib-ria, and some properties related to bifurcations andstability of relative equilibria.

Periodic Orbits from Broucke-Henon Orbit to Figure Eight inThree-Body Problem

Zhifu XieVirginia State University, USATiancheng Ouyang, Duokui Yan

In this talk, we discuss the possible connections ofperiodic solutions in two paradigmatic examples ofthree body problem: Broucke-Henon orbit and thefigure eight orbit. Broucke-Henon orbit was numer-ically discovered by Broucke and Henon (1975) in-dependently and their existence are recently provedvia variational methods with Structural PrescribedBoundary Conditions (SPBC) by Ouyang-Xie-Yan(2015). Figure eight orbit was first numerically dis-covered by Moore (1993) and its existence was provedby Chenciner and Montgomery (2000).

Existence of the Broucke-HenonOrbit

Duokui YanBeihang University, Peoples Rep of ChinaTiancheng Ouyang, Zhifu Xie

In 2003, Venturelli proposed an open problem on theexistence of the Broucke-Henon orbit during a work-shop at American Institute of Mathematics. In thistalk, we introduce our variational method and showthe existence of this orbit.


Special Session 32: Global or/and Blowup Solutions for NonlinearEvolution Equations and Their Applications

Shaohua Chen, Cape Breton University, CanadaRunzhang Xu, Harbin Engineering University, Peoples Rep of China

This session is devoted to the recent developments in global or/and blowup solutions for nonlinear evolu-tion equations and their applications, include reaction di↵usion equations, fluid dynamics, delay, localized,nonlocal, degenerate evolution equations, travelling waves, steady states and their properties.

Blow-Up for the Wave Equationwith Nonlinear Boundary Dampingand Source Term

Md Salik AhmedCollege of Science, Harbin Engineering University,BangladeshQiu Xiaotong

In this paper, we mainly study the initial bound-ary value problem of wave equations with nonlinearsource and boundary damping terms under criticalenergy case.As we know, the dynamical boundaryconditions make the space nature of solutions vary-ing so much, as well as the invariant sets. The clas-sical methods used to study qualitative studies areno longer entirely applicable. Therefore we first in-troduce the previous research results with local exis-tence with the low initial energy. Then we concernwith the global existence and blow-up in the case ofcritical energy. We prove that all the results stillholds in further assumptions by potential well andSobolev embedding theory.This paper is funded by the International Ex-change Program of Harbin Engineering University forInnovation-oriented Talents Cultivation.

Global Well-Posedness for theQuasilinear Hyperbolic Equationwith Nonlinear Damping and SourceTerms

Tianlong ChenCollege of Automation, Harbin Engineering Univer-sity, Peoples Rep of China

In this paper we study the Cauchy problem at threedi↵erent initial energy levels for the quasilinear hy-perbolic equation with nonlinear damping and sourceterms. In the frame of potential well theory, weinvestigate the blowup of solutions with concavitymethod at both low and supercritical initial energylevel. Moreover the global existence and asymptoticbehavior are concerned.This paper is funded by the International Ex-change Program of Harbin Engineering University forInnovation-oriented Talents Cultivation.

Global Existence for a SingularActivator-Inhibitor Model

Shaohua ChenCape Breton University, Canada

We discuss existence results for a singular Gierer-Meinhardt system with zero Dirichlet boundary con-ditions, which originally arose in studies of pattern-formation in biology and has interesting and chal-lenging mathematical properties. The mathemati-cal di�culties are that the system becomes singularnear the boundary and it is non-quasimonotone. Weshow the existence of positive solutions for the gen-eral activator-inhibitor model.

Weighted Gagliardo-Nirenberg In-equalities Involving BMO Normsand Solvability of Strongly CoupledParabolic Systems

Dung LeUniversity of Texas at San Antonio, USA

We will discuss new weighted Gagliardo-Nirenberginequalities with applications to the local/global ex-istence of solutions to nonlinear strongly coupled anduniform parabolic systems. Much weaker su�cientconditions than those existing in literature for solv-ability of these systems will be established.

Non-Simultaneous Blow-Up Solu-tions to Parabolic Equations with NComponents

Bingchen LiuChina University of Petroleum, Peoples Rep ofChinaFengjie Li

In this talk, we present the non-simultaneous blow-up solutions and their blow-up rates for the parabolicequations with n components. Firstly, we show somecriteria for the existence and nonexistence of non-simultaneous blow-up solutions, classified by the con-stant or variable exponents. Secondly, blow-up ratesand blow-up sets of the solutions are considered. Itis interesting that the di↵erent blow-up mechanism,the coupled relationships, and the properties of initialdata, etc. might lead to di↵erent blow-up phenomenaof solutions.


Blow-Up Phenomena in Nonlin-ear Parabolic and PseudoparabolicProblems

Monica MarrasUniversity of Cagliari, ItalyG. Viglialoro

We investigate the question of blow-up for nonneg-ative classical solutions of some nonlinear parabolicproblems defined in a bounded domain. Under con-ditions on data and geometry of the spatial domain,explicit lower bounds for the blow-up time are de-rived. Moreover we extend our results to a class ofnonlinear pseudo-parabolic problems.

Traveling Wave Solutions to theBurgers-↵� Equations

Byungsoo MoonIncheon National University, Korea

The Burgers-↵� equation, which was first introducedby D.D. Holm and M.F. Staley, is considered in thespecial case where ⌫ = 0 and b = 3. Traveling wavesolutions are classified to the Burgers-↵� equationcontaining four parameters b,↵, ⌫, and �, which isa nonintegrable nonlinear partial di↵erential equa-tion that coincides with the usual Burgers equationand viscous b-family of peakon equation, respectively,for two specific choices of the parameter � = 0 and� = 1. Under the decay condition, it is shown thatthere are smooth, peaked and cusped traveling wavesolutions for the Burgers-↵� equation with ⌫ = 0and b = 3 depending on the parameter �. Moreover,all traveling wave solutions without the decay con-dition are parametrized by the integration constantk1

2 R. In an appropriate limit � = 1, the previouslyknown traveling wave solutions of the Degasperis-Procesi equation are recovered.

Degenerate Boundary Layer So-lution to a System of ViscousConservation Laws

Tohru NakamuraKumamoto University, Japan

We consider the existence and asymptotic stabilityof the stationary solution for system of viscous con-servation laws in one-dimensional half space. We es-pecially consider the degenerate stationary solutionwhich verifies algebraic convergence as the space vari-able tends to infinity. With the aid of the centermanifold theory, the existence of the degenerate sta-tionary solution is proved under the situation thatone characteristic is zero and the other characteris-tics are negative. Asymptotic stability of the degen-erate stationary solution is also proved in an alge-

braically weighted Sobolev space provided that theweight exponent is less than 5. The stability analysisis based on deriving the a priori estimate by using theweighted energy method combined with the Hardytype inequality with the best possible constant.

Global Well-Posedness of DampedMultidimensional GeneralizedBoussinesq Equations

Yi NiuCollege of Automation, Harbin Engineering Univer-sity, Peoples Rep of ChinaXiuyan Peng

We study the Cauchy problem for a class of sixth or-der Boussinesq equations with the generalized sourceterm and damping term. By using Galerkin approx-imations and potential well methods, we prove theexistence of global weak solution. Furthermore, wetalk about the condition of damped coe�cient k andobtain the finite time blow up solution.This paper is funded by the International Ex-change Program of Harbin Engineering University forInnovation-oriented Talents Cultivation.

Time Behavior and Uniform Boundsfor Solutions of Evolution Problems

Maria Michaela PorzioSapienza University of Rome, Italy

We will describe the behavior in time of the solu-tions of some nonlinear parabolic problems by meansof a new method which allows to study regularity,uniqueness and asymptotic behavior in a very sim-ple way and to obtain also uniform estimates for thesolutions.

On a Pseudoparabolic Regular-ization of a Forward-Backward-Forward Parabolic Equation

Flavia SmarrazzoUniversity Campus Bio-Medico di Roma, ItalyMichiel Bertsch, Alberto Tesei

In this talk I will consider an initial-boundaryvalue problem for a degenerate pseudoparabolic reg-ularization of a nonlinear forward-backward-forwardparabolic equation, with a bounded nonlinearitywhich is increasing at infinity. I will discuss exis-tence of suitably defined nonnegative solutions of theproblem in a space of Radon measures. The con-structed solutions satisfy several monotonicity andregularization properties; in particular, their singu-lar part is nonincreasing and may disappear in finitetime. Joint work with M. Bertsch and A. Tesei.


Cauchy Problem of the SingularlyPerturbed Sixth-Order Boussinesq-Type Equation

Changming SongZhongyuan University of Technology, Peoples Rep ofChina

In this talk, we will study the existence and unique-ness of the global generalized solution and the globalclassical solution for the Cauchy problem of the sin-gularly perturbed sixth-order Boussinesq-type equa-tion:

utt = uxx + �(u)xx + ↵ux4 + �ux6 .

Time-Periodic Solutions to theDrift-Di↵usion Model for Semicon-ductors

Masahiro SuzukiNagoya Institute of Technology, Japan

We study the existence and asymptotic stability oftime-periodic solutions to the drift-di↵usion modelfor semiconductors. If alternating-current voltage isapplied to PN-junction diodes, a time-periodic cur-rent flow is observed. The main purpose of this talkis mathematical analysis on this periodic flow. Weconstruct a time-periodic solution by utilizing theGalerkin method. The solution is unique in a neigh-borhood of a thermal equilibrium, and it is globallystable. Proofs of the uniqueness and the stability arebased on the energy method employing an energyform.

Behaviour in Time of Solutions toa Class of Fourth Order EvolutionEquations

Stella Vernier-PiroUNICA, ItalyG.A.Philippin

We consider some initial-boundary value problemsfor a class of nonlinear parabolic equations of thefourth order, whose solution u(x, t) may or may notblow up in finite time. Under suitable conditions ondata, a lower bound for t⇤ is derived, where [0, t⇤)is the time interval of existence of u(x, t). Under ap-propriate assumptions on the data, a criterion whichensures that u cannot exist for all time is given, andan upper bound for t⇤ is derived. Some extensionsfor a class of nonlinear fourth order parabolic systemsare indicated.

On Love Wave Dispersion in anOrthotropic Substratum Double-Layered Over Anisotropic PorousMantle: an Analytic Approach

Sumit VishwakarmaBirla Institute of Technology and Science, Pilani,IndiaXu Runzhang

The present study investigate the propagation char-acter of Love wave in an orthotropic substratum be-ing layered doubly over anisotropic porous mantle.Separate displacements have been derived for eachof the three layer and suitable boundary conditionhave been imposed to derive the dispersion equationof the wave in a closed form in terms of Whitakerfunction. It has been found that inhomogeneous pa-rameter associated with the rigidity and the densityof the layered medium has a special bearing on theLove wave propagation. Orthotropic properties andinitial stresses of the media also a↵ects the velocityof the wave to a great extent has been shown graph-ically.

Global Existence and Blowup ofSolutions for the MultidimensionalSixth-Order Good Boussinesq Equa-tion

Runzhang XuHarbin Engineering University, Peoples Rep of China

This paper is concerned with the Cauchy problem ofsolutions for some nonlinear multidimensional goodBoussinesq equation of sixth order at three di↵erentinitial energy levels. In the framework of potentialwell, the global existence and blowup of solutions areobtained together with the concavity method at bothlow and critical initial energy level. Moreover by in-troducing a new stable set, we present some su�cientconditions on initial data such that the weak solutionexists globally at sup-critical initial energy level.

Finite Time Blow Up for the Non-linear Fourth-Order Dispersive-Dissipative Wave Equation at HighEnergy Level

Yanbing YangCollege of Science, Harbin Engineering University,Peoples Rep of ChinaXu Runzhang

This present investigates the initial boundary valueproblem of the nonlinear fourth-order dispersive-dissipative wave equation at sup-critical initial en-ergy level. By using the concavity method, we es-tablish a blow up result for certain solutions witharbitrary positive initial energy, which gives an an-swer to the global nonexistence of solutions for thenonlinear wave equations with dispersive term anddissipative term at sup-critical initial energy level.


A Di↵usion Problem of Kirchho↵Type Involving the Nonlocal Frac-tional P-Laplacian

Binlin ZhangHeilongjiang Institute of Technology, Peoples Rep ofChinaPatrizia Pucci, Mingqi Xiang

In this talk, we present an anomalous di↵usionmodel of Kirchho↵ type driven by a nonlocal integro–di↵erential operator. As a particular case, we areconcerned with an initial–boundary value problem

involving the fractional p–Laplacian. Under some ap-propriate conditions, the well–posedness of solutionsfor the above problem is studied by employing thesub–di↵erential approach. Furthermore, the asymp-totic behavior and extinction of solutions are also in-vestigated.


Special Session 33: Nonlinear Waves in Dispersive Equations

Francois Genoud, TU Delft, NetherlandsStefan Le Coz, Paul Sabatier University (Toulouse 3), France

This session will be devoted to analytical and numerical properties of nonlinear waves in dispersive equationssuch as, for instance, nonlinear Schrodinger equations and Korteweg-de Vries equations. Particular topics ofinterest will include space-periodic solutions and equations from which some usual symmetries (e.g. space-translation or scaling invariance) are absent.

Ground States for the NLS onGraphs

Riccardo AdamiPolitecnico di Torino, ItalyEnrico Serra, Paolo Tilli

The issue of finding the state of a Bose-Einstein con-densate in a branched trap, can be modelled math-ematically as the problem of minimizing a nonlin-ear (quartic) energy functional among functions de-fined on a graph. We give results on the existenceof ground states for the cases of subcritical and criti-cal nonlinearity power. The problem proves di↵erentand richer than the analogous problem on the line.

Nonlinear Schrodinger Equationwith Double Delta-Interaction Wells

Jaime Angulo PavaState University of Sao Paulo, BrazilLuis Andres Rosso Ceron

We study analytically the existence and orbital sta-bility of the peak-standing-wave solutions for the cu-bic nonlinear Schrodinger equation with two inter-actions points determined by Dirac’s deltas. Thisequation admits at least two smooth curve of posi-tive solutions with a profile given by the Jacobi el-liptic function of dnoidal and cnoidal type betweenthe defects point. Via Floquet theory and analyticperturbation, we obtain that in the case of an attrac-tive defect (dnoidal case) the standing wave solutionsare unstable and in the case of an repulsive defect(cnoidal case) the standing wave solutions are stablein H1

per.

Modulations of Dispersive PeriodicWaves

Sylvie Benzoni-GavageUniversity of Lyon, FranceC. Mietka, L.M. Rodrigues

One may consider as a whole a large class ofnonlinear dispersive PDEs that includes NonLin-ear Schrodinger equations, generalized Korteweg-deVries equations, and dispersive perturbations of theEuler equations for compressible fluids. These equa-tions admit families of periodic waves, whose modula-

tions are expected to be governed by averaged equa-tions “a la Whitham”. Whitham equations are stillpoorly understood. However, they are endowed withnice structures that can be investigated at least nu-merically.

On the Multiplier Method forVariable Coe�cients DispersiveEquations.

Federico CacciafestaUniv. Milano Bicocca, ItalyP. D’Ancona, A.S. de Suzzoni, R. Luca

We show how the multiplier method can be developedin the framework of variable coe�cients dispersiveequations to prove local smoothing estimates. In par-ticular we discuss the cases of the Schroedinger (viaKato smoothing theory) and the Dirac equations.

Modulation Equations Via NormalForms

Martina Chirilus-BrucknerUniversity of Leiden, Netherlands

In this talk we will briefly survey the methodof derivation and justification of modulation equa-tions (such as the Korteweg-de Vries or NonlinearSchrodinger equation) in dispersive systems, to thenpresent a novel approach using normal forms, whichallows for a more transparent derivation procedureand a di↵erent viewpoint on higher order correctionterms.

A New Model Describing TidleBore on River

Mathieu ColinIPB, INRIA CARDAMOM, FranceDavid Lannes

The free surface Euler equations are a set of evolu-tion equations on a three-dimensional, free-boundarydomain. The aim of this talk is to present asymptoticmodels for river flows that approximate these equa-tions in relevant physical configurations and that aremuch more simple. For a canal with rectangular sec-tion, we show that there exists at leading order solu-tions that are independent of the transverse variable


y. Their evolution is given by a set of two evolutionequations on the surface elevation and on the longitu-dinal component of the vertically averaged velocity.We will then discuss the case of a river with lateralshorelines.

Dirac Points in PT Symmetric Op-tical Lattices

Christopher CurtisSan Diego State University, USAMark Ablowitz, Yi Zhu

Optical graphene, or an optical honeycomb waveg-uide, has become a material of much interest andexcitement in the optics community. This is due tothe presence of Dirac points in the dispersion rela-tionship which are a result of the symmetry of thelattice. Also of interest in optics are so called parity-time (PT) symmetric perturbations, representing thecareful introduction of gain and loss terms. In thistalk, we examine the impact of introducing PT sym-metritc perturbations into honeycomb lattices andtheir impact on Dirac points. We categorize twotypes of PT perturbations one of which we rigorouslyshow prevents the formation of complex dispersionrelationships. We then track how Dirac points sepa-rate, and thus we show how PT perturbations can beused to introduce band gaps. We also present numer-ical results which show how PT perturbations couldbe used to engineer dispersion relationships which in-duce wave motion in particular directions. The im-pact of nonlinearity in these systems will also be ad-dressed.

Orbital Stability in Infinite Dimen-sion: Geometric Theory

Stephan de BievreUniversity of Lille I - INRIA Lille-Nord Europe,FranceF. Genoud, S. Rota Nodari

In this talk, we (re)consider the energy-momentummethod for proving the orbital stability of relativeequilibria of Hamiltonian dynamical systems on Ba-nach spaces, in the presence of higher dimensionalsymmetry groups. We will highlight the interplaythat is at work in this method between (symplectic)geometry and (functional) analysis.

Existence of Peaked Waves in aFull-Dispersion Bi-Directional Shal-low Water Model

Mats EhrnstromNTNU Norwegian University of Science and Tech-nology, NorwayM.A. Johnson

We consider the existence of periodic traveling wavesin a bidirectional Whitham equation, combining thefull two-way dispersion relation from the incompress-ible Euler equations with a canonical shallow water

nonlinearity. Of particular interest is the existenceof a highest, singular, traveling wave solution, whichwe obtain as a limiting case at the end of the mainbifurcation branch of 2⇡-periodic traveling wave solu-tions. Unlike the uni-directional Whitham equation,containing only one branch of the full Euler disper-sion relation, where such a highest wave behaves like|x|1/2 near its crest, the waves obtained here behavelike |x log |x|| at their crest.

On the Wave Length of SmoothPeriodic Traveling Waves of theCamassa-Holm Equation

Anna GeyerUniversity of Vienna, AustriaJordi Villadelprat

In this talk we are concerned with the wave length� of smooth periodic traveling wave solutions of theCamassa-Holm equation. The set of these solutionscan be parametrized using the wave height a (orpeak-to-peak amplitude). Our main result estab-lishes monotonicity propoerties of the map �(a) i.e.the wave length as a function of the wave height.We obtain the explicit bifurcation values, in terms ofthe parameters associated with the equation, whichdistinguish between the two possible qualitative be-haviours of �(a), namely monotonicity and unimodal-ity. The key point is to relate �(a) to the period func-tion of a planar di↵erential system with a quadratic-like first integral, and to apply a criterion whichbounds the number of critical periods for this typeof systems.

The Energy-Critical Quintic NLS onPerturbations of Euclidean Space.

Casey JaoUCLA, USA

Consider the defocusing quintic nonlinearSchrodinger equation on R3 with initial data in theenergy space. This problem is energy-critical in viewof a certain scaling invariance, which is a main sourceof di�culty in the analysis of this equation. It is anontrivial fact that all finite-energy solutions scatterto linear solutions. We show that this remains trueunder small compact deformations of the Euclideanmetric.


Asmptotics Behavior of Solutionsfor a Class of Nonlinear SchrodingerEquations

Hiroaki KikuchiTsuda college, Japan

In this talk, we consider the following nonlinearSchrodinger equation:

i @ @t

(x, t) + � (x, t) + f( (x, t)) = 0, (x, t) 2 Rd ⇥ R,

where i :=p�1, d � 1, is a complex-valued func-

tion on Rd ⇥R,� is the Laplace operator on Rd andf : C ! C is a continuously di↵erentiable functionin R2-sense to be specified later. We will study thee↵ects of the nonlinearity f to the dynamics of so-lutions (scattering/blowup/existence and stability ofthe standing waves) to (). If time permits, I’d liketo mention about the global dynamics of solutionswhose energy is greater than that of the ground state.

Analysis of Convergence of SchwarzWaveform Relaxation DomainDecomposition Methods for theSchroedinger equation

Emmanuel LorinCarleton University, CanadaX. Antoine

The aim of this talk is to derive and numerically val-idate some asymptotic estimates of the convergencerate of the Classical and Optimized Schwarz Wave-form Relaxation domain decomposition methods ap-plied to the computation of the stationary statesto the one- and two-dimensional linear and nonlin-ear Schroedinger equations. The approach combinessome methods developed by Gander and Halpern,and the theory of inhomogeneous pseudodi↵erentialoperators in conjunction with the associated symbol-ical asymptotic expansions.

Quasilinear Schrodinger Equations

Jeremy MarzuolaUNC Chapel Hill, USAJason Metcalfe, Daniel Tataru

We discuss results, applications and open problems inthe study of quasilinear Schrodinger equations. Re-cently, with J. Metcalfe and D. Tataru, we have beenstudying local in time well-posedness for small andlarge data. I will recall the main ideas of the proofsin such a case and discuss future directions. I will alsodiscuss ongoing work with Jianfeng Lu and LudwigGauckler relating to extensions and phenomenologi-cal studies.

Collision and Blow Up in NLSEquations

Claudio MunozUniversity of Chile, ChileFrank Merle

The purpose of this talk is to describe how collisionbetween two NLS solitons in the unstable regime canlead to di↵erent results, which include blow up in fi-nite time, or decay in time. This is a joint work withF. Merle.

The Fourth-Order Dispersive Non-linear Schrodinger Equation: Or-bital Stability of a Standing Wave

Fabio NataliState University of Maringa, BrazilAdemir Pastor

Considered in this talk is the one-dimensional fourth-order dispersive cubic nonlinear Schrodinger equationwith mixed dispersion. Orbital stability, in the en-ergy space, of a particular standing-wave solution isproved in the context of Hamiltonian systems. Themain result is established by constructing a suitableLyapunov function.

Nonlinear Schroedinger Equationwith a Point Nonlinearity in ThreeDimensions: Scaling Limit from aMean Field theory

Diego NojaUniversity of Milano Bicocca, ItalyClaudio Cacciapuoti, Domenico Finco,Alessandro Teta

In this talk a nonlinear Schrodinger dynamics with anonlinearity concentrated at a point will be definedand constructed. In particular it will be shown thatsuch a model can be recovered as a scaling limit ofa regularized nonlocal dynamics through a suitablerenormalization procedure. The regularized dynam-ics is described by the equation

i @@t "(t) = �� "(t) + g(", µ, |(⇢", "(t))|2µ)(⇢", "(t))⇢"

where ⇢" ! �0

weakly and the function g embodiesthe nonlinearity and the scaling and has to be finetuned in order to have a nontrivial limit dynamics.The generator of the limit dynamics is a nonlinearversion of a delta interaction and it has previouslystudied as regards well posedness, blow up and exis-tence and stability of standing waves.


Space-Modulated Stability andPeriodic Waves of Dispersive Equa-tions

Miguel RodriguesUniversite de Rennes 1, France

Recently, partly motivated by applications to surfacewaves, rapid progresses on the stability theory of pe-riodic waves have been obtained. In particular, forparabolic systems — including those encoding theshallow water description of viscous roll-waves — anessentially complete theory is now available. We shallexpound here some first contributions to a dispersivetheory, still to come.

Orbital Stability in Infinite Dimen-sion: Proving Coercivity

Simona Rota NodariIMB, Universite de Bourgogne, FranceStephan De Bievre

The proof of the orbital stability of relative equilibriaof Hamiltonian dynamical systems on Banach spacescan be reduced to a “coercivity estimate” on an ap-propriately constructed Lyapunov function. In thistalk we show how this estimate can be obtained in thegeneral case of higher dimensional symmetry groups.

Numerical Methods and OrbitalStability of Solitary-Wave Solutionsfor Higher-Order BBM Equations

Juan-Ming YuanProvidence University, TaiwanHongqiu Chen, Shuming Sun

We study the existence and stability of solitary-wavesolutions of a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involves pseudo-di↵erential operators for the linear part. One of suchequations can be derived from water-wave problemsas the second-order approximate equations from fullynonlinear governing equations. Under certain condi-tions on the symbols of pseudo-di↵erential operatorsand the nonlinear terms, it is shown that the generalhigher-order BBM equation has solitary-wave solu-tions. Moreover, under slightly more restrictive con-ditions, the set of solitary-wave solutions is orbitallystable. Here, the equation has a nonlinear part in-volving polynomials of the solution and its derivativeswith di↵erent degrees (i.e., not homogenous), whichhas not been discussed before. We also develop nu-merical methods for the fifth-order BBM-type equa-tions.


Special Session 34: Di↵erential Equations and Applications to BiologicalModels

Nemanja Kosovalic, University of South Alabama, USAXiang-Sheng Wang, Southeast Missouri State University, USA

It is well known that ordinary di↵erential equations, partial di↵erential equations, and functional di↵erentialequations provide a suitable framework for the modelling of various biological phenomena. Moreover, manybiological problems inspire the development of new mathematical ideas to draw qualitative and quantitativeconclusions concerning the underlying models. It is the goal of this session to bring together researchersinterested in either the dynamical modelling of biological phenomena, or the theoretical aspects concerningthe underlying models, to discuss and exchange ideas.

Building a Model of Delay-Di↵erential Equations with State-Dependent Delay for Megakary-opoiesis

Lois BoulluUniversite de Montreal, CanadaNemenja Kosovalic, Laurent Pujo-Menjouet,Jianhong Wu, Jacques Belair

Since the early work of M. C. Mackey and J. Be-lair in the 80s [1], the intricate multiple feedbacksof megakaryopoiesis (the process along which theplatelets are produced) have been the source of manyinvestigations in the field of delay di↵erential equa-tions. In this talk I describe how an emphasis onTPO regulation and progenitor cells proliferation canlead to a model describing megakaryopoiesis with asystem of delay di↵erential equations, the delay be-ing state-dependant and defined by threshold: us-ing a tool presented by L H Smith in 1992 [2], wetransform this system into a functional di↵erentialequation with fixed delay, allowing us to explore well-posedness and fixed point stability.

References

[1] Belair J, Mackey M C (1987) A model for theRegulation of Mammalian Platelet Production,Annals New York Academy of Sciences

[2] Smith L H (1992) Reduction of structured pop-ulation models to threshold-type Di↵erentialequations and funtionnal di↵erential equation :a case study Mathematical Biosciences 113:1-23

Contributions of the Latent Reser-voir and of the Pool of Long-LivedChronically Infected CD4+ T Cellsin HIV dynamics

Ana CarvalhoUniversity of Porto, PortugalCarla M.A. Pinto

In this paper, we study the e↵ect of the size of the la-tent reservoir and of the pool of long-lived chronicallyinfected CD4+ T cells in a model for HIV dynamicswith drug-resistance. We calculate the reproductionnumber and study the local stability of the disease-free equilibrium. The e↵ects of the sizes of the latent

reservoir and of the pool of long-lived chronically in-fected CD4+ T cells were analyzed numerically. Ourresults are biologically reasonable. We found that thelatent reservoir in resting CD4+ T cells appears to besu�cient to the persistence of plasma viral load inpatients under HAART. Moreover, the pool of long-lived chronically infected cells promotes an increasein drug-resistant virus, that escape treatment, whichturns the eradication of the plasma virus an impos-sible goal.

Impulsive Fractional Integro-Di↵erential Equations of Order(1, 2) with Antiperiodic BoundaryConditions

Jaydev DabasIndian Institute of Technology Roorkee, IndiaVidushi Gupta

In the present work our aim is to study the frac-tional order boundary value problem with jump im-pulsive conditions. These type of jump conditionsrepresents a sudden change of values of state variableand its derivative at impulsive points. The impulsivee↵ect considered in the system has memory of thepast states. We establish the existence and unique-ness of solution for a new class of impulsive fractionalintegro-di↵erential equation with separated bound-ary conditions by applying the classical fixed pointtheorems. At last an application is presented to ver-ify our results.

Nonlinear Fractional BoundaryValue Problem with Not Instanta-neous Impulse

Vidushi GuptaIIT Roorkee, IndiaJaydev Dabas

This present work deals with the following bound-ary value problem of a fractional di↵erential equationwith non-instantaneous impulse

cD↵0,ty(t) + f(t, y(t)) = 0,

t 2 (si, ti+1

], i = 0, 1, 2, . . . ,m,↵ 2 (0, 1],

y(t) = Hi(t, y(t)), t 2 (ti, si],

y(0) = µcDqy( ), 0


Dynamics of Mosquito PopulationModels with Or Without StageStructure

Jia LiUniversity of Alabama in Huntsville, USALiming Cai, Yang Li

Mosquitoes undergo complete metamorphosis goingthrough four distinct stages of development dur-ing a lifetime. To study the impact of the ster-ile insect technique on the transmission dynam-ics of mosquito-borne diseases, we formulate stage-structured continuous-time mathematical models forthe interactive dynamics of the wild and sterilemosquitoes. We incorporate di↵erent strategies forthe releases of sterile mosquitoes in the models andinvestigate the model dynamics, including the exis-tence of positive equilibria and their stability. Wecompare the model dynamics with those of simplifiedmodels without stage structure and discuss what wehave learned from this study.

Speed Selection and Stability ofTraveling Waves to Reaction Dif-fusion Equations with NonlinearConvection

Chunhua OuMemorial University, Canada

In this talk, we study the existence of traveling wavesto reaction di↵usion equation with nonlinear convec-tion term. We also establish a mechanism on the lin-ear or nonlinear determinacy of the minimal speed.Stability of these waves is also discussed. Finally wewill present some applications including the Burgers-KPP-Fisher equation.

A Model of Platelet Production:Stability Analysis and Oscillations

Laurent Pujo-MenjouetUniversity of Lyon, FranceM. Adimy, L. Boullu, F. Crauste

We propose here a new model of platelet produc-tion and regulation taking into account the recentbiological discoveries related to this topic, includingthe role played by thrombopoietin (TPO), a plateletregulation cytokine. We consider four di↵erent cellcompartments corresponding to di↵erent cell matu-rity levels : the stem cell, megaka- ryocytic progen-itors, megakaryocytes and platelets compartments.To the best of our knowledge, the progenitor com-partment has never been taken into account in pre-vious platelet production models. We consider alsothe quantity of circulating TPO that influences thedynamics of each cell populations.

Our model consists in a non linear age structuredpartial di↵erential equation system, where each equa-tion corresponds to a compartment. This system canbe reduced to a single non linear delay di↵erentialequation describing the dynamics the platelet popu-lation.After a brief introduction of the model, we prove theexistence of an unique steady state for the delay dif-ferential equation. We set up then conditions to getlocal and global asymptotic stability of this steadystate. We determine then necessary and su�cientconditions for the existence of oscillating solutions.

A Model for Bacteria-Grazers In-teractions in a Chemostat

Paul SalceanuUniversity of Louisiana at Lafayette, USAJ. Kong, H. Wang

We formulate a model of di↵erential equations forthe study of bacteria-grazers interactions in a carbonand nitrogen limited environment, in a chemostat.We provide sharp conditions that di↵erentiate in be-tween persistence and elimination of both bacteriaand grazers from the system. For the case when bac-teria and grazers natural death rates are assumed tobe negligible, as compared to the dilution rate, wealso provide su�cient conditions for global conver-gence to an interior equilibrium.

Bounded Global Hopf Branches forStage-Structured Di↵erential Equa-tions

Hongying ShuTongji University, Peoples Rep of ChinaLin Wang, Jianhong Wu

In this talk, we show that a class of stage-structureddi↵erential equations with unimodal feedback admitthree types of global dynamic properties: a globallystable trivial equilibrium, or a globally stable positiveequilibrium, or globally sustained oscillations. Usingthe delay as a bifurcation parameter, we analyticallyprove that there are finite number of neatly pairedHopf bifurcation values. Continuation of periodic so-lutions bifurcated from each Hopf bifurcation valueforms a global Hopf branch, which is shown to bebounded and joint each Hopf bifurcation value to thecorresponding paired one.

A Hybrid Mathematical Model forCell Motility in Angiogenesis

Nicoleta TarfuleaPurdue University Northwest, USA

The process of angiogenesis is regulated by the inter-actions between various cell types such as endothelialcells and macrophages, and by biochemical factors.In this talk, we present a hybrid mathematical modelin which cells are treated as discrete units in a contin-uum field of a chemoattractant that evolves according


to a system of reaction-di↵usion equations, whereasthe discrete cells serve as sources/sinks in this contin-uum field. It incorporates a realistic model for signaltransduction and VEGF production and release, andgives insights into the aggregation patterns and thefactors that influence stream formation. The modelallows us to explore how changes in the microscopicrules by which cells determine their direction and du-ration of movement a↵ect macroscopic formations. Inparticular, it serves as a tool for investigating tumor-vessel signaling and the role of mechano-chemical in-teractions of the cells with the substratum.

Modeling Optimal Control Treat-ment Strategies for HIV-TB Co-Infected Individuals

Naveen VaidyaUniversity of Missouri - Kansas City, USA

Co-infection of HIV and TB is one of the biggestproblems for properly managing infections of thesediseases. Particularly, individuals co-infected withHIV and TB are often in the situation of makingcritical decision on whether to begin treatments forboth diseases simultaneously or wait to begin HIV-treatment until the completion of TB-treatment. Inthis talk, I will present a mathematical model thathelps evaluate various treatment programs for HIV-TB co-infected individuals suggesting these individ-uals to make better treatment decisions. I will alsopresent an optimal control problem formulated basedon our model and how it can be used to identify thetreatment strategies that result in the minimum bur-dens from this co-infection. This is a joint work withAbhishek Mallela and Suzanne Lenhart.

How Latency and Nonlocality WillImpact the Spread of Some Dis-eases.

Yanyu XiaoUniversity of Cincinnati, USA

The purpose of this work is to study the spatial dy-namics of some delayed nonlocal reaction-di↵usionsystems. We will mathematically examine the e↵ectsof the delay and nonlocality on the spreading speedof the non-quasi-monotone systems.

A Nonlocal Spatial Model for LymeDisease

Xiao YuMemorial Univeristy of Newfoundland, CanadaXiao-Qiang Zhao

In this talk, I will report our recent research on a non-local and time-delayed reaction-di↵usion model forLyme disease with a spatially heterogeneous struc-ture. In the case of a bounded domain, we firstprove the existence of the positive steady state anda threshold type result for the disease-free system,and then establish the global dynamics of the modelsystem in terms of the basic reproduction number.In the case of an unbound domain, we obtain theexistence of the disease spreading speed and its co-incidence with the minimal wave speed. At last, weuse numerical simulations to verify our analytic re-sults and investigate the influence of model parame-ters and spatial heterogeneity on the disease infectionrisk.

Ion Size E↵ects on Ionic FlowsVia Steady-State Poisson-Nernst-Planck Models

Mingji ZhangNew Mexico Tech, USAPeter W. Bates, Guojian Lin, Weishi Liu,Hong Lu, Yingfei Yi

We study boundary value problems of a quasi-one-dimensional steady-state Poisson-Nernst-Planckmodel with a local hard-sphere potential for ionicflows of two oppositely charged ion species throughan ion channel, focusing on e↵ects of ion sizes. Theflow properties of interest, individual fluxes and totalflow rates of the mixture, depend on multiple physicalparameters such as boundary conditions (boundaryconcentrations and boundary potentials) and di↵u-sion coe�cients, in addition to ion sizes and valence.For the relatively simple setting and assumptions ofthe model, we are able to characterize, almost com-pletely, the distinct e↵ects of the nonlinear interplaybetween these physical parameters. The boundariesof di↵erent parameter regions are identified througha number of critical values that are explicitly ex-pressed in terms of the physical parameters. We be-lieve our results will provide useful insights for nu-merical and even experimental studies of ionic flowsthrough membrane channels.


Special Session 35: Control and Optimization Theory for PartialDi↵erential Equations

Lorena Bociu, North Carolina State University, USADaniel Toundykov, University of Nebraska-Lincoln, USA

The session will serve to promote and disseminate recent developments in the mathematical control the-ory for partial di↵erential equations (PDEs). The discussion will address the following questions in thecontext of distributed-parameter systems modeled by evolution PDEs: controllability/observability results,inverse problems, stability and blow-up, optimization and sensitivity analysis, as well as pertinent qualita-tive properties of such systems, e.g., Hadamard well-posedness, regularity of solutions, or structure of globalattractors.

Wellposedness Analysis of a Non-Linear Fluid-Structure PDE Model

George AvalosUniversity of Nebraska-Lincoln, USAPelin G. Geredeli

In this talk we shall focus on current work involv-ing the wellposedness for a coupled partial di↵eren-tial equation model which governs a certain fluid-structure interaction. The basis of our approach isan argumentation thematically similar to that of Z.Yosida and Y. Giga, which was originally invoked forthe (uncoupled) Navier-Stokes equations.

Stabilization of Two Coupled WaveEquations on a Compact Manifoldwith Boundary

Moez DaoulatliUniversity of Dammam, Saudi ArabiaLassad Aloui

In this talk we present a result on stabilization of cou-pled wave equations by an order one term on a com-pact manifold with boundary. Only one of the twoequations is directly damped by a localized damp-ing term. Under natural hypotheses, we show thatthe energy of smooth solutions of the system decayspolynomially.

Shape Di↵erentials and TopologicalSemi-Di↵erentials

Michel DelfourCentre de recherches mathematiques, Universite deMontreal, Canada

Complete metric spaces of shapes and geometries areavailable without appealing to the notions of atlasesor smooth manifolds. Since, at best, they are groups,the issue of characterizing their tangent spaces anddi↵erentials naturally arises. The Hadamard defini-tion of a di↵erentiable function is especially pertinentsince it involves the construction of paths and tangentvectors to paths within the space. In 1937 Frechetdropped the requirement that the di↵erential be lin-ear with respect to the direction while preserving twoimportant properties of the di↵erential calculus: con-tinuity of the function and chain rule. His definition

can be further relaxed to the one of semidi↵erentialwhich handles convex and semiconvex functions whilepreserving the two properties. Since semidi↵erentialsof functions f : A ! B are not required to be linear,it relaxes the requirement that the tangent spaces tothe sets A and B be linear spaces. It is su�cientto work with the Bouligand tangent cones to makesense of semidi↵erentials, shortcircuiting the require-ment of a smooth manifold. It is shown that thetopological derivative of Sokolowski and Zochowskiis in fact a semidi↵erential on the space of character-istic functions and that the tangent space containsnot only elements that create holes but also “cur-rents of Federer and Fleming and Hd-rectifiable sets.

The Free Boundary Euler Equationsin 3D

Marcelo DisconziVanderbilt University, USADavid G. Ebin

We study the incompressible free boundary Eulerequations with surface tension in three spatial di-mensions. After establishing local well-posedness ofthe equations, we show that, under natural hypothe-ses, solutions are near those of the Euler equationsin a fixed domain if the surface tension is su�cientlylarge.

Long Time Behavior of Berger PlatePDE Model Under Partial Nonlin-ear Boundary Dissipation

Pelin Guven GeredeliHacettepe University, TurkeyGeorge Avalos, Justin T. Webster

We consider a (nonlinear) Berger plate in the absenceof rotational inertia acted upon by nonlinear bound-ary dissipation. We assume the boundary to havetwo disjoint components: a clamped (inactive) por-tion and a controlled portion where the feedback isactive via hinged-type condition. We emphasize thedamping acts only in one boundary condition on aportion of the boundary. In this work we address thetechnical issues arising from damping active only ona portion of the boundary, including deriving a nec-essary trace estimate for (�u)

��0. Additionally, we

use recent techniques in the asymptotic behavior of


hyperbolic-like dynamical systems involving a stabi-lizability estimate to show that the compact globalattractor has finite fractal dimension and exhibits ad-ditional regularity beyond that of the state space forfinite energy solutions.

Analytical Problems for Free LiquidFilms

Thomas HagenUniversity of Memphis, USA

The theory of thin free films provides averaged mod-els for the evolution of viscous fluid sheets and filmsforming in the ambient air. Models are given inthe form of nonlinear transport equations for massconservation and heat transfer, coupled with ellip-tic momentum balances. This presentation will fo-cus on central analytical problems pertaining to thesolvability of the governing equations and asymptoticregularity of the linearized semigroups. Of particularmathematical interest is the case of a temperature-induced frost line along the film. To tackle theseproblems, we make use of methods from nonlinearanalysis, spectral and operator theory as well as pdespecific techniques. Part of this work was done jointlywith S. Ceci.

Analysis of a Fluid-Structure In-teraction Problem with the SlipBoundary Condition

Boris MuhaUniversity of Zagreb, CroatiaSuncica Canic

We study a nonlinear, moving boundary fluid-structure interaction (FSI) problem between an in-compressible, viscous Newtonian fluid, modeled bythe 2D Navier-Stokes equations, and an elastic struc-ture modeled by the shell or plate equations. Thefluid and structure are coupled via the Navier slipboundary condition and balance of contact forces atthe fluid-structure interface. The slip boundary con-dition might be more realistic than the classical no-slip boundary condition in situations, e.g., when thestructure is “rough“, and in modeling FSI dynamicsnear, or at a contact. Cardiovascular tissue and cell-seeded tissue constructs, which consist of grooves intissue sca↵olds that are lined with cells, are exam-ples of “rough“ elastic interfaces interacting with anincompressible, viscous fluid. The problem of heartvalve closure is an example of a FSI problem with acontact involving elastic interfaces. We design a sta-ble partitioned numerical scheme for the consideredproblem and prove the existence of a weak solution tothis class of problems by proving that the proposedscheme is convergent.

Regularity and Convergence of So-lutions in Nonlocal Models

Petronela RaduUniversity of Nebraska-Lincoln, USA

The introduction of nonlocal theories in elasticity, dy-namic fracture, biology has been successful in captur-ing real-world phenomena which can not be describedthrough local models. At the theoretical level, manyinteresting questions emerge: what regularity shouldbe expected of nonlocal solutions? What is theconnection between nonlocal models and their localcounterparts? In this talk I will present recent re-search summarizing some of these results and alsotheir role in energy minimization.

Local and Global Solutions for WaveEquations of p-Laplacian Type withGeneralized Robin Boundary Con-ditions

Mohammad RammahaUniversity of Nebraska-Lincoln, USANick Kass

In this talk we will discuss a wave equation with ap-Laplacian

utt � �pu = 0

subject to generalized Robin boundary conditionswith damping and source terms:

|ru|p�2@⌫u+|u|p�2u+g(ut) = f(u) on @⌦⇥(0, T ).

The source feedback f is supercritical, in the sensethat it is not locally Lipschitz W 1,p(⌦) ! L2(@⌦).Under suitable assumptions, we prove the existenceof local weak solutions, which can be extended glob-ally in time, provided the damping term dominatesthe source in some sense.

Implicit Parametrization and Fic-titious Domain Approach in ShapeOptimization

Dan TibaRomanian Academy, Romania

Fixed domain methods play an important role invariable/unknown domains problems like free bound-ary problems or optimal design problems. We useimplicit representations of domains and recent con-structive progress in implcit functions and implicitparametrizations theory, to develop a new approachbased on general functional variations of open sets.The extension to the critical case will be discussedas well. Such variations may involve both boundaryand topological variations simultaneously and maybe very complex and rich from the geometric pointof view. From the analytic point of view, they may beinterpreted as directional derivatives in the class ofimplicit representations and enjoy certain advantagesboth theoretically and computationally (no remesh-ing, no recomputing of the mass matrices in each it-


eration, no Hamilton-Jacobi equation, etc.). We un-derline in this respect that, although shape functionsare used, our method is di↵erent from the level setmethod. The approximating problem is an optimalcontrol problem. Approximation results and compu-tational examples, will be presented.

Modeling and Control of Wing andFlap Flutter: a Free-Clamped Platein a Potential Flow

Justin WebsterCollege of Charleston, USAI. Chueshov, E. Dowell, P.G. Geredeli, I.Lasiecka

We describe the di�cult problem of modeling theflutter phenomenon for plates (or beams), wherea portion of the structure’s boundary is free, andthe subsequent (open) issue of stabilizing—in somesense—the structure asymptotically in time. We dis-

cuss normal and axial flows. Much can be said atthe qualitative level about flag, flap, and wing flut-ter; these phenomena are of great interest in the en-gineering literature, where finite dimensional studiesfor PDE models are compared with known exper-iments for specific configurations. Mathematically,there is a lack of any analysis. We begin by discussingthe recent well-posedness and stabilization results formathematical models of panel flutter: a nonlinear(von Karman or Berger plate) coupled to a perturbedwave equation (where the entire structural boundaryis clamped). We then discuss the ways in which thisanalysis breaks down when a portion of the struc-ture’s boundary is free (e.g., the lack of an explicitformula for the flow Neumann map, and problematicinteractive terms near the free boundary). Many ofthe peculiarities can be viewed as model deficiencies.Various partial results for this open model configu-ration will be presented, as well as many novel andchallenging problems in mathematical aeroelasticity.


Special Session 36: New Trends in Nonlinear Partial Di↵erentialEquations and Applications

Julian Lopez-Gomez, Complutense University of Madrid, SpainSantiago Cano-Casanova, Pontificia Comillas University, Spain

Marcela Molina-Meyer, Carlos III University, Spain

Since the emergence of the metasolutions to describe the dynamics of the most paradigmatic classes ofparabolic equations in population dynamics in the presence of spatial heterogeneities took place, therehave emerged a number of concepts and techniques, both theoretical and numerical, that are provokinga revolution in the theory of Nonlinear Partial Di↵erential Equations whose consequences are far fromunderstood yet. This session will try to give a unified version of some of the most significative recentadvances from a number of di↵erent perspectives.

Traveling Waves in Parabolic Oper-ators with a Flux-Limited Operator.

Juan CamposGranada University, Spain

We are going to analize the travelling wave problemof

ut = (c(u, ux)u)x + f(u)

where c(u, ux) = up�1�(um�1ux) for a bounded in-creasing function � : R ! R and f and look likef(u) = u(1 � u) of Fisher type. There is a regu-lar travelling wave solution for � > �smooth with forsome value �smooth > 0. This is the analogous tothe minimal speed of propagation with regular wavesand it is not always a minimum. For

Metasolutions in Logistic Problemswith Spatial Heterogeneities andNonlinear Mixed Boundary Condi-tions

Santiago Cano-CasanovaComillas Pontifical University, Spain

In this talk will be analyzed the existence of meta-solutons in a very general class of logistic problemswith nonlinear mixed boundary conditions and spa-tial heterogeneities in the PDE and on the boundaryconditions. All the results will be given in terms ofthe nodal behavior of the potentials appearing in thePDE and on the boundary conditions. The maintecniques used to achieve the results are bifurcation,monotonicity, continuation and blow up methods.The results obtained are strongly based in the previ-ous results in the field due to R. Gomez-Renasco andJ. Lopez-Gomez and they complement them.

Fitness Based Prey Dispersal andPrey Persistence in Intraguild Pre-dation Systems

Robert Stephen CantrellUniversity of Miami, USAKing-Yeung Lam, Tian Xiang, Xinru Cao

We establish prey persistence in intraguild persis-tence systems in bounded habitats under mild con-ditions when the prey disperses using its fitness asa surrogate for the balance between resource acquisi-tion and predator avoidance. The model is realized asa quasilinear parabolic system where the dimensionof the underlying spatial habitat is arbitrary.

A Free Boundary Approach toSpreading Problems

Yihong DuUniversity of New England, Australia

In this talk I‘ll describe some recent results on thespreading of species based on the equation

ut � d�u = f(u), x 2 ⌦(t), t > 0,

where ⌦(t) is a varying domain in RN , whose bound-ary evolves according to a Stefan type free boundarycondition. The results will be compared with classi-cal results for the corresponding problem without afree boundary, namely

ut � d�u = f(u), x 2 RN , t > 0.

A Class of Degenerate ParabolicEquations

Patrick GuidottiUniversity of California, Irvine, USAYuanzhen Zhao

In this talk a class of degenerate parabolic equationswill be introduced which is motivated by applicationsto Image Processing. It will be shown how lack ofuniqueness a↵ects the linear equations and their nu-merical approximations. Uniqueness in a special classof functions will be shown for the nonlinear equa-tions.


A Two-Species Competition Systemwith Slow Di↵usion

Georg HetzerAuburn University, USALourdes Tello

Two-species competition is considered in anisolated habitat (homogeneous Neumann conditions)in case that dispersal is modeled by a p-Laplacianwith p > 2 (slow di↵usion). Slow di↵usion accounts,e.g., for e↵ects of filtration. If none of the specieshas adapted to spatial conditions, the convergence topositive equilibria without persistence is established.

Nodal Solutions for a Class of De-generate Boundary Value Problems

Julian Lopez-GomezComplutense University of Madrid, SpainM. Molina-Meyer, P. H. Rabinowitz

In non-degenerate one-D sublinear boundary valueproblems the structure of the set of solutions wherethe problem admits a solution with n nodes is wellknown since the pioneering findings of P. H. Rabi-nowitz in the early seventies. Actually, these resultstremendously facilitated the development of globalbifurcation theory as used today. However, thisparadigm changes dramatically in the context of sin-gular boundary value problems, where the solutionset can admit a large number of components makingthe analysis of these problems a real challenge.

Entire Solutions to GeneralizedParabolic k-Hessian Equations

Kazuhiro TakimotoHiroshima University, JapanSaori Nakamori

About a hundred years ago, Bernstein proved that iff = f(x, y) 2 C2(R2) and the graph of z = f(x, y) isa minimal surface in R3, then f is necessarily an a�nefunction of x and y. This theorem gives the charac-terization of entire solutions to the minimal surfaceequation in R2. For Monge-Ampere equation, thefollowing result is known; if u 2 C4(Rn) is a convexsolution to detD2u = 1 in Rn, then u is a quadraticpolynomial. In this talk, we shall obtain this type oftheorem for entire solutions to parabolic k-Hessianequation of the form

ut = µ⇣Fk(D

2u)1k

⌘in Rn ⇥ (�1, 0],

where µ : (0,1) ! R is a function, under some as-sumptions.

Acceleration of Fisher-KPP Propa-gation in Presence of Reaction andDi↵usion Heterogeneities

Andrea TelliniEHESS, Paris, FranceL. Rossi, E. Valdinoci

In this talk I will present some new propagation phe-nomena in the context of reaction-di↵usion systemsof Fisher-KPP type, which model the dynamics of apopulation in a 2 dimensional environment with oneor more lines of fast di↵usion. I will also consider thee↵ect of reaction heterogeneities, again in 1D � 2Dcases, but also in 2D � 2D ones.

Multiplicity of Stationary Solutionsfor a Nonlinear Climate Model.

Lourdes TelloUniversidad Politecnica de Madrid, Spain

We study a nonlinear energy balance model arisingin Climatology. Time and space scales are relativelylarge. The space domain is a 2D-manifold. The maindi�culties in the mathematical analysis of the modelhave been the nonlinear di↵usion and the coalbedoterm which is represented by a bounded maximalmonotone graph. One of the characteristic of thiskind of global climate model is the sensitivity to theterrestrial and solar parameters. We analyze thenumber of stationary solutions according those pa-rameters.

A Loop Type Subcontinuum ofPositive Solutions of an IndefiniteConcave-Convex Problem with theNeumann Boundary condition

Kenichiro UmezuIbaraki University, JapanHumberto Ramos Quoirin

In this talk we consider the concave-convex ellip-tic equation ��u = �b(x)|u|q�2u + a(x)|u|p�2u in⌦ coupled with the Neumann boundary condition@u@n = 0 on @⌦, where ⌦ is a smooth bounded domain

of RN , N � 2, � is a real parameter, 1 < q < 2 < p,and a, b 2 C↵(⌦), ↵ 2 (0, 1), change sign. Weuse variational and bifurcation combined argumentsto discuss existence of a loop type subcontinuumof positive solutions and investigate its propertieswhen � varies. The loop type subcontinuum means abounded one joining (�, u) = (0, 0) to itself and nevermeeting any (�, 0) with � 6= 0.


Patch E↵ects on the Pattern For-mation in a Population Model withCross Di↵usion

Zhifu XieVirginia State University, USA

My talk is concerned with the patch e↵ects on thepattern formation in a population model with cross-di↵usion. In previous paper without the patch, crossdi↵usion induces the existence of various patternssuch as stripe pattern, spot pattern, and non-steadypatterns. In particular, we are interested in how thesize of the patch changes the dynamical propertiesof the positive stationary solutions. We also numeri-cally illustrate how the patch a↵ects the pattern for-mations due to the presence of the cross di↵usion andthe patch.

Multiple Spreading Phenomenafor a Free Boundary Problem forDi↵usion Equations with BistableNonlinearity

Yoshio YamadaWaseda University, Japan

This talk is concerned with a free boundary prob-lem for di↵usion equations with bistable nonlinearitywhich allows two positive equilibrium states as anODE model. The problem models the invasion ormigration of a biological species and the free bound-ary represents the spreading front of the habitat.

Our main interest is to study large time behaviorsof solutions of the free boundary problem. It will beshown that our problem exhibits two di↵erent typesof spreading behaviors of solutions. Moreover, we willgive precise information on these two types of spread-ing behaviors such as asymptotic profiles of solutionsand asymptotic speeds of free boundaries when timegoes to infinity.

Sharp Asymptotic Profiles of So-lutions to a Simplified Attraction-Repulsion Chemotaxis Model in theWhole Space

Tetsuya YamadaNational Institute of Technology, Fukui College,Japan

In this talk we consider the Cauchy problem for asimplified version of attraction-repulsion chemotaxismodel in high dimensions which was introduced inLuca et al to describe the aggregation of microglia inAlzheimer’s disease. Recently, Shi and Wang provedthe existence of global classical solution and more-over gave the boundedness of solution and its decayestimate in Lp-norm if the repulsion prevails over theattraction. On the other hand, the solution may blowup in finite time in two dimensions under suitableconditions if the attraction prevails over the repul-sion. We shall focus on the case where the repulsionprevails over the attraction and give more preciseasymptotic profiles of the decaying solution, intro-ducing a spatial shift and correction term.


Special Session 37: Recent Advances in Dynamical Systems withApplications to Ecology and Epidemiology

Hongying Shu, Tongji University, Peoples Rep of ChinaYanyu Xiao, University of Cincinnati, USA

Guihong Fan, Columbus State University, USA

Dynamical systems have been playing an important role in modeling biological phenomena in ecology andepidemiology. Over the past decades, this research area has gained an increasing attention and received agrowing momentum in mathematical biology. This special session will focus on recent advances in dynamicalsystems with applications to ecology and epidemiology. We will invite researchers from various backgroundsto work together and contribute to the study of dynamical systems. Speakers and talks are carefully selectedto make the session attractive to a diverse audience.

Mathematical Assessment of theRole of Temperature and Rainfallon Mosquito Population Dynamics

Ahmed AbdelrazecArizona State University, USAAbba B. Gumel

A new stage-structured model for the population dy-namics of the mosquito (a major vector for numer-ous vector-borne diseases), which takes the form ofa deterministic system of non-autonomous nonlineardi↵erential equations, is designed and used to studythe e↵ect of variability in temperature and rainfallon mosquito abundance. Three functional forms ofeggs oviposition rate, namely a constant, logistic andMaynard-Smith-Slatkin functions, are used. Rigor-ous analysis of the autonomous version of the modelshows that, for any of the three oviposition functionsconsidered, the trivial equilibrium of the model islocally- and globally-asymptotically stable if a cer-tain vectorial threshold is less than unity. Condi-tions for the existence and globally-asymptotic sta-bility of non-trivial equilibrium solutions of the modelare also derived. The model, subject to the logisticand Maynard-Smith-Slatkin oviposition functions, isshown to undergo a Hopf bifurcation under certainconditions.The analyses reveal that the Maynard-Smith-Slatkinoviposition function sustains more oscillations thanthe other two oviposition functions considered(hence, it is more suited, from ecological viewpoint,for modeling the oviposition process).The non-autonomous model is shown to have aglobally-asymptotic stabile trivial periodic solution,for all forms of the oviposition functions, when theassociated reproduction threshold (R⇤

0

) is less thanunity. Furthermore, the model, with the logistic andMaynard-Smith-Slatkin oviposition functions, has aunique and globally-asymptotic stable periodic so-lution under certain conditions. Numerical simula-tions of the non-autonomous model, using mosquitosurveillance and weather (temperature and rainfall)data from the Peel region of Ontario, Canada, show apeak mosquito abundance for temperature and rain-fall in the ranges [20 – 25]oC and [15 – 35] mm,respectively. These ranges are recorded in the Peelregion between July and August (hence, this studysuggests that anti-mosquito control e↵ects should beintensified during this period).

Global Threshold Dynamics of aHeroin Epidemic Model with Age ofInfection and Nonlinear Incidence

Yuming ChenWilfrid Laurier University, CanadaJunyuan Yang, Zhen Jin

In this talk, we propose and analyze a heroin modelwith a general nonlinear incidence rate and age ofinfection. Whether the use of heroin spread or notis determined by the basic reproduction number R

0

.If R

0

1 then the drug-user steady state is globallyasymptotically stable. This threshold dynamics isestablished by employing the fluctuation lemma andthe Lyapunov functionals. Our results imply thatimproving the detection rate and drawing up e�-cient prevention ways play an more important rolethan just increasing the treatment rate for drug users.This is joint work with Professors Junyuan Yand andZhen Jin.

Avian Species-Diversity and Dilu-tion and The Transmission Dynam-ics of West Nile Virus

Guihong FanColumbus State University (Georgia), USAHuaiping Zhu

West Nile virus is a typical mosquito-borne diseasewhich vector mosquitoes play a critical rule in thetransmission and spreading of the diseases. Thereare many compartmental models in literature aboutWest Nile virus (WNv), but most of them ignore theimpact of bird species diversity and impact of speciesdilution. In this work, we formulate a system of dif-ferential equations to model the transmission of WNvwith an emphasis on the impact of bird species di-versity. We classify birds into n species and for eachspecies, we define a competent index D(Rj) which isa function of species specific parameters and relatedbasic reproduction numberRj . We also find the basicreproduction number R

0

for the model with n speciesof birds. Study shows that if one more species of birdswith the lowest competent index D(Rj) is added tothe model, then the basic reproduction number R

0

decreases. Analytical analysis and numerical simula-tions show that conservation of avian species diver-sity can help reduce human exposure to the virus.


But on the contrary, bird species diversity also in-creases the chance for virus to survive and may beresponsible for the repeated outbreak of the diseasefrom year to year. This is a joint work with Prof.Huaiping Zhu (York University).

Recovery Rates in Epidemiologicaland Ecological Models

Scott GreenhalghQueen’s University, CanadaTroy Day

Di↵erential equation models of infectious diseasehave undergone many theoretical extensions thathave proved invaluable for the evaluation of diseasespread. For instance, while one traditionally uses abilinear term to describe the incidence rate of infec-tion, physically more realistic nonlinear generaliza-tions exist. However, such theoretical extensions ofrecovery rates have yet to be developed. This is de-spite the fact that a constant recovery rate does notperfectly describe the dynamics of recovery, and thatthe recovery rate is arguably as important as anyincidence rate in governing the dynamics of a sys-tem. In this talk, a derivation of state dependent andtime varying recovery rates in di↵erential equationmodels of infectious disease is provided. We justifyour derivation through an intimate connection be-tween integral equations, di↵erential equations, andstochastic processes. Finally, we apply our deriva-tion to a obtain a novel recovery rate where infectedindividuals can only contribute to disease spread fora finite amount of time, which is in contrast to thebehavior under constant recovery rates.

Bifurcation Behaviours in Convales-cent Blood Transfusion System

Xi HuoRyerson University, CanadaXiaodan Sun, Yanni Xiao, Kunquan Lan,Jianhong Wu

Convalescent blood transfusion refers to the therapythat transfuses blood donations from survivors of aninfectious disease to the ills. This treatment strategywas historically practiced in the pre-antibiotic era,and is recommended for investigation to become anempiric treatment in many emerging infectious dis-eases - such as Ebola, H1N1, and MERS. We estab-lish a di↵erential equation system with single delayto describe a population level blood transfusion ser-vice and study the bifurcation behaviours of the threedimensional system. We will present the asymptoticstability of equilibria, challenges in analyzing the bi-furcation behaviour of the system, and public healthimplications of such phenomenon.

Stability and Bifurcation in a Stoi-chiometric Producer-Grazer Modelwith Knife Edge

Yang KuangArizona State University, USAXianshan Yang, Xiong Li, Hao Wang

All organisms are composed of multiple chemical ele-ments such as nitrogen (N), phosphorus (P), and car-bon (C). P is essential to build nucleic acids (DNAand RNA) and N is needed for protein production.To keep track of the mismatch between P require-ment in the grazer and the P content in the pro-ducer, stoichiometric models have been constructedto explicitly incorporate food quality and quantity.Most stoichiometric models have suggested that thegrazer dynamics heavily depend on P content in theproducer when the producer has low nutrient con-tent (low P:C ratio). Motivated by recent lab exper-iments, researchers explored the e↵ect of excess pro-ducer nutrient content (extremely high P:C ratio) onthe grazer dynamics. This phenomenon is called thestoichiometric knife edge. However, the global analy-sis of this knife-edge model has is challenging becausethe phase plane is separated into many di↵erent re-gions in which the governing equations are di↵erent.The aim of this paper is to present a sample of acomplete mathematical analysis of the dynamics ofthis model and to perform a bifurcation analysis forthe model with Holling type II functional response.

Hopf Bifurcation in Reaction-Di↵usion Population Model withSpatial-Temporal Nonlocal DelayedGrowth Rate

Junping ShiCollege of William and Mary, USAWenjie Zuo

We consider the existence of spatially inhomogeneoustime-periodic orbit in a reaction-di↵usion populationmodel with spatial-temporal nonlocal delayed growthrate and Dirichlet boundary condition. When thedispersal kernel is of strong or weak type, the scalarreaction-di↵usion equation with distributed delay isconverted into a system of two or three reaction-di↵usion equations without delay. We prove the ex-istence of periodic orbits for the system which areequivalent to periodic orbits for the original scalarmodel.


Turing-Hopf Bifurcation in theReaction-Di↵usion Equations andIts Applications

Yongli SongTongji University, Peoples Rep of ChinaT.Zhang, Y.Peng

In this talk, we consider the Turing-Hopf bifurcationarising from the reaction-di↵usion equations. It is adegenerate case and where the characteristic equa-tion has a pair of simple purely imaginary roots anda simple zero root. First, the normal form theory forpartial di↵erential equations (PDEs) with delays de-veloped by Faria is adopted to this degenerate caseso that it can be easily applied to Turing-Hopf bi-furcation. Then, we present a rigorous procedure forcalculating the the normal form associated with theTuring-Hopf bifurcation of PDEs. We show that thereduced dynamics associated with Turing-Hopf bifur-cation is exactly the dynamics of codimension-twoordinary di↵erential equations (ODE), which impliesthe ODE techniques can be employed to classify thereduced dynamics by the unfolding parameters. Fi-nally, we apply our theoretical results to an auto-catalysis model governed by reaction-di↵usion equa-tions; for such model, the dynamics in the neighbour-hood of this bifurcation point can be divided into sixcategories, each of which is exactly demonstrated bythe numerical simulations; and then according to thisdynamical classification, a stable spatially inhomoge-neous periodic solution has been found.

Explicitly Separating Growth andMotility in a Glioblastoma TumorModel

Tracy StepienArizona State University, USAErica Rutter, Meng Fan, Yang Kuang

Glioblastoma multiforme is an aggressive brain can-cer that is extremely fatal. Gliomas are character-ized by highly di↵usive growth patterns, which makesthem impossible to remove with surgery alone. Togain insight on the mechanisms most responsible fortumor growth and the di�cult task of forecasting fu-ture tumor behavior, we investigate a mathematicalmodel that is inspired by a bacteriophage infection-spreading model in which tumor cell motility andcell proliferation are considered as separate processes.Numerical and analytical results are compared to ex-perimental data.

Mathematical Model of PathogenPersistence in a Water DistributionNetwork

Benjamin VaughanUniversity of Cincinnati, USABenjamin L. Vaughan, Jr.

In industrialized nations, the availability of clean wa-ter is provided through sophisticated water distribu-tion systems that consist of a network of pipes, pump-ing stations, reservoirs, and other components. Bio-logical contamination of the water distribution sys-tem, due to factors such as a breach in a pipe or othercauses, can have an adverse e↵ect on water quality. Inthis talk, we will discuss the development and analy-sis of a mathematical model pertaining to the releaseof pathogens into water distribution systems. Bac-teria within a water distribution system frequentlyform biofilms on the interior surfaces of pipes thatpathogens can then attach to and grow. Our modelinvestigates the interaction between planktonic andbiofilm-bound pathogens under time-constant andtime-periodic flow regimes and is analyzed using Lya-punov stability and Floquet theory as well as numer-ical simulations. Often, water distribution networkshave a significant number of connections, making di-rect calculations expensive. So, we have developedand analyzed an e�cient approach for predicting thelong-time behavior of the pathogen populations forlarge water-distribution networks. The analytical re-sults are validated using numerical simulations of thefull non-linear system on a range of water distribu-tion network sizes.

Competition for Two Essential Re-sources with Internal Storage andPeriodic Input

Feng-Bin WangDepartment of Natural Science, Center for GeneralEducation, Chang Gung University, TaiwanSze-Bi Hsu, Xiao-Qiang Zhao

We study a mathematical model of two species com-peting in a chemostat for two internally stored essen-tial nutrients, where the nutrients are added to theculture vessel by way of periodic forcing functions.Persistence of a single species happens if the nutrientsupply is su�cient to allow it to acquire a thresholdof average stored nutrient quota required for growthto balance dilution. More precisely, the population iswashed out if a sub-threshold criterion holds, whilethere is a globally stable positive periodic solution ifa super-threshold criterion holds. When there is mu-tual invasibility of both semitrivial periodic solutionof the two-species model, both uniform persistenceand the existence of periodic coexistence state areestablished.


Sensitivity of the GeneralRosenzweig–MacArthur Modelto the Mathematical Form of theFunctional Response: a BifurcationTheory Approach

Gail WolkowiczMcMaster University, CanadaGunog Seo

The Rosenzweig–MacArthur predator-prey modelhas been shown to be sensitive to the mathematicalform used to model the predator response functioneven if the forms have the same basic shape: zero atzero, monotone increasing, concave down, and sat-urating. We revisit this model to help explain thissensitivity in the case of Holling type II Monod ,Ivlev, and hyperbolic trigonometric response func-tions. We consider the local and global dynamicsand determine the possible bifurcations with respectto variation of the carrying capacity of the prey, ameasure of the enrichment of the environment. Wegive analytic expressions that determine the critical-ity of the Andronov-Hopf bifurcation, and prove thatalthough all three forms can have supercritical Hopfbifurcation, only the hyperbolic trigonometric formcan give rise to subcritical Hopf bifurcation, saddle-node bifurcation of periodic orbits, and multiple limitcycles, providing a counterexample to a conjecture ofKooij and Zegeling (1996) and a result in a paperby Attili and Mallak (2006). We revisit the ranking

of responses, according to their potential to desta-bilize dynamics, and show that given data, not onlythe choice of functional form, but the choice of num-ber and position of data points influences predicteddynamics.

Bridging the Behaviors of IntestinalStem Cells and Their MolecularControl Networks with Mathemati-cal Modeling

Tongli ZhangUniversity of Cincinnati, USARichard Ballweg, Emma Teal, Toru Matsu-ura, Benjamin Vidourek, Meredith Barone,Christian Hong

Intestinal stem cells continuously di↵erentiate intomature functional cells (e.g. Enterocytes, Gobletcells, Enteroendocrine cells and Paneth cells). Thisessential process is tightly regulated in healthy organ-isms and its deregulation leads to diseases. As cur-rent molecular and cellular biology reveals the reg-ulatory details of the pathways (e.g. Wnt, Notch,MAPK, BMP) controlling this process, the accumu-lated information proposes the challenge of integrat-ing these details together into a logical and dynamicalframework. In order to cope with this challenge, wehave converted several key pathways into computa-tional models. These model have been constrainedwith available data reported in the literature, andthe novel predictions generated by these models aretested in intestinal enteroids.


Special Session 38: Evolution Equations and Integrable Systems

Alex Himonas, University of Notre Dame, USADionyssis Mantzavinos, SUNY at Bu↵alo, USA

The theme of this session is nonlinear evolution equations and integrable systems including the NLS equation,the KdV equation, the Camassa-Holm equation, and the Euler equations of hydrodynamics. Topics coveredfor these equations include, among others, local and global well-posedness, scattering and stability issues,integrability and solitary waves.

Wellposedness of the Fully Non-Linear KdV Equation

Timur AkhunovUniversity of Rochester, USADavid Ambrose, Doug Wright

Dispersion is a phenomenon that describes waves ofdi↵erent frequency traveling at di↵erent speed. Thise↵ect is related to a rainbow forming with light pass-ing through a prism and in uncertainty principle inquantum mechanics. One of the most famous disper-sive equations is the Korteweg-de Vries (KdV) equa-tion, which was originally proposed for the propaga-tion of long waves in a shallow canal. Several gener-alizations of KdV, like K(m,n) model with compactsolitary waves do not have linear dispersion and aremuch less understood. In this talk I would discussnew wellposedness and illposedness results.

Ill-Posedness of Some Water WaveModels

Jerry BonaUniversity of Illinois at Chicago, USADavid Ambrose, Mimi Dai, Timur Milgrom

We discuss ill-posedness of some water wave models.These models are members of a class of Boussinesq-type systems that arise in approximating surfacewaves in relatively shallow water. They include somewell known systems of equations. The ill-posednessdoes not stem from attempting to work in a very largefunction class, but rather is intrinsic to the model.

Semiclassical Dynamics of theThree-Wave Resonant InteractionEquations.

Robert BuckinghamUniversity of Cincinnati, USARobert Jenkins, Peter Miller

The three-wave resonant interaction equations de-scribe the evolution of three electrical pulses in adispersive medium with quadratic linearity. We usethe inverse-scattering method to analyze the small-dispersion (or semiclassical) behavior. We presentanalytic results on the WKB approximation of thescattering data, as well as a numerical study ofexact solutions that suggests semiclassical behavior

(i.e. approximation of solutions by modulated ellip-tic functions) similar to that seen in other nonlin-ear wave equations such as the KdV, NLS, and sine-Gordon equations. This work is joint with RobertJenkins and Peter Miller.

An Inverse Problem for the Mod-ified Camassa-Holm (or calledFORQ) Equation and Multi-PointPade Approximants

Xiangke ChangUniversity of Saskatchewan, CanadaJacek Szmigielski

An spectral and inverse spectral problem associatedto multipeakons for the modified Camassa-Holm (orcalled FORQ) equation are studied. It is shown thatthe inverse problem is solvable in terms of multi-pointPade approximations, which leads to an explicit con-struction of peakon solutions.

Existence and Stability of SolitaryWave Solutions to a Coupled BBMSystem

Hongqiu ChenUniversity of Memphis, USAXiaojun Wang

Considered here is a class of system of BBM-typenonlinear dispersive equations

Ut + c0

Ux �AUxxt + (rH(u))x = 0,

where U = U(x, t) is an R2-valued function, c0

is apositive constant, A is a 2 ⇥ 2 real positive definitematrix, and rH is the gradient of a homogeneouspolynomial function H : R2 ! R of degree p. Inthis work, we present existence and stability crite-ria for the solitary wave solutions to the system thatcontains coupled nonlinear terms. Using the idea byBona, Chen and Karakashianand exploiting the accurate point spectrum informa-tion of the associated Schrodinger operator, we foundthat the stability of solitary wave solution is deter-mined by the degree p in a simple algebraic relation.


The Geometry of Fourier-MellinTransforms, and How to Use It

Darren CrowdyImperial College London, England

The construction of novel Fourier/Mellin-type trans-form pairs that are tailor-made for given planar re-gions within the special class of circular domainsis described. Circular domains are those havingboundary components that are either circular arcsor straight lines. The new transform pairs generalizethe classical Fourier and Mellin transforms. Thesegeometry-fitted transform pairs can be used to greatadvantage in solving boundary value problems de-fined in these domains.

The Unified Transform Method forSystems

Bernard DeconinckUniversity of Washington, USA

The Unified Transform Method (UTM) due to Fokashas been successfully applied to a great number ofvery di↵erent problems. Most of these problems havebeen scalar problems. When dealing with systemsof equations, the dispersion relation of the problemis of the same order as the system, which leads tobranched frequency functions appearing in the globalrelation. Most systems that have been approachedusing the UTM avoid have dispersion relation solu-tions that are not branched, and as such their so-lution is not typical. A few systems with branchedsolutions have been considered, but on an ad hoc ba-sis. I will outline how the UTM can systematicallydeal with systems of equations without much addede↵ort, compared to the scalar case.

A Priori Symmetry and DecayProperties of a Nonlocal ShallowWater Wave Equation

Mats EhrnstromNTNU Norwegian University of Science and Tech-nology, NorwayG. Brull, L. Pei

We prove the following interesting connection: thattraveling solitary waves of a nonlocal wave equationare necessarily symmetric, monotone on a half-line,and of exponential decay rate; and that symmetricsolutions of the initial-value problem for the sameequation are necessarily traveling. Whereas the sec-ond proof is based on a quite general structural prop-erty (which we extend here to a nonlocal setting), theproof of the first three facts relies on a detailed analy-

sis of the Fourier transform of m(⇠) =q

tanh ⇠⇠

, which

we prove is completely monotone. More precisely, westudy the Whitham equation

ut + uux +

ZK(x� y)ux dx = 0,

where the integral kernel K has the symbol m(⇠),arising naturally in the study of water waves.The talk is based on recent results joint with GabrieleBrull and Long Pei; Anna Geyer; and Erik Wahlen.

Scattering for a 3D Coupled Non-linear Schrodinger System

Luiz gustavo FarahUniversidade Federal de Minas Gerais (UFMG),BrazilAdemir Pastor

We consider the three-dimensional cubic nonlinearSchrodinger system.

⇢i@tu+ �u+ (|u|2 + �|v|2)u = 0,i@tv + �v + (|v|2 + �|u|2)v = 0.

Let (P,Q) be any ground state solution of theabove Schrodinger system. We show that forany initial data (u

0

, v0

) in H1(R3) ⇥ H1(R3)satisfying M(u

0

, v0

)A(u0

, v0

) < M(P,Q)A(P,Q)and M(u

0

, v0

)E(u0

, v0

) < M(P,Q)E(P,Q), whereM(u, v) and E(u, v) are the mass and energy (in-variant quantities) associated to the system, the cor-responding solution is global in H1(R3) ⇥ H1(R3)and scatters. Our approach is in the same spiritof Duyckaerts-Holmer-Roudenko, where the authorsconsidered the 3D cubic nonlinear Schrodinger equa-tion. The author was partially supported byCNPq/Brazil and FAPEMIG/Brazil.

Regularity Results for Generaliza-tions of the Wave Maps Equation

Dan GebaUniversity of Rochester, USA

In this talk, we will present an overview of recent re-sults for semilinear and quasilinear generalizations ofthe wave maps equation, which have a strong phys-ical motivation and present many challenges from amathematical point of view. The emphasis will be onlocal and global regularity theories for these problemsand the new analytic tools developed in connectionto these theories.

The Nonlinear Schrodinger Equa-tion on the Half-Line

Alex HimonasUniversity of Notre Dame, USAAthanassios S. Fokas, Dionyssios Mantzavinos

The initial-boundary value problem (ibvp) for thenonlinear Schrodinger (NLS) equation on the half-line with data in Sobolev spaces is analyzed viathe formula obtained through the unified transformmethod, and a contraction mapping approach. First,


the linear Schrodinger (LS) ibvp with data in Sobolevspaces is solved and the basic space and time esti-mates of the solution are derived. Then, using theseestimates well-posedness of the corresponding non-linear ibvp is proved for data belonging in Sobolevspaces with appropriate exponents.

An Ill-Posedness Result forNovikov’s Equation

Curtis HollimanThe Catholic University of America, USAAlex Himonas

Novikov produced a new integrable equation in 2009that has many similarities to the Camassa-Holmequation. Well-posedness for this equation has beenestablished in Sobolev spaces with exponents s >3/2, and we now show that this result is sharp.

The Hunter-Saxton Equation inBesov Spaces on the Circle

John HolmesThe Ohio State University, USAFeride Tiglay

We show that the Hunter-Saxton equation is well-posed (existence, uniqueness and continuous depen-dence) in Besov spaces Bs

2,r on the circle when s >3/2 and r > 1 as well as the case s = 3/2 and r = 1.Furthermore, we construct an example which demon-strates that the continuity of the data-to-solutionmap is not uniformly continuous from any boundedsubset of Bs

2,r to Bs2,r. Our results are improve-

ments upon earlier results in Sobolev spaces, and takeadvantage of some properties of functions in Besovspaces.

Stability of Closed Solutions to theVFE Hierarchy

Thomas IveyCollege of Charleston, USAStephane Lafortune

The Vortex Filament Equation (VFE) is part of anintegrable hierarchy of geometric evolution equationsfor space curves, several of which have been shown todescribe vortex filament motion in various situations.We develop a general framework for studying the ex-istence and linear stability of closed solutions of theVFE hierarchy, based on the correspondence betweenthe VFE and the nonlinear Schrodinger (NLS) hier-archies. Our results show that it is possible to estab-lish a connection between the AKNS Floquet spec-trum and the stability properties of the solutions ofthe filament equations. We apply our machinery tosolutions of the filament equation associated to theHirota equation.

Local Energy Conservation andSpectral Projection for an Inte-grable Long Wave Model

Henrik KalischUniversity of Bergen, Norway

The Kaup-Boussinesq system is a coupled systemof nonlinear partial di↵erential equations which hasbeen derived as a model for surface waves in the con-text of the Boussinesq scaling. The system is knownto be linearly ill-posed, but it is also known to becompletely integrable, so that solutions of the sys-tem may be constructed more or less explicitly. Thiswork presents a derivation of the energy density andenergy flux of the Kaup-Boussinesq system. It is alsoshown that the total energy of the wave system isequal to the Hamiltonian function found by Craigand Groves. In addition, a spectral method for thenumerical discretization of the Kaup-Boussinesq sys-tem is put forward, and shown to be convergent andstable for certain classes of solutions.

The Korteweg-De Vries Equationon the Half-Line

Dionyssios MantzavinosSUNY Bu↵alo, USAAthanassios S. Fokas, A. Alexandrou Himonas

A new method for the well-posedness of initial-boundary value problems for nonlinear dispersiveevolution equations was recently introduced via theanalysis of the nonlinear Schrodinger (NLS) and the“good“ Boussinesq (GB) equations on the half-line.This talk will be concerned with the further extensionand implementation of this method for the Korteweg-de Vries (KdV) equation on the half-line with datain Sobolev spaces. As already known from the anal-ysis of the Cauchy problem, the KdV nonlinearityrequires special attention and, in particular, intro-duces the need for additional linear estimates thatemanate from the relevant bilinear estimate. More-over, the unified transform method solution formula,which assumes the role held by Duhamel’s formula inthe Cauchy problem, involves genuinely complex con-tours of integration that stay away from the real andimaginary axes of the Fourier plane. Consequently,several modifications and generalizations of the tech-niques previously employed for NLS and GB will nowbe necessary.

Two-Component Generalizations ofthe Camassa-Holm Equation

Vladimir NovikovLoughborough University, EnglandAndrew N.W. Hone, Jing Ping Wang

We present a classification of integrable two-component systems of non-evolutionary partial dif-ferential equations that are analogous to theCamassa-Holm equation via the perturbative sym-metry approach. Independently, we perform a clas-


sification of compatible pairs of Hamiltonian opera-tors, which leads to bi-Hamiltonian structures for thesame systems of equations. We also consider exactsolutions and Lax representations for the obtainedintegrable systems.

Global Analyticity Well-Posednessfor a Generalized Camassa-HolmEquation

Gerson PetronilhoFederal University of Sao Carlos, BrazilRafael F. Barostichi, Alex A. Himonas, Ger-son Petronilho

We prove that the Cauchy problem for a general-ized Camassa-Holm equation with initial data in ananalytic space has a unique global analytic solutionu 2 C!(Rx ⇥ [0,1)t) provided that (1� @2

x)u0

doesnot change sign.

Negative Integrable Hierarchy Re-lated to Peakon Equation

Zhijun (George) QiaoUniversity of Texas - Rio Grande Valley, USABaoqiang Xia

Integrable peakon equations are derived from thenegative order hierarchy through Lax formulation.Some multi-peakon soltions are given through somefeasible procedure.

Initial-To-Interface Maps for theHeat Equation on Composite Do-mains

Natalie SheilsUniversity of Minnesota, USABernard Deconinck

A map from the initial conditions to the values ofthe function and its first spatial derivative evaluatedat the interface is constructed for the heat equationon finite and infinite domains with n interfaces. The

existence of this map allows changing the problemat hand from an interface problem to a boundaryvalue problem which allows for an alternative to theapproach of finding a closed-form solution to the in-terface problem.

Classical Solutions of the General-ized Camassa-Holm Equation

Ryan ThompsonUniversity of North Georgia, USAJohn Holmes

In this presentation, well-posedness in C1(R) (a.k.a.classical solutions) is investigated for a generalizedCamassa-Holm equation (g-kbCH) having (k + 1)-degree nonlinearities and containing as its integrablemembers the Camassa-Holm, the Degasperi-Procesiand the Novikov equations.

Nonuniform Dependence on InitialData for Compressible Gas Dynam-ics

Feride TiglayOhio State University, Newark, USABarbara Keyfitz

Once the well posedness of a Cauchy problem is es-tablished, a natural question to ask is whether thedata to solution map is better than continuous. Westudy the data to solution map of the Cauchy prob-lem for compressible gas dynamics.


Special Session 40: Polymer Dynamics Models and Applications toNeurodegenerative Disease

Leon Matar Tine, Universite de Lyon, FranceLaurent Pujo-Menjouet, Universite de Lyon, France

This session is dedicated to the study of models dealing with polymer dynamics (coagulation / fragmentation/ polymerization) such as the Smoluchowski equation, Becker-Doring model for instance but not only.Biological applications will mainly be focused on neurodegenerative diseases such as prion disease, Alzheimer,Parkinson but not only.

Asymptotic Behaviour of theBecker-Doring Equations

Jose CanizoUniversity of Granada, SpainBertrand Lods, Amit Einav

The Becker-Doring equations appear often in mod-els of polymerisation but their asymptotic behaviouris still not well understood. We present some recentadvances in the study of the asymptotic behaviour ofsubcritical solutions, where entropy methods can beapplied both in the linearised regime and in the non-linear one. We obtain explicit exponential rates ofrelaxation to equilibrium for rapidly decaying initialconditions, and algebraic rates for initial conditionswith a larger tail.

Modelling Interactions BetweenAlzheimer’s Disease and Prions

Pauline MazzoccoICJ University Lyon 1, FranceMatthieu Dumont, Abdelkader Lakmeche,Laurent Pujo-Menjouet, Human Rezaei, LeonMatar Tine

Alzheimer’s disease (AD) is a fatal incurable dis-ease leading to progressive neuron destruction. ADis caused by the presence of A� peptides inside thebrain, especially A�-40 and A�-42 monomers. Theyhave the faculty to aggregate into oligomers and fib-rils, which eventually form amyloid plaques. Theselatter were thought to cause neuron loss for years.However it seems that oligomers are the most toxicstructures as they can interact with neurons throughmembrane receptors, including PrPC proteins. Thisinteraction may lead to the misconformation of PrPC

into pathogenic prions, PrPSC .In this work, we developed a model describing A�polymerization process into oligomers and fibrils.We included the interactions between oligomers andPrPC . The model consists of 13 equations, includingsize structured transport equations, ordinary di↵er-ential equations and delayed di↵erential equations.We provide theoretical results regarding existence ofsolutions, and numerical simulations of the model.

Dynamics of Intermediate Filaments

Stephanie PortetUniversity of Manitoba, Canada

Intermediate filaments are one of the cytoskeletoncomponents. The cytoskeleton is made of structuralproteins polymerized in filaments that are organizedas networks in the cytoplasm. The organization of acytoskeletal network is the main determinant of itsfunction in cells. The cytological signature of somehuman diseases is the misorganization of cytoskele-tal networks. Here, models based on experimentaldata combining assembly/disassembly processes andtransport of intermediate filaments are presented toinvestigate the major contributors to their organiza-tion.

A Micellar On-Pathway Intermedi-ate Step Explains the Kinetics ofPrion Amyloid Formation

Laurent Pujo-MenjouetUniversity of Lyon, FranceErwan Hingant, Pascaline Fontes, MariaTeresa Alvarez-Martinez, Jacques-DamienArnaud, Jean- Pierre Liautard

In a previous work by Alvarez-Martinez et al. in2011, the authors pointed out some fallacies in themainstream interpretation of the prion amyloid for-mation. It appeared necessary to propose an originalhypothesis able to reconcile the in vitro data withthe predictions of a mathematical model describingthe problem. Here, a model is developed accordinglywith the hypothesis that an intermediate on-pathwayleads to the conformation of the prion protein intoan amyloid competent isoform thanks to a structure,called micelles, formed from hydrodynamic interac-tion. The authors also compare data to the predic-tion of their model and propose a new hypothesis forthe formation of infectious prion amyloids.


Prion Quasi-Species and MolecularBasis of Auto-Perpetuation of PrionStructural Information

Human RezaeiInstitut National de la Recherche Agronomique,FranceDavy Martin, Joan Torrent i Mas, StephaniePrigent, Vincent Beringue

The prion phenomenon is based on autonomousstructural information propagation towards singleor multiple protein conformation changes. Since thislast decade, the prion concept referring to the trans-mission of structural information has been extendedto several regulation systems and pathologies includ-ing Alzheimer and Parkinson’s diseases. The unifiedtheory in Prion replication implies structural infor-mation transference (SIT) from the prion to a non-prion conformer through a mechanism also called im-properly, with regards to biophysical considerations“seeding” phenomenon. Therefore considering prionreplication as a structural information transductionfrom a donor (i.e. template) to an acceptor (i.e.substrate) through a transduction interface a newquestioning arises: what are molecular mechanismsof the auto-perpetuation of the Prion structural in-formation and its faithfulness?

Considering the Prion propagation as more or lessfaithful perpetuation of structura information, in thepresent work, we explored the concept of prion quasi-species (i.e. existence of prion network heterogeneousassemblies) and highlighted the existence of prionnetwork, which has an autopoietic behaviour (auto-replicative). Our observations strongly suggest thatspecific criteria in term of: protein structure, delay-process and thermo-kinetics should be collated beforea system become dissipative and autopoietic.

Modeling in Vivo Prion AggregateDynamics in Yeast

Suzanne SindiUniversity of California, Merced, USAJason K. Davis

Prion proteins are responsible for a variety ofneurodegenerative diseases in mammals such asCreutzfeldt-Jakob disease in humans and “mad-cow“disease in cattle. While these diseases are fatal tomammals, a host of harmless phenotypes have beenassociated with prion proteins in S. cerevisiae, mak-ing yeast an ideal model organism for prion diseases.Most mathematical approaches to modeling prion dy-namics have focused on either the protein dynamicsin isolation, absent from a changing cellular environ-ment, or modeling prion dynamics in a populationof cells by considering the “average“ behavior. How-ever, such models have been unable to recapitulatein vivo properties of yeast prion strains including ex-perimentally observed rates of prion loss. We havedeveloped physiologically relevant models by consid-ering both the prions and their yeast host. We first

generalize a previously developed nucleated polymer-ization model for aggregate dynamics. We next dis-cuss a stochastic model of prion protein dynamics inthe context of a growing yeast culture. Our modelis based on a stochastic chemical reaction networkwithin a cell and a Crump–Mode–Jagers branchingprocess model of population growth. In order to si-multaneously conform to observations of two distinctprion strains we uncovered several novel aspects ofprion biology.

Analysis and 3D Numerical Simu-lation of a Polymerization Modelwith Aggregation

Leon Matar TineUniversite Lyon 1, FranceLeye Babacar

In this talk we present an analytical and numericalmodeling of a general polymerization process withpossible lengthening by coagulation mechanism. Theproposed model take into account the 3D spatial dif-fusion for the mass transfer between monomers andpolymers. We will discuss about the well-posednessof this general polymerization model and we willpropose for the simulations, a 3D numerical schemebased on a generalization of the anti-dissipative strat-egy method for the flux approximation.

Optimal Growth for Linear Growth-Fragmentation Processes

Leon Matar TineUniversite Lyon 1, FranceVincent Calvez, Pierre Gabriel, StephaneGaubert

I will present recent advances in the optimization oflinear, finite dimensional models of growth- fragmen-tation processes. Consider a linear di↵erential inclu-sion which preserves positivity. When the matricesare uniformly irreducible and bounded, there existsa unique Lyapunov exponent which characterizes theinfinite horizon optimal growth of the linear system.Moreover this exponent is related to the critical vis-cosity solution of a Hamilton-Jacobi-Bellman equa-tion. Existence of such a critical viscosity solution isknown as Fathi’s weak-KAM theorem in Lagrangiandynamics. The corresponding Aubry set informs usabout the optimal trajectories of the linear di↵er-ential inclusion. The talk will be illustrated withnumerical simulations of three dimensional systems.This is a joint work with Pierre Gabriel (Universityof Versailles, France) and Stephane Gaubert (Inria,Saclay, France).


Recovering the History of a FibrilPopulation Undergoing Fragmenta-tion from Its Asymptotic Profile

Magali TournusAix-Marseille Universite, FranceMarie Doumic, Miguel Escobedo, Wei-FengXue

Using measurements of the size distribution in a pop-ulation in order to infer the characteristics of theirgrowth is a field of growing interest in populationdynamics. Such techniques allow one to assess ona solid ground an empirical model without a prioriinformation on the microscopic laws for each indi-viduals growth and division. Linked to recent devel-opments in experimental biology, which gives accessto the size distribution of amyloid fibrils, we focushere on a pure fragmentation process. The quantityof interest is the density f(t, x) � 0 of particles ofsize x 2 R+ at time t � 0, expressed as the solu-tion of a continuous fragmentation equation. Basedon the knowledge of the system at equilibrium, wepresent a methodology to estimate the parameters.The mathematical tools used are the Mellin trans-form and functional equations.

Nucleation Time in StochasticBecker-Doring Model

Romain YvinecINRA Tours, FranceSamuel Bernard, Tom Chou, Julien De-schamps, Maria R. D’Orsogna Erwan Hin-gant, Laurent Pujo-Menjouet

This work is motivated by protein aggregation phe-nomena in neurodegenerative diseases. A key ob-servation of in-vitro polymerization experiments ofprion protein is the large variability of the so-called“nucleation time”, which is experimentally definedas the lag time before the polymerization of proteinstrully stars. In this context, we study a stochasticversion of a well-known nucleation model in physics,namely the Becker-Doring model. In this model,aggregates may increase or decrease their size one-by-one, by capturing or shedding a single particle.We will present numerical and analytical investiga-tion of the nucleation time as a first passage timeproblem [1,2].

Finally, we will present limit theorem techniques tostudy the link from the discrete size Becker-Doringmodel to a continuous size version (the Lifshitz-Slyozov model). For general coe�cients and ini-tial data, we introduce a scaling parameter andshow that the empirical measure associated to theBecker-Doring system converges in some sense to theLifshitz-Slyozov equation when the parameter goesto 0 [3].

Contrary to previous studies, we use a weak topologythat includes the boundary of the state space, allow-ing us to rigorously derive a boundary value for theLifshitz-Slyozov model. It is the main novelty of thiswork and it answers to a question that has been con-jectured or suggested by both mathematicians andphysicists. We emphasize that the boundary valuedepends on a particular scaling (as opposed to a mod-eling choice) and is the result of a separation of timescale and an averaging of fast (fluctuating) variables.

References

[1] R. Y., M. R. D’Orsogna, and T. Chou. Firstpassage times in homogeneous nucleation andself-assembly. Journal of Chemical Physics,137:244107, (2012).

[2] http://arxiv.org/abs/1510.04730

[3] http://arxiv.org/abs/1412.5025


Special Session 41: Stochastic Partial Di↵erential Equations

Benjamin Gess, University of Bielefeld, GermanyMichael Rockner, University of Bielefeld, Germany

Stochastic Partial Di↵erential Equations (SPDE) and their applications is a relatively young field of mathe-matics. In the past two decades and it has, however, become one of the main research directions of ProbabilityTheory, with rising activity across its entire spectrum. In particular, modern SPDE techniques and theircombination with ideas from rough path theory led to Martin Hairer’s theory of regularity structures forwhich he was awarded the Fields medal in 2014. The field of SPDE combines the classical area of partialdi↵erential equations with modern branches of probability theory, in particular, stochastic analysis, and thusconstitutes one of the most prominent contact points between analysis and stochastics. Besides various otherconnections to pure mathematics (e.g. Di↵erential Geometry, Dynamical Systems) one main focus of SPDEare its applications to the Sciences, in particular Physics, but also Biology and Chemistry. Another mainarea of applications is Economics, in particular Mathematical Finance. The aim of the session is to give anupdate on recent developments on SPDE and at the same time identify new frontiers with challenging openproblems for the field, with emphasis on both theory and applications.

Burgers Equation with RandomForcing.

Yuri BakhtinCourant Institute, NYU, USA

Burgers equation with random forcingI will talk about the ergodic theory of randomlyforced Burgers equation (a basic nonlinear evolutionPDE related to fluid dynamics and growth models) inthe noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the in-viscid case) and polymer measures on one-sided infi-nite trajectories (in the positive viscosity case). Jointwork with Eric Cator, Kostya Khanin, Liying Li.

Regularization by Noise for SomeStochastic Di↵erential Equations.

Remi CatellierUniversity of Rennes 1, France

We first study the linear transport equation

@@tu(t, x) + b(t, x) ·ru(t, x) +ru(t, x) · @

@tX(t) = 0, u(0, x) = u

0

(x)

where b is a vectorfield of limited regularity and X avector-valued Holder continuous driving term. Usingthe theory of controlled rough paths we give a mean-ing to the weak formulation of the PDE and solvethat equation for smooth vectorfields b. In the caseof the fractional Brownian motion a phenomenon ofregularization by noise is displayed.By using the same techniques, and thanks to thescalar conservation laws formulation for equations ofthe form

@tu(t, x) + divx

⇣A�x, u(t, x)

�⌘+ru(t, x) · @

@t

X(t) = 0, u(0, ·) = u0

,

the same kind of regularization by noise phenomenonis shown in the latter case.

Singular SPDEs on Manifolds

Joscha DiehlTU Berlin, GermanyAntoine Dahlqvist, Bruce Driver

We show how the theories of paracontrolled dis-tributions and regularity structures can be imple-mented on manifolds, to solve singular SPDEs likethe parabolic Anderson model. This is ongoing workwith Bruce Driver (UCSD) and Antoine Dahlqvist(Cambridge)

Malliavin Calculus and RegularityStructures

Paul GassiatUniversity Paris-Dauphine, FranceG. Cannizzaro, P. Friz

Many nonlinear stochastic PDEs of interest are ill-posed in the sense that one cannot give a canonicalmeaning to the nonlinearity. Hairer’s theory of regu-larity structures allows to give a good notion of solu-tion for a large class of such equations. In this talk Iwill explain how this can be combined with classicaltools from Malliavin calculus, which allows in par-ticular to obtain information on the densities of themarginal laws of the solutions.


A Stochastic Equation of Ginzberg-Landau Type

Wilfried GreckschMartin-Luther-University Halle-Wittenberg, Ger-many

We introduce for a complex valued process(X(t, x))t2[0,T ],x2G the following Ito equation in thesense of the variational solution

dX(t, x) = (a1

+ ia2

)�xX(t, x) +

+(b1

+ ib2

)f(X(t, x), X(t.x))X(t, x)

+g(t, x(t, x))dw(t) X(0, x) = X0

(x).

The Laplacian operator� x is defined on a boundedset G ⇢ Rm with homogeneous Neumann boundaryconditions and a W denotes a cylindrical Wiener pro-cess and a

1

� 0, a2

, b1

, b2

, c1

, c2

2 R1.Concrete examples are discussed and the existenceand uniqueness of a variational solution are proved.The solution process of this problem is approximatedby solutions of linear problems.

Hypocoercivity for SPDEs

Martin GrothausUniversity of Kaiserslautern, Germany

Recently we provided hypocoercivity concepts for de-generate Kolmogorov (backward) equations. In thistalk we plan to apply them to stochastic partial dif-ferential equations.

Rough Gronwall Lemma and WeakSolutions to RPDEs

Martina HofmanovaTechnical University Berlin, Germany

We put forward a general framework for the studyof a wide class of rough path driven PDEs. To bemore precise, we introduce an intrinsic notion of dis-tributional solution, i.e. weak solution in the PDEsense, and develop new a priori estimates based ona rough Gronwall lemma argument leading to theproof of existence and uniqueness. Our approachdoes not rely on any sort of transformation formula(flow transformation, Feynman-Kac representationformula etc.) and is therefore rather flexible and ap-plicable in various contexts, also when such a trans-formation is no longer available. The talk is basedon a joint work with Aurelien Deya, MassimilianoGubinelli and Samy Tindel.

Multiplicative Stochastic HeatEquations on the Whole Space

Cyril LabbeUniversity Paris Dauphine, FranceMartin Hairer

The theory of regularity structures allows one to givea meaning to some singular stochastic PDEs suchas the Kardar Parisi Zhang (KPZ) equation or theparabolic Anderson model (PAM) in dimension 2 or3. However, the original framework of the theorydeals with equations on bounded domains while itis very natural to consider these equations on thewhole space. Actually, to measure the regularity ofthe white noise on the whole space, one needs toadd weights in the spaces of Holder distributions.These weights do not behave well under multiplica-tion: this induces some serious di�culties for solv-ing these stochastic PDEs. In collaboration withMartin Hairer, we have adapted the theory of reg-ularity structures to construct the solutions of somemultiplicative stochastic heat equations on the wholespace, such as (PAM) in dimension 3. Moreover, ourresult allows one to consider irregular initial condi-tions: in the case of (PAM), we can start the equationfrom a Dirac mass, which is a natural initial conditionfor this model.

Perturbation and Homogenization

Xue-Mei LiThe University of Warwick, England

I will discuss stochastic homogenization motivatedby geometry and mathematical physics. In particu-lar we consider the family of operators of the form(1/✏2)L

1

+ (1/✏)L0

with a parameter ✏, where L0

is a first order di↵erential operator, L1

a sum ofsquares Hormander type second order operator sat-isfying Hormander’s conditions. These can be con-sidered as random perturbation to a conserved quan-tity taking value in a non-linear space. Examples fallinto this category include: random perturbation togeodesic equations, approximation of Brownian mo-tion by a random Hamiltonian system, and collapsingof Riemannian manifolds.

Linking Regularity Structures andParacontrolled Analysis

Joerg MartinHU Berlin, GermanyNicolas Perkowski

Quite recently the study of singular SPDEs has pro-duced remarkable new theories such as Hairer’s regu-larity structures and the usage of paracontrolled dis-tributions by Gubinelli, Imkeller and Perkowski. Wedemonstrate how both theories can be combined andshow a universal link between them, thereby enablingthe usage of classical, analytical tools in Hairer’s al-gebraic framework.


Regularization by Noise for Stochas-tic Scalar Conservation Laws

Mario MaurelliWIAS Berlin & TU Berlin, ItalyBenjamin Gess

We say that a regularization by noise phenomenonoccurs for a possibly ill-posed di↵erential equationif this equation becomes well-posed (in a pathwisesense) under addition of noise. Most of the resultsin this direction are limited to SDEs and associatedlinear SPDEs.In this talk, we show a regularization by noise resultfor a nonlinear SPDE, namely a stochastic scalar con-servation law on Rd with a space-irregular flux:

@tv + b ·r[v2] +rv � W = 0,

where b = b(x) is a given deterministic, possibly ir-regular vector field, W is a d-dimensional Brown-ian motion (� denotes Stratonovich integration) andv = v(t, x,! ) is the scalar solution. More preciselywe prove that, under suitable Sobolev assumptionson b and integrability assumptions on its divergence,the equation admits a unique entropy solution. Theresult is false without noise.The proof of uniqueness is based on a careful com-bination of arguments used in the linear case: firstwe show the renormalization property for the kineticformulation of the equation, then we use second orderPDE estimates and a duality argument to conclude.If time permits, we will discuss also some open ques-tions.

The Enskog Process

Barbara RuedigerUniversity Wuppertal, GermanyS. Albeverio, P. Sundar

The existence of a weak solution to a McKean-Vlasovtype stochastic di↵erential system corresponding tothe Enskog equation of the kinetic theory of gases isestablished under natural conditions. The distribu-tion of any solution to the system at each fixed timeis shown to be unique. The existence of a probabilitydensity for the time-marginals of the velocity is ver-ified in the case where the initial condition is Gaus-sian, and is shown to be the density of an invariantmeasure.

Generalized Couplings and Conver-gence of Transition Probabilities

Michael ScheutzowTU Berlin, GermanyAlexei Kulik

We provide su�cient conditions for the uniquenessof an invariant measure of a Markov process as wellas for the weak convergence of transition probabil-ities to the invariant measure. Our conditions areformulated in terms of generalized couplings.

Exponential Stability of SPDEDriven by a Fractional BrownianMotion

Bjorn SchmalfußFriedrich Schiller University of Jena, Germany

We consider the S(P)DE on some Hilbert space V

du = Audt+ F (u)dt+G(u)d!, u(0) = u0

2 V

where u ⌘ 0 solves this equation. A, F, G satisfy par-ticular regularity conditions. ! is a fractional Brow-nian motion with a special Hurst parameter. We dis-cuss conditions ensuring the asymptotic stability ofthe trivial solution.

Vector Analysis for Dirichlet Formsand Related Questions

Alexander TeplyaevUniversity of Connecticut, USA

The talk will give an overview of several questionsrelated to the recent progress in vector analysis forDirichlet forms. One of the applications of such anal-ysis is the existence and uniqueness of solutions of thenon-linear heat equation that appears in the hydro-dynamic limit of weakly non-symmetric simple exclu-sion processes on non-smooth spaces. In fact, a largeclass of nonlinear vector PDEs can be defined andstudied on spaces that have no di↵erential structurebut only a Dirichlet form. If time permits, Hodge the-orem, Dirac semigroup, magnetic Schrodinger semi-group and related stochastic analysis, and some esti-mates for SPDEs and infinite dimensional SDEs willbe discussed. This work has been done in collabora-tion with Joe Chen, Masha Gordina, Michael Hinz,Dan Kelleher, Michael Roeckner.


Stability of Solutions and Ergodicityfor Stochastic Local and Nonlocalp-Laplace Equations

Jonas TolleAalto University, FinlandBenjamin Gess, Jonas M. Tolle

We provide a general framework for the stability ofsolutions to stochastic partial di↵erential equationswith respect to perturbations of the drift. More pre-cisely, we consider stochastic partial di↵erential equa-tions with drift given as the subdi↵erential of a con-vex function and prove continuous dependence of thesolutions with regard to random Mosco convergenceof the convex potentials. To this aim, we identify theconcept of stochastic variational inequalities (SVI) asa well-suited framework to study such stability prop-erties. In particular, we provide an SVI treatmentfor stochastic singular nonlocal p-Laplace equationsand prove their convergence to the respective localmodels. Furthermore, ergodicity for local and nonlo-cal stochastic singular p-Laplace equations is proved,without restriction on the spatial dimension and forall p 2 [1, 2). This generalizes previous results from[Benjamin Gess, J. M. T.; JMPA (2014)], [Wei Liu,J. M. T.; ECP (2011)], [Wei Liu; JEE (2009)]. Inparticular, the results include the multivalued case ofthe stochastic nonlocal and local total variation flows.Under appropriate rescaling, the convergence of theunique invariant measure for the nonlocal stochasticp-Laplace equation to the unique invariant measureof the local stochastic p-Laplace equation is proved.

The presentation is based on the following works:1. Benjamin Gess, J. M. T., Stability of solutions tostochastic partial di↵erential equations, Journal ofDi↵erential Equations, Volume 260, Issue 6 (2016),pp. 4973–5025.2. Benjamin Gess, J. M. T., Ergodicityand local limits for stochastic local and nonlo-cal p-Laplace equations, preprint (2015), 28 pp.,http://arxiv.org/abs/1507.04545

Stochastic Maximal Regularity forEquations with Adapted Drift

Mark VeraarTU Delft, Netherlands

In this talk we present recent results on optimal regu-larity estimates and existence and uniqueness resultsfor parabolic SPDEs with multiplicative noise. Theresults are a far reaching extension of previous re-sults of Krylov and collaborators and results of vanNeerven and Weis and the speaker.


Special Session 42: Dynamics of Evolution Equations in Viscoelasticityand Thermoelasticity

To Fu Ma, University of Sao Paulo, BrazilMarcelo Moreira Cavalcanti, State University of Maringa, Brazil

Yuming Qin, Donghua University, Peoples Rep of China

This section is devoted to the dynamics of classical and new systems in viscoelasticity and thermoelastic-ity. They include models with, for instance, fading memory, heat waves, localized damping, delay, nonau-tonomous forcing and nonlocal terms. On these models we discuss qualitative properties as, for instance,exponential stability, analyticity of linear system, global attractor and singular limits. Such problems areimportant to the real world applications and represent a major research subject in PDEs.

Exponential Decay Estimates for theDamped Defocusing SchrodingerEquation in Exterior Domains

Wellington CorreaFederal Technological University of Parana, BrazilNicolas Burq, Marcelo Moreira Cavalcanti,Valeria Neves Domingos Cavalcanti

In this work, we study the well-posedness as well asthe exponential stability in H1�level for the dampeddefocusing Schrodinger equation posed in a two di-mensional exterior domain ⌦ with smooth boundary@⌦. The proofs of the well-posedness are based onproperties of pseudo-di↵erential operators introducedin Dehman, Gerard and Lebeau and Brezis – Gal-louet’s inequality, while the exponential stability isachieved combining arguments firstly considered byZuazua for the wave equation adapted to the presentcontext and a global uniqueness theorem.

New Decay Rates for the Magneto-Thermo-Elastic System

Cleverson da LuzFederal University of Santa Catarina, BrazilJauber Cavalcante de Oliveira

In this work we study the asymptotic behavior ofsolutions for a Cauchy problem associated to amagneto-thermo-elastic system. We improve resultson decay rates considering weaker regularity on theinitial data when compared to previous works in theliterature. We also improve the method developed byR. C. Charao, C. R. da Luz and R. Ikehata (2013) inworks for the wave equation and plates, extending itfor this coupled system of mixed hyperbolic-parabolicpartial di↵erential equations.

Stability for a Transmission Prob-lem in Thermoelasticity with SecondSound.

Hugo Fernandez SareUniv. Federal do Rio de Janeiro, BrazilReinhard Racke, Jaime E. Munoz Rivera

We consider a semilinear transmission problem for acoupling of an elastic and a thermoelastic material.The heat conduction is modeled by the Cattaneo lawremoving the physical paradox of infinite propaga-tion speed of signals. The damped, totally hyperbolicsystem is shown to be exponentially stable, and theexistence of a global attractor is shown.

Attractors for a Class of ExtensibleBeam Models with Nonlocal Non-linear Damping

Marcio Jorge SilvaState University of Londrina, BrazilVando Narciso

In this talk we consider some results on well-posedness and long-time dynamics for a class of ex-tensible beam models with nonlocal nonlinear damp-ing posed on bounded domains. This kind of non-linear damping given by the product of two nonlin-ear terms constitutes a mathematical generalizationof nonlocal frictional dissipations first proposed forplate models. Our main results encompass the exis-tence of attractors to the dynamical system associ-ated with the model. Moreover, we also analyze thequalitative properties of such attractors.

Vanishing Viscosity Limit of theCompressible Isentropic Navier-Stokes Equations with DegenerateViscosities

Yachun LiShanghai Jiao Tong University, Peoples Rep of China

In this talk I will first establish the local-in-timewell-posedness of the unique regular solution to thecompressible isentropic Navier-Stokes equations withdensity-dependent viscosities in a power law and withvacuum appearing in some open set or at the far field,then after establishing uniform energy-type estimates


with respect to the viscosity coe�cients for the regu-lar solutions we prove the convergence of the solutionof the Navier-Stokes equations to that of the Eulerequations with arbitrarily large data containing vac-uum. This is a joint work with Yongcai Geng andShengguo Zhu.

Dynamics of Wave Equations withDegenerate Memory

To Fu MaUniversity of Sao Paulo, BrazilM. M. Cavalcanti, L. H. Fatori

This talk is concerned with the long-time dynamicsof viscoelastic wave equations with some mild degen-eracy in the memory kernel. The existence of a globalattractor is obtained by adding locally complemen-tary frictional damping.

Uniform Analyticity and Expo-nential Decay of the SemigroupAssociated with a ThermoelasticPlate Equation with Perturbedboundary conditions

Louis TebouFlorida International University, USA

In a bounded domain, we consider an Euler-Bernoullitype thermoelastic plate equation with perturbedboundary conditions. The boundary conditions aresuch that when the perturbation parameter goes toinfinity, we recover the hinged boundary conditions,while one recovers the clamped boundary conditionswhen the perturbation parameter goes to zero. Re-lying on resolvent estimates, we show that the un-derlying semigroup is uniformly, with respect to theperturbation parameter, analytic and exponentiallystable. The main features of our proof are: appropri-ate decompositions of the components of the systemand the use of Lions‘ interpolation inequalities.

Large Time Behavior for the Three-Dimensional Generalized Benjamin-Bona-Mahony Equation with LargeInitial Data

Yutong WangShanghai Jiao Tong University, Peoples Rep ofChinaWeike,Wang

In this talk, we consider the global existence as wellas the optimal decay estimates of the Cauchy prob-lem for the 3-dimensional Benjamin-Bona-Mahonyequation with large initial data. The results are ob-

tained by Green’s function method, Fourier analysismethod and energy method combined with the time-frequency decomposition method. We mainly focuson some di�culties in our problem. First of all, welose a good term- a dissipation term� 2u in our equa-tion, which tend to make the decay better, so we haveto divide it into two parts to consider them sepa-rately. What’s more, there is no maximum principlelike the viscous Burgers equation, which increases thedi�culty. And even worse, we even can not simplyuse the energy method to get the L1 bounded es-timate. To overcome the di�culties, we first provethe convolution property of the Green’s function G,then by combining it with variable substitution andGreen’s function method, we obtain the L1 and L1

bounded estimate. Now since we get the Lp boundedestimate, the optimal decay can be obtained by thetime-frequency decomposition method.

Zero Dielectric Constant Limitfor Systems of an ElectromagneticFluid

Xin XuShanghai Jiao Tong University, Peoples Rep of China

The magnetohydrodynamic equations were obtainedas the singular limit of the complete equations for anelectrically conducting compressible fluid at the van-ishing of the dielectric constant. We want to give arigorous justification to this singular limit.

Pullback Attractors for 2D Navier-Stokes Equations with Inhomo-geneous Boundary Conditions OrDelay on Lipschitz Domain

Xinguang YangHenan Normal University, Peoples Rep of China

The Navier-Stokes equations give the essential law ofthe fluid flow. Based on the global wellposedness,we derive the pullback attractors for incompressible2D Navier-Stokes equations with nonhomogeneousboundary conditions or multi-delays on Lipschitz do-main.


Special Session 43: Long Time Dynamics, Numerical Analysis andControl of Evolutionary Systems

Louis Tebou, Florida International University, USACiprian Gal, Florida International University, USA

Theodore Tachim, Florida International University, USA

Recent developments in the mathematical analysis of distributed parameter systems will be explored. Aspecial attention will be paid to physically relevant models. Problems to be addressed include, but are notlimited to, well-posedness, global behavior, numerical analysis, and control of such models. Possible areasof application encompass fluid dynamics, heat transfer, acoustics,...

On the Upper Semicontinuity ofthe Global Attractor for NonlinearParabolic Equations with LargeDi↵usion

Maria AstudilloState University of Maringa, BrazilMarcelo M. Cavalcanti

In this talk we discuss the asymptotic behavior (interms of attractors) of a class of nonlinear parabolicproblems involving porous medium type equations asthe di↵usion coe�cient becomes large. We prove theconvergence of the solutions of a homogeneous Neu-mann problem associated to a class of porous mediumtype equations with relative general conditions on thereaction term, as the di↵usion coe�cient goes to in-finity. Moreover, we prove the upper semicontinuityof the associated global attractor.

Analysis and Sensitivity in the Lam-ina Cribrosa

Lorena BociuNC State University, USAGiovanna Guidoboni, Riccardo Sacco, JustinWebster

We consider a nonlinear system of PDEs that modelsfluid flow through poro-visco-elastic material. Theability of the fluid to flow within the solid is de-scribed by the permeability tensor, which is assumedto vary nonlinearly with the volumetric solid strain.We study the problem of existence of weak solutionsin bounded domains with mixed boundary condi-tions, accounting for non-zero volumetric and bound-ary forcing terms. Moreover, we investigate the in-fluence of viscoelasticity on the smoothness of thesolution and on the regularity requirements for theforcing terms. The theoretical results are comple-mented with numerical simulations, and interpretedin the context of the lamina cribrosa in the eye andthe connection between its biomechanics and the de-velopment of glaucoma.

Nonlinear Dynamics and Stabilityin an Asset Flow Model

Gunduz CaginalpUniv of Pittsburgh, USA

Modeling traders who focus on price trend as well asfundamentals yields a curve of equilibria, rather thana single point that one has in classical economics.With di↵erent groups assessing value and using dif-ferent strategies, there is a curve of equilibria thatcontains both stable and unstable points. Instabilityis fostered by momentum traders focused on a smalltime scale. Stability and bifurcation properties arecharacterized by the strategies of the traders.Most of this work is in collaboration with M. DeSan-tis and D. Swigon.

Unilateral Problem for the WaveEquation with Spatial-Time Degen-erate Nonlinear Damping: Well-Posedness and exponential stability

Marcelo CavalcantiState University of Maringa, BrazilValeria Domingos Cavalcanti, Marcio A.Jorge Silva, Luci H. Fatori

A unilateral problem related to a wave model with aspatial-time degenerate damping on a compact Rie-mannian manifold is considered. Our results arenew and concern two main issues: (a) to establishthe well-posedness of the variational problem; (b) toshow that the corresponding energy decays exponen-tially to zero under sharp conditions of zone for thee↵ect of dissipativity. These conditions are used bymeans of an observability condition related to thesharp zone where the localized damping is acting.


Well-Posedness and Uniform Sta-bility for Nonlinear SchrodingerEquations with Dynamic/WentzellBoundary Conditions

Wellington CorreaFederal Technological University of Parana, BrazilM. M. Cavalcanti, I. Lasiecka, C. Lefler

Schrodinger equation with a defocusing nonlinearterm and dynamic boundary conditions defined on asmooth boundary of a bounded domain in dimensionsN = 2, N = 3 is considered. Local well-posednessof strong H2 solutions is established. In the caseN = 2 local solutions are shown to be global. Inaddition, existence of weak H1 solutions in dimen-sions N = 2, N = 3 is also shown. The energycorresponding to weak solution is shown to satisfyuniform decay rates under appropriate monotonicityconditions imposed on the nonlinear terms appear-ing in the dynamic boundary conditions. The proofof well-posedness is critically based on converting theequation into Wentzell boundary value problem as-sociated with Schrodinger dynamics. The analysis ofthis later problem with nonhomogeneous boundarydata allows to build a theory suitable for the treat-ment of the dynamic boundary conditions.

New Phenomena for the Null Con-trollability of Parabolic Systems:Minimal Time and GeometricalDependence.

Luz de TeresaUNAM, Mexico

We consider the distributed null controllability prob-lem for two coupled parabolic equations with a space-depending coupling term. We exhibit a minimal timeof T

0

2 [0,1) such that the corresponding system isnull controllable at any time T > T

0

and is not nullcontrollable if T¡T

0

. This minimal time depends onthe relative position of the control interval and thesupport of the coupling term. We also prove that,for a fixed control interval and a time ⌧

0

2 [0,1),there exist coupling terms such that the associatedminimal time is ⌧

0

Exponential Stability for the WaveEquation with Degenerate NonlocalWeak Damping

Valeria Domingos CavalcantiState University of Maringa, BrazilM. M. Cavalcanti, M. A. Jorge Silva, C. M.Webler

A damped nonlinear wave equation with a degener-ate and nonlocal damping term is considered. Wellposedness results are discussed, as well as the expo-nential stability of the solutions. The degeneracy ofthe damping term is the novelty of this stability ap-proach.

Exact Boundary Control for 1-DWave and Schrodinger Equations

Julian EdwardFlorida International University, USASergei Avdonin

We consider the problem of boundary control for aone dimensional wave equation with N interior pointmasses. We assume the control is at the left end,and the string is fixed at the right end. Singulari-ties in waves are smoothed out to one order as theycross a point mass. We show that the reachable setfor a L2 control equals (L2 ⇥ H�1) ⇥ (H1 ⇥ L2) ⇥... ⇥ (HN ⇥ HN�1) plus some compatibility condi-tions. We prove exact controllability results. Theproof reduces the control problem to a moment prob-lem, which is then solved using the theory of expo-nential divided di↵erences in tandem with a uniqueshape controllability result. The methods are thenextended to Schrodinger-wave type equations withstrong potential singularities. This is work done incollaboration with Sergei Avdonin.

Semi-Lagrangian Forward Methodsfor Time-Dependent Partial Di↵er-ential Equations

Daniel GuoUniversity of North Carolina Wilmington, USA

One-step semi-Lagrangian forward method is investi-gated for computing the numerical solutions of time-dependent partial di↵erential equations with initialand boundary conditions. This method is based onLagrangian trajectory or the integration from the de-parture points (regular nodes) to the arrival points.The arrival points are traced forward from the depar-ture points along the trajectory of the path. Mostlikely the arrival points are not on the regular gridnodes. The convergence and stability are studied forthe explicit methods. The numerical examples showthat those methods work very e�cient for the time-dependent partial di↵erential equations.

Null Boundary Controllability ofa Heat Equation with an InternalPoint Mass

Scott HansenIowa State University, USAJose de Jesus Martinez

We consider a one dimensional heat equation with aninternal point mass. The dynamics at the point massare obtained as a limit of a sequence of heat equa-tions with densities that tend to a delta function. Weshow that the system is null controllable when a con-trol acts at one end. We also describe some relatedresults for the analogous Schrodinger system.


Suspension Bridge: a Semi LinearModel

Salim MessaoudiKing Fahd University of Petroleum and Minerals,Saudi ArabiaSoh Edwin

In a statistics appeared in the University of Cam-bridge 2004, an estimate of 400 suspension bridgeshave collapsed after construction. Some of these fail-ures are due to human error while others from natu-ral disaster such as overloading of building materials,tsunami and earthquake. Thus, there is a necessityfor reliable mathematical models to give a precise de-scription of its instability and structural behavior. Afirst attempt to model a suspension bridge through aplate is due to Ferrero-Gazzola, who introduced re-cently the hyperbolic equation with some unique andnovel boundary conditions and proved some existenceand regularity results. In this talk, we discuss twovariants of this model and establish some existenceand stability results.

Cahn-Hilliard Inpainting

Alain MiranvilleUniversity of Poitiers, France

Our aim in this talk is to discuss variants of the Cahn-Hilliard equation in view of applications to image in-painting. We will present theoretical results as wellas numerical simulations.

The Inverse Spectral Problem forthe Wave Equation on Finite Trees

Gulden MurzabekovaKazakh Agrotechnical University, KazakhstanSergei Avdonin

We consider the inverse spectral problem for di↵eren-tial equations on graphs. Leaf-peeling method allowsrecalculate the inverse data from the original tree tosmaller tree, and so on to the roots edge. We describethe main step of the spectral version of peeling algo-rithm. This is work done in collaboration with SergeiAvdonin.

The Problem of Recovering Func-tion with Leaf-Peeling Method

Karlygash NurtazinaEurasian National University, KazakhstanSergei Avdonin, Jonathan Bell

In this talk we discuss how unknown coe�cients andsource terms for a parabolic equation can be re-covered from the dynamical Neumann-to-Dirichletmap associated with the boundary vertex. We show

that with a companion wave equation problem thetopology of the graph and lengths of the edges canbe recovered from the same dynamical Neumann-to-Dirichlet map. This is work done in collaborationwith Sergei Avdonin and Jonathan Bell.

Long Time Dynamics of Au-tonomous and Non-AutonomousReaction- Di↵usion Equations withRobin Boundary Condition

Eylem OzturkHacettepe University, Turkey

We investigated the long-time behavior and solvabil-ity of the reaction-di↵usion equation, which has apolynomial growth nonlinearity of arbitrary order,with Robin boundary condition on the bounded do-main.The problem that we investigate as the following:8<

:

ut � �u+ a(x)|u|⇢u� b(x)|u|⌫u = h(x, t), (x, t) 2 QT (1)( @u@⌘

+ k(x0)u)|@⌦ = 0, (x0, t) 2 ⌃T (2)u(x, 0) = u

0

(x), x 2 ⌦ (3)

here⌦ ⇢ Rn(n � 3), is a bounded domain suchthat @⌦ su�ciently smooth boundary, T > 0, QT =⌦ ⇥ (0, T ) and ⌃T = @⌦ ⇥ [0, T ]. In super linearcase, for the existence and uniqueness of the gen-eralized solution of problem (1)-(3) in correspond-ing spaces, we obtain su�cient conditions for func-tions a, b and k and relations between ⇢ and ⌫.Forthe long-time behavior, firstly we prove that solu-tions has an absorbing set in L

2

(⌦). Also in au-tonomous case, we prove some asymptotic regular-ity of solutions and the existence of global attractorin W 1

2

(⌦)\L⇢+2

(⌦) by using Moser’s iteration tech-nique and corresponding stationary problem. In non-autonomous case, the existence of uniform attractoris obtained in W 1

2

(⌦) \ L⇢+2

(⌦).

Longtime Behavior of Solutionsfor Nonlocal Wave Equations withDamping

Petronela RaduUniversity of Nebraska-Lincoln, USAGrozdena Todorova, Borislav Yordanov

Nonlocal wave equations with damping have only re-cently started to be explored in the context of peridy-namics and other theories that allow solutions to bediscontinuous. In this talk I will focus on results thatconnect the asymptotic behavior of solutions to dissi-pative wave equations to solutions of the correspond-ing di↵usion equations, more precisely, show that theabstract di↵usion phenomenon takes place. The re-sults hold true in fact for systems that involve twonon-commuting self-adjoint operators in a Hilbertspace. When the di↵usion semigroup has the Markovproperty and satisfies a Nash-type inequality, we ob-tain precise estimates for the consecutive di↵usionapproximations and remainder. Also, I will presentsome applications including sharp decay estimates fordissipative hyperbolic equations with variable coe�-cients on an exterior domain. To our knowledge we


have obtained the first decay estimates for nonlocalwave equations with damping terms; the decay ratesare sharp. Some of these results have been obtainedin collaboration with Grozdena Tododrova and BorisYordanov.

Some Robust Control ProblemsAssociated with the Cahn-Navier-Stokes System

Theodore Tachim MedjoFlorida International University, USA

In this work we study a class of robust control prob-lems associated with a coupled Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded domain.The model consists of the Navier-Stokes equations forthe velocity, coupled with the Cahn-Hilliard modelfor the order (phase) parameter. We prove the ex-istence and uniqueness of solutions and we derive afirst-order necessary optimality condition for theserobust control problems. We also present an adjoint-based iterative method for the numerical approxima-tion of these control problems.

Long Time Stability of the ImplicitEuler Scheme for an IncompressibleTwo-Phase Flow Model

Florentina ToneUniversity of West Florida, USATheodore Tachim Medjo

In this talk we present results on the stability for allpositive time of the fully implicit Euler scheme foran incompressible two-phase flow model. More pre-cisely, we consider the time discretisation scheme andwith the aid of the discrete Gronwall lemma and ofthe discrete uniform Gronwall lemma we prove thatthe numerical scheme is stable.

The Discrete Inf-Sup Inequalityfor a Finite Element Hydro-ElasticModel

Daniel ToundykovUniversity of Nebraska-Lincoln, USAGeorge Avalos

A seminal result concerning mixed finite element ap-proximations of the Stokes equation was the discreteinf-sup inequality, uniform with respect to the (small)discretization mesh parameter h. This inequality ul-timately leads to error estimates for the convergenceof the FEM scheme. An essential requisite ingredientfor this result was the no-slip condition imposed onthe entire boundary of the fluid domain. On the otherhand, in fluid-structure interaction problems, the in-terface between the fluid and the solid is subject tovelocity and stress matching constraints which do notenforce the no-slip condition. Accordingly, we estab-lish the discrete inf-sup estimate for the case whenthe no-slip condition holds only on a portion of theboundary, thus paving a way to FEM convergenceestimates for fluid-structure interaction problems.


Special Session 44: Fractal Geometry, Dynamical Systems, and TheirApplications

Michael Barnsley, Australian National University, AustraliaJames Keesling, University of Florida, USA

Mrinal Kanti Roychowdhury, University of Texas Rio Grande Valley, USA

The aim of this session is to bring together scientists including the young researchers to discuss and ex-change ideas in the areas of fractal geometry, dynamical systems, and their recent advances and emergingapplications.

Interleaving of Path Sets

William AbramHillsdale College, USAArtem Bolshakov, Je↵rey Lagarias, DanielSlonim

We study an interleaving operation on path sets,which are spaces of one-sided infinite symbol se-quences corresponding to the one-sided infinite walksbeginning at a fixed initial vertex in a labeled graphG. Path sets are useful for the study of intersec-tions of fractals, and have been used to study inter-sections of multiplicative translates of 3-adic Cantorsets. They have also been used by Ban and Chang tostudy the mosaic solutions to the initial value prob-lem of multi-layer cellular neural networks. Inter-leaving is a kind of multiplication for path sets thathas proved useful in computations. We study its dy-namical and algebraic properties and provide severalexamples.

Some Complexity Results in theTheory of Normal Numbers

Dylan AireyUniversity of Texas at Austin, USAStephen Jackson, Bill Mance

Let N (b) be the set of real numbers that are normalin base b. G. Rauzy characterized the set N?(b) ={r : for all x 2 N (b), r + x 2 N (b)} in terms ofan entropy condition. Using this characterization wedetermine the descriptive set theoretic complexity ofN?(b) and related sets.

Topological Speedups of Odometersand Substitutions

Lori AlvinBradley University, USADrew Ash, Nic Ormes

Given a minimal Cantor system (X,T ), one can de-fine a topological speedup of (X,T ) as a system(X,S) where S = T p(·) is a minimal homeomorphismfor some p : X ! Z+. We investigate when theminimal Cantor system (X,T ) is topologically conju-gate to a non-trivial speedup of itself. We show thatwhen (X,T ) is an odometer and (X,S) is a bounded

speedup of (X,T ), then (X,S) is topologically conju-gate to (X,T ). On the other hand, when (X,T ) is asubstitutive system and (X,S) is a bounded speedupof (X,T ), then (X,S) is a substitutive system thatis not topologically conjugate to (X,T ).

Approximation of Fractal Functionsand Fractal Flows

Michael BarnsleyAustralian National University, AustraliaC. Bandt, A. Vince

I will describe a constellation of results and ideas con-cerning the approximation of rough objects and flowson fractals.

Nonequilibrium Stationary State ina Weakly Driven System

Federico BonettoGeorgia Tech, USAJoel Lebowitz

We study a system of one or more particles movingin a chaotic billiard under the influence of an exter-nal electric field. A friction term, in the form of aGaussian thermostat, is added to keep the total ki-netic energy exactly constant. We study the invariantmeasure of this system both analytically and numeri-cally. We also introduce a simplified stochastic modelto better understand the long time evolution of thesystem.

On Dynamics of the Sierpinski Car-pet

Jan BoronskiIT4 Innovations, Ostrava, Czech RepP. Oprocha

In 1993, Aarts and Oversteegen proved that theSierpinski carpet S admits a transitive homeomor-phism, answering a question of Gottschalk. Theyalso showed that it does not admit a minimal one.Earlier, in 1991 Kato proved that S does not admitexpansive homeomorphisms. In 2007 Bis, NakayamaandWalczak proved that S admits a homeomorphismwith positive entropy, and that it admits a minimalgroup action. We show that S admits homeomor-phisms with strong mixing properties. Namely, thereis a homeomorphism H : S ! S that has a fully sup-ported measure m, such that (H,m) is Bernoulli, H


has a dense set of periodic points, and H does nothave Bowen’s specification property. In particular, Sadmits a topologically mixing homeomorphism. Thestarting point of our construction is Arnold’s catmap.

Ramsey-Type Results for Dynami-cal Systems

Will BrianBaylor University, USA

In certain dynamical systems, arbitrary sequences ofpoints can exhibit surprising amounts of structure. Inother words, it seems that complete disorder is im-possible inside some dynamical systems, even highlychaotic ones. We will discuss several results of thistype.

Non-Linear Analysis of LogisticMaps

Renu ChughMaharishi Dayanand University, Rohtak, India

The logistic map describes all possible behaviors of anonlinear system. The idea of logistic map rx(1� x)was given by the Belgian mathematician Pierre Fran-cois Verhulst around 1845 and worked as basic modelto study the discrete dynamical system. It is a modelof population growth that exhibits di↵erent types ofbehavior depending on the value of a few parameters.Above a certain parameter value, the logistic mapshows the chaotic behavior. For choosing x between0 and 1 and 0 < r 4, the logistic map has found acelebrated place in chaos, fractal and discrete dynam-ics. The aim of this talk is to study the periodicityand chaotic behavior of logistic map using Mann iter-ative process (a two-step feedback method). We seethat the nature of chaotic behavior of logistic mapsincreases drastically as compared to Picard iterativeprocess and the chaotic behavior disappears for someranges of parameter values.

Orbit Portraits for Non-Autonomous Iteration

Mark ComerfordUniversity of Rhode Island, USA

The combinatorics associated with angles of externalrays has been one of the most important tools used tounderstand parameter spaces associated with fami-lies of rational functions, most particularly quadraticpolynomials. We show how the notion of an orbitportrait as first introduced by Milnor can be gener-alized to the setting of non-autonomous polynomialiteration where one studies iterates which are com-positions of a sequence of polynomials with suitablybounded degrees and coe�cients. In the case of se-quences of constant degree, all portraits are eventu-ally periodic which is similar to, though not exactlythe same as, the classical case. On the other hand, if

the degrees of the polynomials in the sequence are al-lowed to vary, we show that one can obtain portraitswith complementary intervals whose angles are irra-tional multiples of 2⇡ which are fundamentally dif-ferent from the classical ones.

How Sticky Is the Chaos/orderBoundary?

Carl DettmannUniversity of Bristol, England

In dynamical systems with divided phase space, thevicinity of the boundary between regular and chaoticregions is often “sticky,” that is, trapping orbits fromthe chaotic region for long times. Here, we inves-tigate the stickiness in the simplest mushroom bil-liard, which has a smooth such boundary, but sur-prisingly subtle behaviour. As a measure of sticki-ness, we investigate P (t), the probability of remain-ing in the mushroom cap for at least time t givenuniform initial conditions in the chaotic part of thecap. The stickiness is sensitively dependent on theradius of the stem r via the Diophantine proper-ties of ⇢ = (2/⇡) arccos r. Almost all ⇢ give rise tofamilies of marginally unstable periodic orbits (MU-POs) where P (t) ⇠ C/t, dominating the stickinessof the boundary. Here we consider the case where⇢ is MUPO-free and has continued fraction expan-sion with bounded partial quotients. We show thatt2P (t) is bounded, varying infinitely often betweenvalues whose ratio is at least 32/27. When ⇢ hasan eventually periodic continued fraction expansion,that is, a quadratic irrational, t2P (t) converges toa log-periodic function. In general, we expect lessregular behaviour, with upper and lower exponentslying between 1 and 2. The results may shed light onthe parameter dependence of boundary stickiness inannular billiards and generic area preserving maps.

Algebro-Geometric Solutions of theSchlesinger Systems

Vladimir DragovicThe University of Texas at Dallas, USAVasilisa Shramchenko

A new method to construct algebro-geometric solu-tions of rank two Schlesinger systems is presented.For an elliptic curve represented as a ramified dou-ble covering of CP 1, a meromorphic di↵erential isconstructed with the following property: the com-mon projection of its two zeros on the base of thecovering, regarded as a function of the only mov-ing branch point of the covering, is a solution of aPainleve VI equation. This di↵erential provides aninvariant formulation of a classical Okamoto trans-formation for the Painleve VI equations. A gener-alization of this di↵erential to hyperelliptic curves isalso constructed. The corresponding solutions of therank two Schlesinger systems associated with ellipticand hyperelliptic curves are constructed in terms ofthis di↵erential. The initial data for construction ofthe meromorphic di↵erential include a point in the


Jacobian of the curve, under the assumption thatthis point has nonvariable coordinates with respectto the lattice of the Jacobian while the branch pointsvary. The research has been partially supported bythe NSF grant 1444147.

Cantor Sets Within Cantor Sets:Hausdor↵ Dimensions of P-AdicJulia Sets

Joanna FurnoIUPUI, USA

Haar measure and Hausdor↵ dimension are two pos-sible methods of measuring size in the set of p-adicnumbers and its finite extensions. We use these twotools to measure the size of the Julia set for somep-adic polynomials.

Climate Change and The FractalGeometry of Arctic Melt Ponds

Kenneth GoldenUniv Utah, USA

During the Arctic melt season, the sea ice surfaceundergoes a remarkable transformation from vast ex-panses of snow covered ice to complex mosaics ofice and melt ponds. Sea ice reflectance or albedo,a key parameter in climate modeling, is largely de-termined by the complex evolution of melt pond con-figurations. In fact, ice-albedo feedback has played amajor role in the recent declines of the summer Arc-tic sea ice pack. However, understanding melt pondevolution remains a significant challenge to improv-ing climate projections. I will discuss recent findingson the evolution of melt pond geometry. In partic-ular, as the ponds grow and coalesce, their fractaldimension undergoes a transition from 1 to about2, around a critical length scale of 100 square me-ters in area. As the ponds evolve they take complex,self-similar shapes with boundaries resembling space-filling curves. Moreover, I will outline how math-ematical models of composite materials and statis-tical physics, such as percolation and Ising models,are being used to describe this evolution and makepredictions of key geometrical parameters that agreevery closely with observations.

Fractal Curves Arising from Cuttingand Resewing Pillow Cases

Patrick HooperCity College of New York, USA

I will discuss the dynamics of a fairly simple piecewiseisometry of a square pillowcase. We cut the pillow-case along two horizontal edges we obtain a cylinder,which we can rotate and then sew back together. Wecan then do the same in the vertical direction. Thecomposition of these two cutting and resewing oper-ations yields a piecewise isometry of the pillowcasewith interesting dynamics. We will describe how insome cases the collection of aperiodic points forms

a fractal curve, and the dynamics on this curve istopologically conjugate to a rotation (modulo con-cerns related to discontinuities). Properties of thismap such as the existence of this curve depend onthe even continued fraction expansions of the param-eters.

Mappings with a Single CriticalPoint and Applications to RationalDi↵erence Equations

Sue HuangPace University, USAPeter Knopf

Convergence properties of maps xn + 1 = f(xn) areestablished for a general class of mappings f ; wheref has at most one critical point. Using these results,necessary and su�cient conditions are obtained forthe convergence of the solutions for a very generalclass of rational quadratic di↵erence equations.

Wavelets on Fractals

Palle JorgensenUniversity of Iowa, USADorin Dutkay

Palle Jorgensen. The class of fractals referred toare those which may be specified by a finite sys-tem of a�ne transformations, assuming contractivescaling; and their corresponding selfsimilar measures,mu. They include standard Cantor spaces such as themiddle third, and the planar Sierpinski caskets in var-ious forms, and their corresponding selfsimilar mea-sures, but the class is more general than this; includ-ing fractals realized in Rd, for d > 2. In part 1, wemotivate the need for wavelets in the harmonic anal-ysis of these selfsimilar measures mu. While classesof the Hilbert spaces L2(µ) have Fourier bases, itis known (the speaker and Pedersen) that many donot, for example the middle third Cantor can haveno more than two orthogonal Fourier frequencies. Inpart 2 of the talk, we outline a construction by thespeaker and Dutkay to the e↵ect that all the a�nesystems do have wavelet bases; this entails what wecall thin Cantor spaces.

Postcritically Finite Maps, LattesMaps, and Symmetrization in CPk

Scott KaschnerButler University, USAThomas Gauthier

We discuss various types of postcritically finite mapsof CPk and present examples of each type. Symmet-ric products have been used to produce examples ofendomorphisms of CPk (k � 2) with certain charac-teristics. We use them in this talk to produce en-domorphisms of CPk that are strongly postcriticallyfinite. We also use them to characterize families ofLattes maps.


Little’s Law Analysis of a StochasticNetwork

James KeeslingUniversity of Florida, USACeleste Vallejo

Little’s Law is a principle that is used in the analysisof stochastic networks. Little’s Law states that if ↵is the arrival rate at a facility, W is the average wait-ing time of an individual in the facility, and n is theaverage number in the facility, then ↵ · W = n pro-vided that the numbers are finite. There are no as-sumptions about the distributions giving rise to thesenumbers or even if any such distributions exist. Theonly assumption is that the limits exist. Little’s Lawis used largely in making some di�cult calculationin a network. In this talk we suggest that the anal-ysis of the flow through a network can be analyzedstarting with Little’s Law rather than starting withvarious stochastic assumptions. The motivation forthis approach was the analysis of the flow of patientsthrough an Emergency Department and through anentire hospital. We give other applications as well.

Topological Entropy in GeneralizedInverse Limits

Judy KennedyLamar University, USAGoran Erceg

We generalize the definition of topological entropydue to Adler, Konheim, and McAndrew to set-valuedfunctions from a closed subset A of the interval toclosed subsets of the interval. We view these set-valued functions, via their graphs, as closed subsetsof [0, 1]2. We show that many of the topological en-tropy properties of continuous functions of a compacttopological space to itself hold in our new setting,but not all. We also compute the topological entropyof some examples, and give an idealized applicationto impulse functions, as these are not functions inthe usual sense, but do occur in models coming fromother branches of science.

Ground States on the Boundary ofRotation Sets

Tamara KucherenkoThe City College of New York, USAChristian Wolf

Ground states are accumulation points of equilibriumstates when the temperature goes to zero. They playa fundamental role in statistical physics. We con-sider rotation sets associated with a continuous dy-namical system on a compact metric space and amulti-dimensional continuous potential. We studythe question for which boundary vectors of the rota-tion set one can realize an entropy maximizing mea-sure as a ground state associated with a certain lin-ear combination of the potential. We show that atan exposed point there always exists a ground state

that maximizes entropy in its rotation class. We alsoconstruct examples of rotation sets (in any dimen-sion ) that have exposed boundary points withouta ground state in its rotation class. Finally, we con-sider non-exposed points and show that the followingtwo phenomena exist: a) boundary points without anassociated ground state; b) boundary points with aunique ground state that is not ergodic.

Estimating the Hausdor↵ Dimen-sion of Minimal Sets in SmoothCounterexamples to the SeifertConjecture

Krystyna KuperbergAuburn University, USA

The smooth counterexamples to the Seifert Conjec-ture posses a large minimal set. We describe amethod of estimating Hausdor↵ dimension of thisminimal set using flow boxes in a similar fashion asin the box-counting method of Minkowski and Bouli-gand.

Fractals: from Laminations to JuliaSets

John MayerUA-Birmingham, USA

Julia sets of complex analytic functions are well-known examples of (almost) self-similar fractals inthe complex plane or Riemann sphere. Laminationsof the unit disk were introduced by William Thurstonas a topological/combinatorial model for understand-ing the (connected) Julia sets of polynomials. Thisunderstanding can be extended, at least partially, tolaminations corresponding to connected Julia sets fordegree d � 2 polynomials; the Julia set is the mono-tone image of the lamination and with semiconju-gate dynamics. Laminations whose critical chordsare not in the lamination are, for that very reason,self-similar. The corresponding Julia set is thus alsoself-similar; the polynomial on its Julia set is a cover-ing map. We focus on laminations of degree 2 and 3,and in particular on degree 3 laminations that con-tain an identity return leaf or triangle. The topologi-cal correspondence between laminations and polyno-mial Julia sets is thus a correspondence of fractals.

Topological Transitivity of Exten-sions of Hyperbolic Systems withInfinite Dimensional Fiber.

Viorel NiticaWest Chester University, USA

We will discuss topological transitivity of extensionsof hyperbolic systems with infinite dimensional fiber.


The E↵ect of Projections on FractalSets and Measures in Banach Spaces

William OttUniversity of Houston, USAZijie Zhou

Many infinite-dimensional dynamical systems ofinterest in the physical sciences admit finite-dimensional invariant attracting sets. This motivatesthe following: Given a compact subset A of a Banachspace X and a typical C1 map ' : X ! Rn, what isthe relationship between the Hausdor↵ dimension ofA and that of '(A)? Do A and '(A) have the sameHausdor↵ dimension (the ideal situation)? If not,what determines the gap between the two values?Ott, Hunt, and Kaloshin answered these questionswhen the ambient space X is a Hilbert space. Inparticular, they showed that Hausdor↵ dimension istypically preserved up to a factor involving the thick-ness exponent ⌧(A). In this talk we provide answersin the general Banach space case. We formulate tworesults - one involving the thickness exponent andone involving the dual thickness exponent. The lat-ter result answers a question posed by Robinson.

A Class of Cubic Rauzy Fractal

Tatiana RodriguesUNESP, BrazilJe↵erson Bastos

In this work we study arithmetical and topologicalproperties for a class of Rauzy fractals Ra given bythe polynomial x3�ax2+x�1 where a � 2 is an inte-ger. In particular, we prove the number of neighborsof Ra in the periodic tiling is equal to 8. We also giveexplicitly an automaton that generates the boundaryof Ra, for a = 2. In this case we calculate the Haus-dor↵ dimension. We also give explicitly an automa-ton that generates the boundary of Ra for a > 2 andusing an exotic numeration system we prove that Ra

is homeomorphic to a topological disk.

Quantization

Mrinal RoychowdhuryUniversity of Texas Rio Grande Valley, USA

Quantization for probability distributions refers tothe idea of estimating a given probability by a dis-crete probability with finite support. Quantizationdimension gives the speed how fast the specified mea-sure of the error goes to zero as the number of pointsin the underlying support goes to infinity. Recently,several works have been done in this direction. I willtalk about it.

Dimension Calculation for an In-variant Measure Supported on aSubfractal

Elizabeth SattlerNorth Dakota State University, USA

In this talk, we will examine properties, includingfractal dimensions, of a subfractal induced by a sub-shift of finite type or sofic subshift. We will con-struct an invariant measure supported on a subfrac-tal induced by a subshift of finite type and discussa method for calculating the Hausdor↵ dimension ofsuch a measure.

Topological Measures of Order forPattern-Forming Systems

Patrick ShipmanColorado State University, USAR. M. Bradley, F. C. Motta, R. Neville, D.Pearson

Exploiting theory and methods from computationaltopology, we introduce several measures which quan-tify the order of nearly hexagonal, planar lattices pro-duced by dynamical systems or systems of PDEs. Forexample, when the surface of a nominally flat binarymaterial is bombarded with a broad, normally inci-dent ion beam, disordered hexagonal arrays of nan-odots can form. Defects, such as dislocations in ripplepatterns or penta-hepta pairs in hexagonal arrays,limit the utility of patterns produced by ion bom-bardment. To evaluate the e�cacy of this method, ameans of quantifying the degree of order is needed.We compare time-honored methods, such as the auto-correlation function, with the new topological meth-ods.

Hereditarily Non Uniformly PerfectSets

Rich StankewitzBall State University, USAToshiyuki Sugawa, Hiroki Sumi

We introduce the concept of hereditarily non uni-formly perfect sets, compact sets for which no com-pact subset is uniformly perfect, and compare thefollowing properties: hereditarily non uniformly per-fect sets, Hausdor↵ dimension zero sets, logarithmiccapacity zero sets, Lebesgue 2-dimensional measurezero sets, and porous sets. In particular, we give anexample of a compact set of positive Hausdor↵ di-mension and positive logarithmic capacity which ishereditarily non uniformly perfect.


Hausdor↵ Dimension of the Ju-lia Sets of Postcritically BoundedPolynomial Semigroups and TheTransversality Condition

Hiroki SumiOsaka University, Japan

We consider the dynamics of the semigroups gener-ated by two polynomial maps {f

1

, f2

} of degree twoor more on the Riemann sphere. Let B be the setof (f

1

, f2

) for which the planar postcritical set of thesemigroup hf

1

, f2

i generated by {f1

, f2

} is bounded.Let C be the set of (f

1

, f2

) for which the Julia setof hf

1

, f2

i is connected. Let H be the set of (f1

, f2

)for which hf

1

, f2

i is hyperbolic. Let I be the set of(f

1

, f2

) for which the Julia sets of f1

, f2

intersect.We show that there exists an open dense subset A of((@C) \ B \H) \ I such that for every (f

1

, f2

) 2 A,there exists an open neighborhood U of (f

1

, f2

) suchthat for almost every (g

1

, g2

) 2 U with respect tothe Lebesgue measure on U , the Hausdor↵ dimen-sion of the Julia set of the semigroup generated by{g

1

, g2

} is equal to the Bowen parameter of (g1

, g2

)(i.e. Bowen’s formula holds). Note that we cannotexpect the open set condition in U. The key idea inthe proof is to show that the transversality conditionholds in U.

S-Adic Dynamical Systems andRauzy Fractals

Jorg ThuswaldnerLeoben University, AustriaP. Arnoux, V. Berthe, M. Minervino, W.Steiner

We study dynamical systems that are defined interms of a sequence of substitutions. We relate RauzyFractals to these dynamical systems. Tiling proper-ties of these fractals allow to conclude that they areconjugate to rotations on the torus. Moreover, thesedynamical systems have relations to generalized con-tinued fraction algorithms like Brun’s algorithm.

Rational Maps from PolynomialMatings Using Combinatorial Meth-ods and Thurston’s Algorithm

Mary WilkersonCoastal Carolina University, USA

‘Topological mating‘ describes an operation thatcombines two complex polynomials to obtain a newdynamical system on the quotient of a 2-sphere. Thedynamics of the mating are then dependent uponthe two polynomials that are mated. While in manycases, we wouldn‘t expect this process to yield a mapor quotient space that is ‘nice‘, the topological matingof two postcritically finite polynomials is frequentlyThurston equivalent to a rational map on the Rie-mann sphere. Given a postcritically finite quadraticpolynomial pair, there is a simple parameter test todetermine whether such a rational map F exists—butthis test is not constructive of F . We present an iter-ative method that uses combinatorial techniques andThurston’s algorithm to construct an approximationto F , and follow with the discussion of an exampleusing this method.

Hitting Time Distribution and Ex-treme Value Law for Flows

Fan YangUFRJ, BrazilMaria Jose Pacifico

For flows whose return map on a cross section hassu�cient mixing property, we show that the hittingtime distribution of the flow to balls is exponential inlimit. We also establish a link between the extremevalue distribution of the flow and its hitting time dis-tribution, generalizing a previous work by Freitas etal in the discrete time case. Finally we show thatfor maps that can be modeled by Young’s tower withpolynomial tail, the extreme value law holds. As ex-amples we consider the classic Lorenz attractor andthe Rovella attractor. This is a joint work with MariaJose Pacifico.

Multi-Chaos

James YorkeUniversity of Maryland, USAS. Das, Y. Saiki

We investigate systems on a torus that have a denseset of periodic saddles and a dense set of periodic re-pellers. Such systems may be a prototype of systemswith higher dimensional chaos.


Special Session 45: Nonlinear Waves and Singularities in Optical andHydrodynamic Systems

Sergey Dyachenko, University of Illinois, USAPavel Lushnikov, University of New Mexico, USA

The session is devoted to recent advances in nonlinear optics, free surface hydrodynamics and others nonlinearsystems. The emphasis is given to the works on singularity formation in Hamiltonian and non-Hamiltoniansystems described by partial di↵erential equations. The examples include but are not restricted to new ad-vances in free surface hydrodynamics, self-focusing of laser beam, filamentation and laser-plasma interaction.

Analyzing the Stability Spectrumfor Elliptic Solutions to the Focus-ing NLS Equation

Bernard DeconinckUniversity of Washington, USA

The one-dimensional focusing cubic nonlinearSchroedinger (NLS) equation is one of the mostimportant integrable equations, arising in a multi-tude of applications. The stability of the stationaryperiodic solutions of NLS is well studied, leading to,for instance, the iconic figure-eight spectrum for itscnoidal wave solutions. We present an explicit ex-pression for the linear stability spectrum of both thetrivial- and nontrivial-phase solutions. We use thisexpression to generate many explicit results aboutthe spectrum.

Finding the Stokes Wave: fromSmall Steepness to the Wave ofGreatest Height

Sergey DyachenkoUniversity of Illinois, USAPavel Lushnikov, Alexander Korotkevich

A Stokes wave is a fully nonlinear wave that trav-els over the surface of deep water. We solve Eu-ler equations with free surface in the framework ofconformal variables via Newton Conjugate Gradientmethod and find Stokes waves in regimes dominatedby nonlinearity. By investigating Stokes waves withincreasing steepness we observe peculiar oscillationsoccur as we approach Stokes limiting wave. Finallyby analysing Pade approximation of Stokes waves weinfer that analytic structure associated with thosewaves has branch cut nature.

Numerical Methods for the Chemo-taxis Models

Yekaterina EpshteynUniversity of Utah, USA

In this talk, I will first introduce and review severalchemotaxis models including the classical Patlak-Keller-Segel model. Chemotaxis is the phenomenonin which cells, bacteria, and other single-cell or multi-cellular organisms direct their movements accordingto certain chemicals (chemoattractants) in their en-vironment. Chemotaxis is an important process inmany medical and biological applications including

bacteria/cell aggregation, pattern formation mecha-nisms, and tumor growth. The mathematical mod-els of chemotaxis are usually described by highlynonlinear time dependent systems of partial di↵er-ential equations (PDEs). Therefore, accurate ande�cient numerical methods are very important forthe validation and analysis of these systems. Fur-thermore, a known property of the existing chemo-taxis models is their ability to describe a concentra-tion phenomenon that mathematically results in so-lutions rapidly growing in small regions of concentra-tion points or curves. The solutions can blow up infinite-time or can develop a singular, spiky behavior.This blow-up represents a mathematical descriptionof a cell concentration phenomenon that happens inreal biological systems. Hence, capturing such solu-tions numerically is a very challenging problem. Inthis presentation, we will introduce and discuss sev-eral recently developed numerical methods for the ap-proximation and simulation of the chemotaxis and re-lated models. Numerical experiments to demonstratethe stability and accuracy of the proposed methodsfor chemotaxis models will be presented. Ongoingresearch projects will be discussed as well.

Stable Solitons of the Cubic-QuinticNLS with a Delta-Function Poten-tial

Francois GenoudTU Delft, NetherlandsBoris Malomed, Rada Weishaupl

This talk is about the one-dimensional nonlinearSchrodinger equation with a combination of cubicfocusing and quintic defocusing nonlinearities, andan attractive delta-function potential. This modelcomes from nonlinear optics, and the delta-functionpotential describes the interaction of a broad soli-tonic beam with a narrow defect. I will show thatall spatial solitons can be determined explicitly interms of elementary functions. Using bifurcation andspectral-theoretic arguments, I will then prove thatthey are all (nonlinearly) stable. A noteworthy fea-ture of the model is a regime of ‘bistability‘, wheretwo solitons with same propagation constant coexist.


Circular Instability of a StandingGravity-Capillary Wave: Theoryand Wave Tank Experiment

Alexander KorotkevichUniversity of New Mexico, USALukaschuk S.

We provide a theory and compare numerical simula-tion of instability of weakly nonlinear standing waveson the surface of deep fluid in the framework of theprimordial dynamical equations and in a laboratorywave tank experiment. The instability o↵ers a newapproach for generation of nearly isotropic spectrumusing parametric excitation. Direct measurements ofspacial Fourier spectrum confirm existence of the in-stability in a real life conditions for gravity-capillarywaves.

Nonlinear Schrodinger Systemswith Non-Zero Boundary Condi-tions

Gregor KovacicRensselaer Polytechnic Institute, USAGino Biondini, Daniel Kraus

The study of scalar and vector nonlinear Schrodinger(NLS) systems with non-zero boundary conditions atinfinity has received renewed interest recently. Thistalk will report on recent results on focusing scalarand vector NLS equations with non-zero boundaryconditions. It will be shown how the inverse scatter-ing transform can be constructed in both cases, and anumber of explicit soliton solutions will be discussed.

Studies of (in)stability of the Nu-merical Method of CharacteristicsApplied to Conservative HyperbolicPDEs

Taras LakobaUniversity of Vermont, USAZ. Deng

The numerical Method of Characteristics (MoC) iswidely used to solve hyperbolic evolution equations.This method reduces the solution of the partial dif-ferential equation (PDE) to that of a small systemof ordinary di↵erential equations (ODEs), where thesolution of each ODE is obtained along ”its” charac-teristic direction. The ODEs are solved by an ODEnumerical solver, such as, e.g., a simple Euler or aRunge–Kutta method.

This study was motivated by our numerical solutionof a set of PDEs describing propagation of light inbirefringent optical fibers:

S+

t +S+

x = S+⇥JS�, S�t �S�

x = S�⇥JS+. (1)

Here S± are Stokes vectors of the fields propagat-ing along each of the principal polarization axes ofthe fiber and J is some diagonal matrix.These equa-tions support propagation of solutions of the domainwall type. It is important to mention that the S±

2

components of these solutions asymptotically tendto *nonzero* constants, while the other componentsasymptotically vanish:

S+

2

(x ! ± 1) ! ±a, S�2

(x ! ± 1) ! ⌥a; S±1, 3(|x| ! 1) ! 0.

(2)It can be shown that Eqs. (1) conserve the norm,|S±|2, of each field. We numerically solved Eqs. (1),supplemented with *non-reflecting* boundary con-ditions (BCs), by the MoC. We used the Leapfrogmethod as the ODE solver, because it is known tonearly conserve energy in Hamiltonian ODEs andthus is expected to be a ”good” method for Eqs. (1).However, we observed a conspicuous numerical in-stability in our simulations. This is the **first unex-pected result** of our study. When we replaced theLeapfrog method with the modified Euler method,which is known *not* to conserve energy in ODEsand hence was expected to be a ”bad” method for thisproblem, the numerical instability of the MoC wasstrongly reduced. This is the **second unexpectedresult** of our study. Before we had attempted toanalyze this behavior, we repeated simulations withthe same two numerical ODE solvers in the MoC butused *periodic* BCs. The reason was that for peri-odic BCs, one could use the von Neumann analysis toinfer stability of the numerical scheme. On the otherhand, as written in many textbooks, we expectedthe BCs to have little to no e↵ect on the numeri-cal (in)stability. Here we observed our **third unex-pected result** : while the instability of the Leapfrogmethod remained the same as for non-reflecting BCs,the near stability of the modified Euler method wasreplaced with an instability about as strong as thatfor the Leapfrog method. We have found analyticalexplanations to all three unexpected results. Sincethe results reported above occurred not only for thedomain-wall solutions but also for the constant solu-tions (2), we carried out our analyses for those sim-pler solutions, for the sake of clarity and simplicity.Thus, first, by means of the von Neumann analysis,we explained why an energy-preserving ODE solver,such as Leapfrog, when combined with the MoC, canproduce a strong instability for periodic BCs. Sec-ond, we explained why changing BCs from periodicto non-reflecting suppresses a relatively strong nu-merical instability of the modified Euler method.


Formation of Limiting Stokes Wavefrom Non-Limiting Stokes Wave:Merging of Square Root BranchPoints from the Infinite Set ofsheets of Riemann surface to form2/3 singularity of limiting wave

Pavel LushnikovUniversity of New Mexico, USA

Stokes wave is the fully nonlinear periodic gravitywave parameterized by its height. Wave of greatestheight has the limiting form with 120 degrees angleon the crest. Assume z(⇣) provides a conformal mapof a free fluid surface of Stokes wave into the real linewith fluid domain mapped into the lower complexhalf-plane of ⇣. Then Stokes wave is fully character-ized by the complex singularities in the upper com-plex half-plane. The only singularity in the physicalsheet of Riemann surface of non-limiting wave is thesquare-root branch point located at ⇣ = i⇣c. Cor-responding branch cut defines the second sheet ofthe Riemann surface if we cross the branch cut. Wefound the infinite number of square root singularitiesin infinite number of non-physical sheets of Riemannsurface. These singularities located both symmet-rically ⇣ = ±i⇣c and on diagonals (with respect tovertical axis) corresponding to di↵erent non-physicalsheets of Riemann surface. Increase of the heightof the Stokes wave means that all these singularitiessimultaneously approach the real line from di↵erentsheets of Riemann surface and merge together form-ing 2/3 power law singularity of the limiting wave. Itwas conjectured (P.M. Lushnikov, ArXiv:1507.02784)that non-limiting Stokes wave z(⇣) at the leading or-der consists of the infinite product of nested squareroot singularities which form the infinite number ofsheets of Riemann surface.

Polarization Switching in a Reso-nant Optical Medium

Katie NewhallUNC Chapel Hill, USA

Optical-pulse polarization switching along an activeoptical medium in the ⇤-configuration is describedusing the idealized integrable Maxwell-Bloch model.Analytical results for the final limiting polarizationshow the dependence on the initial preparation of themedium. Right- and left-circularly polarized light arestable even in the case of spatially disordered occupa-tion numbers in the lower energy sub level pair. How-ever, arbitrary elliptical polarization can be madestable by modifying the initial virtual polarizationbetween the two degenerate lower energy states.

Obtaining Stokes Wave with High-Precision Using Conformal Mapsand Spectral Methods on Non-Uniform Grids

Denis SilantyevUniversity of New Mexico, USAPavel Lushnikov, Sergey Dyachenko

Two-dimensional potential flow of the ideal incom-pressible fluid with free surface and infinite depthhas a class of solutions called Stokes waves whichis fully nonlinear periodic gravity waves propagat-ing with the constant velocity. The increase of thescaled wave height H/L, where H is the wave heightand L is the wavelength, from H/L= 0 to the criti-cal value Hmax/L marks the transition from almostlinear wave to a strongly nonlinear limiting Stokeswave. Fully nonlinear Euler equations describing theflow can be reformulated in terms of conformal mapof the fluid domain into the complex lower half-plane,with fluid free surface mapped into the real line. Thisdescription is convenient for analysis and numericalsimulations since the whole problem is now reducedto a single equation on the real line. We use spec-tral method together with an iterative scheme to ob-tain solutions. Extending solutions to the rest of thecomplex plane one can see that the distance Vc fromthe closest singularity in the upper half-plane to thereal line goes to zero as we approach the limitingStokes wave, which is the reason for the wideningof the solution’s spectrum. This makes us seek anew approach that allows one to overcome this dif-ficulty. We improve performance of our numericalmethod drastically by introducing second conformalmap that pushes the singularity higher into the upperhalf-plane and correspondingly shrinks the spectrumof the solution.

Modeling of Novel Parity-TimeSymmetric Systems.

Alexey SukhininSouthern Methodist University, USA

Rapid emergence of PT-symmetric systems is a newtrend in nonlinear optics. In this talk I will describemodeling techniques in non-Hermitian systems withbalanced gain and loss and discuss the nonlinear gov-erning equations that are used for the analysis.


Cascades in Wave Turbulence

Natalia VladimirovaUniversity of New Mexico, USAGregory Falkovich

We consider developed turbulence in the 2D Gross-Pitaevsky model, which describes wide classes ofphenomena from atomic and optical physics to con-densed matter, fluids and plasma. The well-knowndi�culty of the problem is that the hypothetical lo-cal spectra of both inverse and direct cascades in the

weak-turbulence approximation carry fluxes whichare either zero or have the wrong sign; such spectracannot be realized. We analytically derive the exactflux constancy laws (analogs of Kolmogorov’s 4/5-law for incompressible fluid turbulence), expressedvia the fourth-order moment and valid for any non-linearity. We confirm the flux laws in direct numer-ical simulations. We show that a constant flux isrealized by non-local wave interaction in both thedirect and inverse cascades. Wave spectra (second-order moments) are close to slightly (logarithmically)distorted thermal equilibrium in both cascades.


Special Session 47: Mathematical Contribution Towards theUnderstanding of the Dynamics of the 2014 Ebola Epidemic in West

Africa

Miranda Teboh-Ewungkem, Lehigh University, USAAbba Gumel, Arizona State University, USA

Folashade Agusto, Austin Peay State University, USA

The 2014 West African Ebola epidemic riveted the world, and is the worst yet on record. Even though therehas been a slowdown of the epidemic, there is a risk of resurgence in the near future or sometime later. Thusresearch to fully understand factors that contributed to its spread and control e↵orts that were crucial inmitigating its spread need to continue. Mathematical models were crucial in some of these regards. Theaim of this minisymposium is to highlight the mathematical contributions towards the understanding of thedynamics, spread and control of the 2014 Ebola epidemic.

Mathematical Assessment of theRole of Traditional Belief Systemsand Customs and Health-Care Set-tings on the Transmission Dynamicsof the 2014 Ebola Outbreaks

F AgustoUniversity of Kansas, USAMiranda Teboh-Ewungkem, Abba Gumel

A mathematical model is designed and used to assesspopulation-level impact of basic non-pharmaceuticalcontrol measures on the 2014 Ebola outbreaks. Themodel incorporates the e↵ects of traditional beliefsystems and customs, disease transmission withinhealth-care settings and by Ebola-deceased individu-als. A sensitivity analysis is performed to determinemodel parameters that most a↵ect disease transmis-sion. The model is parameterized using data fromGuinea, one of the three Ebola-stricken countries.Three e↵ectiveness levels of basic public health con-trol measures are considered. The distribution ofthe basic reproduction number (R

0

) for Guinea (inthe absence of basic control measures) is such thatR

0

2 [0.77, 1.35], for the case when the belief systemsdo not result in more unreported Ebola cases. Whensuch systems inhibit control e↵orts, the distributionincreases to R

0

2 [1.15, 2.05]. The total Ebola casesare contributed by Ebola-deceased individuals (22%),symptomatic individuals in the early (33%) and lat-ter (45%) infection stages. The 2014 outbreaks arecontrollable using a moderately-e↵ective basic publichealth intervention strategy alone. A much higher(> 50%) disease burden would have been recorded inthe absence of such intervention.

Modeling the Ebola Outbreak andContact Tracing

Cameron BrowneUniversity of Louisiana at Lafayette, USAHayriye Gulbudak, Glenn Webb

The 2014-2015 Ebola outbreak in West Africa re-sulted in over 28,000 cases. Previous Ebola outbreakshad been rapidly controlled with contact tracing andisolation strategies. However, the failure of initialcontainment and the subsequent unprecedented scaleof the epidemic challenged public health authorities

to employ e↵ective control measures. Mathematicalmodeling can be an important tool to evaluate thee�cacy and implementation of interventions, alongwith generally providing insights into the dynamicsof Ebola or other emerging pathogens. In this talk,I discuss recent work in which we develop a mech-anistic di↵erential equation model incorporating thekey features of contact tracing during a disease out-break. We characterize the impact of contact tracingon the e↵ective reproduction number Re, and formu-late Re completely in terms of reported case and con-tact tracing observables. Data from the West AfricaEbola outbreak is utilized to form real-time estimatesof Re and evaluate the impact of contact tracing onthe epidemic. Taken together, our model and resultsquantify contact tracing as both a dynamic interven-tion strategy impacting disease spread and a probeinto the epidemic status at the population level.

Transmission Dynamics and Fi-nal Epidemic Size of Ebola VirusDisease Outbreaks with VaryingInterventions

Attila DenesBolyai Institute, University of Szeged, HungaryMaria Vittoria Barbarossa, Gabor Kiss,Yukihiko Nakata, Gergely Rost, Zsolt Vizi

The 2014 Ebola Virus Disease outbreak in WestAfrica was the largest and longest ever reported sincethe first identification of this disease. We propose acompartmental model for Ebola dynamics, includingvirus transmission in the community, at hospitals andat funerals. Using time-dependent parameters, weincorporate the increasing intensity of interventione↵orts. Estimating the parameter values by generat-ing sample sets from previously proposed parameterranges, we fit the system to the early phase of the2014 West Africa Ebola outbreak, and estimate thebasic reproduction number as 1.44. By PRCC anal-ysis, we find that the most important factor for thespread of the epidemic is virus transmission duringtraditional burial practices. We derive a final size re-lation which allows us to forecast the total numberof cases during the outbreak when e↵ective interven-tions are in place. Our model predictions show that,as long as cases are reported in any country, interven-tion strategies cannot be dismissed. Since the main


driver in the slowdown of the epidemic is not thedepletion of susceptibles, future waves of infectionmight be possible, if control measures or populationbehaviour are relaxed. We also show the importanceof timely intervention, showing that few weeks delaycan result into twice as large total number of cases.

Period-Doubling Bifurcation andChaos in an Autonomous Model forMalaria

Calistus NgonghalaHarvard Medical School, USAMiranda I. Teboh-Ewungkem, Gideon A.Ngwa

An autonomous model for the dynamics of malariatransmission is presented. The model di↵ersfrom standard malaria transmission models in thatmosquito demography, feeding and reproduction pat-terns are modeled explicitly. The e↵ects of variousmosquito birth functions on the dynamics of the sys-tem are examined. It is demonstrated that the sys-tem transitions from a stable disease-free equilibriumto a subcritical bifurcation when the basic reproduc-tion number is less than unity and then to a stable en-demic equilibrium when the basic reproduction num-ber is bigger than unity. For one of the birth func-tions, it is shown that further increases in the basicreproduction number drives the system into period-doubling bifurcations, closely followed by chaotic dy-namics and then period-halving bifurcations. Theoccurence of a subcritical bifurcation indicates thatmalaria intervention strategies have to be appliedadequately long to ensure that the basic reproduc-tion number falls below the saddle-node bifurcationpoint, where the subcritical bifurcation occurs. Onthe other hand, the chaotic dynamics indicates that

modeling mosquito demography, feeding and repro-duction patterns explicitly might be important in un-derstanding the complexity involved the dynamics ofmalaria. We conclude that malaria data may requiremore careful examination for complex dynamics.

A Mathematical Model with Quar-antine States for the Dynamics ofEbola Virus Disease in Human Pop-ulations

Miranda Teboh-EwungkemLehigh University, USAGideon Akumah Ngwa

A deterministic ordinary di↵erential equation modelfor the dynamics and spread of Ebola Virus Diseaseis derived and studied. The model contains quaran-tine and non-quarantine states and can be used toevaluate transmission both in treatment centers andin the community. Possible sources of exposure to in-fection, including cadavers of ebola virus victims, areincluded in the model derivation and analysis. Ourmodel’s results show that there exist a threshold pa-rameter, R

0

, with the property that when its valueis above unity, an endemic equilibrium exists whosevalue and size is determined by the size of this thresh-old parameter, and when its value is less than unity,the infection does not spread into the community.The equilibrium state, when it exists is locally andasymptotically stable with oscillatory returns to theequilibrium point. The basic reproduction number,R

0

is shown to be strongly dependent on the initialresponse of the emergency services to suspected casesof ebola infection. When intervention measures suchas quarantining are instituted fully at the beginning,the value of the reproduction number reduces andany further infections can only occur at the treat-ment centers. E↵ective control measures, to reduceR

0

to values below unity, are discussed.


Special Session 48: Uncertainty Quantification in Dynamical Systems

Marcos A. Capistran, CIMAT A.C., Mexico

This session aims is to bring together researchers of applied mathematics to show advances on uncertaintyquantification of dynamical systems. Uncertainty quantification encompasses methods to account for acascade of errors arising while observing, modeling, discretizing, etc., physical models of reality. We areinterested on both, forward propagation of uncertainty as well as inverse assessment of model and parameteruncertainty.

A Randomized Misfit Approach forBig Data in Large-Scale BayesianInverse Problems

Tan Bui-ThanhThe University of Texas at Austin, USAEllen Le, Aaron Myers

We present a randomized misfit approach (RMA) fore�cient data reduction in large-scale inverse prob-lems. The method is a random transformation ap-proach that generates reduced data by randomlycombining the original ones. The main idea is to firstrandomize the misfit and then use the sample averageapproximation to solve the resulting stochastic opti-mization problem. At the heart of our approach isthe blending of the stochastic programming and therandom projection theories, which brings togetherthe advances from both sides and exploits opportu-nities at their interfaces. This allows us to conduct amore complete analysis of the RMA method, whichis unlikely possible using sole theory from either ofthe communities separately. One of the main resultsof the paper is the interplay between the Johnson-Lindenstrauss lemma and large deviation theory. Inparticular, the former provides sharp bounds on thereduced data dimensions for a large class of inter-esting sparse random transformations, while the lat-ter introduces a new look and proof of the former.To justify the RMA approach, a detailed theoreti-cal analysis is carried out for both linear and non-linear inverse problems. A tight connection betweenthe Morozov discrepancy principle and the Johnson-Lindenstrauss lemma is presented. It is this con-nection that allows us to explain the ability of theRMA method in significantly reducing observationdata with acceptable accuracy lost for the solutionof inverse problems. Various numerical results tomotivate and to verify our theoretical findings arepresented for inverse problems governed by ellipticpartial di↵erential equations in one, two, and threedimensions.

Numerical Posterior DistributionError Control and Expected BayesFactors in the Bayesian UncertaintyQuantification of Dynamical Sys-tems

Andres ChristenCIMAT, MexicoMarcos Capistran, MIguel Angel Moreles

In the bayesian analysis of ODEs Inverse Problemsmost relevant cases have intractable analytical solu-tions. These necessarily involve a numerical methodto find approximate versions of such solutions andlead to a numerical/approximate posterior distribu-tion. Recently several results have been publishedon the regularity conditions required on such numer-ical methods to ensure converge of the numerical tothe theoretical posterior. However, more practicalguidelines are needed to ensure a suitable workingnumerical posterior. Capistran, Christen and Don-nett (2013) (arXiv:1311.2281) prove for ODEs thatthe Bayes Factor of the approximate vs the theoret-ical model tends to 1 in the same order as the nu-merical method order. In this work we generalize thelatter paper in that we consider 1) correlated obser-vations, 2) practical guidelines in a multidimensionalsetting and 3) explore the use of expected Bayes Fac-tors. This permits us to obtain bounds on the ab-solute global errors to be tolerated by the numeri-cal solver, which we illustrate with some examples.Since the Bayes Factor is kept above 0.95 we expectthat the resulting numerical posterior is basically in-distinguishable from the theoretical posterior. Themethod is illustrated with some examples using syn-thetic data.

Characterizing Marginal Distri-butions Using MCMC with LocalPolynomial Approximations

Andrew DavisMIT, USAYoussef Marzouk, Patrick Heimbach

We develop methods to characterize selectedmarginal distributions of high-dimensional probabil-ity distributions in a setting where evaluating thejoint (high-dimensional) density is computationallyexpensive. We use importance sampling to estimatethe selected marginal density, and combine the result-ing noisy estimates with local polynomial regressionto approximate the smooth underlying marginal den-sity. The approximation is continually and infinitely


refined in conjunction with a Markov chain MonteCarlo (MCMC) algorithm that enables asymptot-ically exact sampling from the marginal distribu-tion of interest. By exploiting regularity of thelow-dimensional marginal, the overall scheme sig-nificantly reduces computational expense relative toboth regular MCMC and pseudo-marginal MCMC.We use our approach to solve a Bayesian inverseproblem in a goal-oriented setting. In particular,we examine a dynamical model of an ice stream inWest Antarctica, where significant uncertainty is as-sociated with high-dimensional parameter fields (e.g.,basal topography) which translates into uncertaintyin future evolution of ice stream volume above floata-tion. The targeted local approximation scheme al-lows us to directly and e�ciently characterize theposterior distribution of this ice volume.

Iterative Updating of Model Errorin Bayesian Inversion

Matthew DunlopUniversity of Warwick, EnglandDaniela Calvetti, Errki Somersalo, AndrewStuart

One of the computational challenges associated withlarge-scale inverse problems is the cost of forwardmodel evaluations. Often a compromise must bemade between accuracy (fine models) and speed(coarse models). We outline an algorithm that iter-atively estimates the distribution of the model errorarising from using a coarse model, allowing for more

accurate sampling of the posterior distribution. Theconvergence of the algorithm in the linear Gaussiancase is analyzed, wherein the posterior remains Gaus-sian and can be characterized by the evolution of itsmean and covariance. We show that these both con-verge exponentially fast, with the limiting covariancebeing non-degenerate. In the non-linear case, nu-merically all measures will be approximated by en-sembles of particles; we show convergence of somedi↵erent particle approximations in the large parti-cle limit. Finally we present some numerical resultsto illustrate the behavior of the algorithm.

A Bayesian Approach for ParameterIdentification in Aquifers

Miguel MorelesCIMAT, MexicoLiliana Guadalupe

An inverse problem of interest in Geohydrology isto estimate phenomenological parameters in aquifersfrom hydraulic potential data. We regard the pa-rameters as functions in a Hilbert space, and ex-plore bayesian estimation for gaussian priors as wellas gaussian noise. First we consider the ellipticcase, and show estimation of transmissitivity in anisotropic confined aquifer for which Darcy’s law nadthe two dimensional approximation hold. Then wediscuss the joint estimation of transmissitivity andstorativity in the underlying parabolic partial di↵er-ential equation.


Special Session 49: Recent Advances of Di↵erential Equations withApplications in Life Sciences

Ping Liu, Harbin Normal University, Peoples Rep of ChinaYing Su, Harbin Institute of Technology, Peoples Rep of ChinaFengqi Yi, Harbin Engineering University, Peoples Rep of China

Di↵erential equations have been playing important roles in explaining the rich phenomena arising from lifesciences. This special session is to concentrate on the recent advances of di↵erential equations of varioustypes (with or without delay) with applications in life sciences. We will invite researchers in this field fromaround the world to Orlando, Florida for the purpose of providing an excellent forum to exchange ideas,create new research collaborations, and rekindle old connections. Speakers and talks are carefully selectedto make the session attractive to a diverse audience

Hopf Bifurcation and Optimal Con-trol in a Di↵usive Predator-PreySystem with Time Delay and PreyHarvesting

Xiaoyuan ChangHarbin University of Science and Technology, Peo-ples Rep of ChinaJunjie Wei

In this paper, we investigated the dynamics of a di↵u-sive delayed predator-prey system with Holling typeII functional response and nozero constant prey har-vesting on no-flux boundary condition. At first, weobtain the existence and the stability of the equilibriaby analyzing the distribution of the roots of associ-ated characteristic equation. Using the time delay asthe bifurcation parameter and the harvesting termas the control parameter, we get the existence andthe stability of Hopf bifurcation at the positive con-stant steady state. Applying the normal form theoryand the center manifold argument for partial func-tional di↵erential equations, we derive an explicit for-mula for determining the direction and the stabilityof Hopf bifurcation. Finally, an optimal control prob-lem has been considered.

A Spatial SIS Model in AdvectiveHeterogeneous Environments

Renhao CuiRenmin University, Peoples Rep of ChinaYuan Lou

We study the e↵ects of di↵usion and advection fora susceptible-infected-susceptible epidemic reaction-di↵usion model in heterogeneous environments. Thedefinition of the basic reproduction number R

0

isgiven. If R

0

1 is studied. The e↵ects of di↵usion andadvection rates on the stability of the DFE are fur-ther investigated. Among other things, we find thatif the habitat is a low-risk domain, there may existone critical value for the advection rate, under whichthe DFE changes its stability at least twice as dIvaries from zero to infinity, while the DFE is unsta-ble for any dI when the advection rate is larger thanthe critical value. These results are in strong con-trast with the case of no advection, where the DFEchanges its stability at most once as dI varies fromzero to infinity.

Bifurcation Analysis and ChaosSwitchover Phenomenon in a Non-linear Financial System with DelayFeedback

Yuting DingNortheast Forestry University, Peoples Rep of ChinaJun Cao

In this talk, we study dynamics in delayed nonlinearfinancial system, with particular attention focused onHopf and double Hopf bifurcations. Firstly, we iden-tify the critical values for stability switches, Hopf anddouble Hopf bifurcations. We show how the param-eters e↵ect the dynamical behavior of the system.Secondly, the normal forms near the Hopf and dou-ble Hopf bifurcations, as well as classifications of lo-cal dynamics are analyzed. These bifurcations lead achaotic system to be stable states, such as the coex-istence of a pair of stable equilibria or a pair of stableperiodic oscillations, and the chaos disappears. Nu-merical simulations are presented to verify the ana-lytical predictions. Furthermore, detailed numericalanalysis using MATLAB extends the local bifurca-tion analysis to a global picture, namely, a familyof stable periodic solutions exist in a large region ofdelay and “chaos switchover“ phenomenon appears.Therefore, in accordance with above theoretical anal-ysis, reasonable parameters can be designed in orderto achieve various applications.

Pattern Formation in a Cross-Di↵usive Schnakenberg System

Gaetana GambinoUniversity of Palermo, ItalyMaria Carmela Lombardo, Salvatore Lupo,Marco Sammartino

In this talk the Turing pattern formation mechanismof a two components reaction-di↵usion system model-ing the Schnakenberg chemical reaction is considered.Linear and nonlinear cross-di↵usion terms, charac-terised by a gradient in the concentration of onespecies inducing a flux of the other chemical species,are introduced in the system. Cross-di↵usion leads tothe destabilization of the constant steady state and isresponsible for the initiation of spatial patterns evenif the di↵usion constant of the inhibitor is smalleror equal to the di↵usion constant of the activator.


The Turing and the Hopf instability boundaries arealso determined, showing that the presence of cross-di↵usion extends the range of di↵usion coe�cientsover which Turing patterns can occur. The processof pattern formation is studied both in 1D and 2Dspatial domains. Through a weakly nonlinear mul-tiple scales analysis the equations for the amplitudeof the stationary patterns are derived. The analysisof the amplitude equations shows the occurrence of anumber of di↵erent phenomena, including travelingpatterning waves, stable subcritical Turing patternsor multiple branches of stable solutions leading tohysteresis.

Amplitude Death and Spatiotem-poral Bifurcations in a NonlocallyDelay-Coupled System

Yuxiao GuoHarbin Institute of Technology, Peoples Rep of China

Amplitude death and spatiotemporal oscillations areremarkable patterns in coupled systems arising frombiology, neuroscience, etc. We consider a ring ofn identical oscillators with distance dependent cou-plings and time delay. The amplitude death regionis the intersection of three stable regions. Employ-ing the method of multiple scales and normal formtheory, the stability and criticality of spatiotemporaloscillations are determined. Around the amplitudedeath boundary there exist one branch of synchro-nized oscillations, n-3 branches of co-existing phase-locked oscillations, n branches of mirror-reflecting os-cillations, n branches of standing-wave oscillations,one branch of quasiperiodic oscillations and twobranches of co-existing synchronized oscillations. Itis proved that amplitude death is robust to small in-homogeneity of couplings, and the stability of syn-chronized or phase-locked oscillations inherits that ofthe individual decoupled oscillator. For the arbitraryform of coupling functions, some general results arealso obtained for the thermodynamic limit. Finally,two examples are given to support the main results.

Hopf Bifurcation and Turing Insta-bility in the Reaction-Di↵usionHolling-Tanner Predator-PreyModel

Weihua JiangHarbin Institute of Technology, Peoples Rep ofChinaXin Li, Junping Shi

In this talk, we will report our research results onHopf bifurcation and Turing instability in the reac-tion - di↵usion Holling-Tanner predator-prey modelwith Neumann boundary condition. We perform adetailed stability and Hopf bifurcation analysis andderive conditions for determining the direction of bi-furcation and the stability of the bifurcating periodicsolution. For partial di↵erential equation, we con-sider the Turing instability of the equilibrium solu-

tions and the bifurcating periodic solutions. Throughboth theoretical analysis and numerical simulations,we show the bistability of a stable equilibrium solu-tion and a stable periodic solution for ordinary di↵er-ential equation and the phenomenon that a periodicsolution becomes Turing unstable for partial di↵er-ential equation. We also will introduce our recentresearch on on codimension-two bifurcations.

Pattern Formation Close and O↵Equilibrium Driven by NonlinearDi↵usion

Maria Carmela LombardoUniversity of Palermo, ItalyGaetana Gambino, Marco Sammartino

In this talk we will explore the pattern forming prop-erties of some classical reaction-di↵usion system inthe presence of nonlinear di↵usion and cross di↵usionterms. Weakly nonlinear analysis is employed to dis-tinguish between super- and sub-critical bifurcationsand to construct solutions close to equilibrium. Insubcritical regimes we shall also show the emergenceof far-from-equilibrium oscillating patterns whose ex-istence cannot be not predicted by the weakly nonlin-ear theory. In the final part of the talk we shall dis-cuss, using classical methods of asymptotic analysis,how it is possible to construct significant far-from-equilibrium solution such as spots or mesa-patterns.

Two-Parameter Bifurcations ina Neutral Functional Di↵erentialEquation

Ben NiuHarbin Institute of Technology, Peoples Rep of China

Neutral functional di↵erential equations are oftenused for describing evolutionary systems, which arewidely applied to lossless transmission line, ecosys-tem and control theory. In this talk, we mainly studytwo-parameter bifurcations in neutral functional dif-ferential equation, by extending the center manifoldreduction method and the method of multiple scales,including Bogdanov-Takens bifurcation, Hopf-zerobifurcation and double Hopf bifurcation. We investi-gate van der Pol’s equation with extended delay feed-back as a second order neutral equation. Bifurcationsets are drawn by analyzing the roots‘ distribution ofthe characteristic equation, thus the stability domainis obtained. Universal unfoldings near Bogdanov-Takens bifurcation are obtained, thus homoclinic bi-furcation and three coexisted periodic solutions arefound by analyzing the normal forms. Around Hopf-pitchfork bifurcation, we find two stable coexistedquasi-periodic solutions. Near the double-Hopf bi-furcation, invariant two-torus, three-torus and an at-tractor of Ruelle-Takens-Newhouse type are found.


The Delay E↵ect on Some Di↵usivePopulation Models

Ying SuHarbin Institute of Technology, Peoples Rep ofChinaJunping Shi, Junjie Wei

In this talk, we will show the existence and stabilityof the spatially inhomogeneous periodic solutions forsome di↵usive population models subject to Dirichletboundary condition. We demonstrate that the spa-tially inhomogeneous periodic solutions can be bifur-cated from the positive steady state for both Logisticand weak Allee type population models. For a specialLogistic type model, such bifurcated periodic solu-tions are shown to be persistent when the parameteris far away from the bifurcation values.

Existence of Anti-Periodic Solutionsfor Sturm-Liouville Equations

Jiebao SunHarbin Institute of Technology, Peoples Rep ofChinaWenjuan Yao, Shengzhu Shi, Boying Wu

This article is devoted to the anti-periodic boundaryvalue problem for fractional Sturm-Liouville equa-tions. By applying Schaefer’s fixed point theorem,we establish the existence of anti-periodic solutionsunder certain nonlinear growth conditions of the non-linearity term. Finally, we give an example to illus-trate our result.

Coexistence and Competitive Exclu-sion in an SIS Model with StandardIncidence and Di↵usion

Necibe TuncerFlorida Atlantic University, USAMaia Martcheva, Yixiang Wu

In this talk, we introduce a two-strain spatiallyexplicit SIS epidemic model with space-dependenttransmission parameters. We define reproductionnumbers of the two strains, and show that thedisease-free equilibrium will be globally stable if bothreproduction numbers are below one. We also intro-duce the invasion numbers of the two strains whichdetermine the ability of each strain to invade thesingle-strain equilibrium of the other strain. Themain question that we address is whether the pres-ence of spatial structure would allow the two strainsto coexist, as the corresponding spatially homoge-neous model leads to competitive exclusion.We showanalytically that if both invasion numbers are largerthan one, then there is a coexistence equilibrium. Wedevise a finite element numerical method to numeri-cally confirm the stability of the coexistence equilib-

rium and investigate various competition scenariosbetween the strains. Finally, we show that the nu-merical scheme preserves the positive cone and con-verges of first order in the time variable and secondorder in the space variables.

Periodic Systems with Time Delayand Avian Influenza Dynamics

Xiang-Sheng WangSoutheast Missouri State University, USAJianghong Wu

Modeling the spread of avian influenza by migra-tory birds between the winter refuge ground and thesummer breeding site gives rise to a periodic systemof delay di↵erential equations exhibiting both thecooperative dynamics (transition between patches)and predator-prey interaction (disease transmissionwithin a patch). Such a system has two importantbasic reproductive ratios, each of which being thespectral radius of a monodromy operator associatedwith the linearized sub-system (at a certain trivialequilibrium): the (ecological) reproduction ratio forthe birds to survive in the competition of birth andnatural death, and the (epidemiological) reproduc-tion ratio for the disease to persistent. We calculatethese two ratios by our recently developed finite di-mensional reduction and asymptotic techniques, andwe show how these two ratios characterize the non-linear dynamics of the full system.

Hopf - Transcritical Bifurcation inToxic Phytoplankton-ZooplanktonModel with Delay

Hongbin WangHarbin Institute of Technology, Peoples Rep ofChinaYong Wang, Weihua Jiang

In this talk, we will report our research onHopf - transcritical bifurcation in a phytoplankton-zooplankton model with toxic liberation delay.Firstly, the critical values of Hopf bifurcation, tran-scritical bifurcation and Hopf-transcritical bifurca-tion are given, and to give more detailed informa-tion about the periodic oscillations, the directionand stability of Hopf bifurcation is studied by us-ing the normal-form theory and center manifold the-orem. Then, we give the detailed bifurcation set bycalculating the universal unfoldings near the Hopf-transcritical bifurcation point. Finally, we show thatthe plankton system may exhibit quasi-periodic os-cillations, which are verified both theoretically andnumerically, and explain the experimental observedfluctuation phenomenon of planktonpopulation.


Hopf Bifurcation for a Partial Neu-tral Functional Di↵erential Equation

Chuncheng WangHarbin Institute of Technology, Peoples Rep of China

A partial neutral functional di↵erential equation in-volving a stable operator D is considered. Semigroupproperties for the linear equation and decompostiontheorem of solution opertors for the nonlinear equa-tion are establised for this equation. The decompo-sition of phase space using an alternative theorem isalso proved. Based on these properties, Hopf bifur-cation theorem for nonlinear equation and an algo-rithm for computing the bifurcation properties areprovided.

A PDE System Modeling theGrowth of Phytoplankton withInternal Storage in a Water Column

Feng-Bin WangChang Gung University, TaiwanSze-Bi Hsu, Linfeng Mei

In this talk, we will analyze a PDE system describ-ing the growth of phytoplankton in a water column,where growth of species increases monotonically withthe nutrient quota stored within individuals. We es-tablish a threshold result on the global extinctionand persistence of phytoplankton. Basically, condi-tion for extinction/persistence is shown to dependon the principal eigenvalue of an eigenvalue problem,which is related to the di↵usivity, the sinking speed,nutrient uptake rate, and growth rate.

Attractor of Two-Patch BrusellatorModel

Yan WangCollege of William and Mary, USAJunping Shi

Many dynamic problems in physics, chemistry, biol-ogy and other fields interest in modeling individualmovement between coupled, nonlinear, discrete cells.The dynamic behavior of two coupled Brusselator re-action is described. The reactors are coupled througha permeable wall through which all species can dif-fuse. The initial conditions and the refuelling concen-tration of all reactants in each reactor are identical.The Brusselator model is the minimal mathemati-cal model possess nonlinear oscillation. The coupledsystem maintain the nonlinear oscillation either in asynchronize way or asynchronize way. After the di↵u-sion is coupled to the Brusselator chemical reactions,it can destabilize the stable symmetric equilibrium,generating four more asymmetric equilibria throughtwo saddle-node bifurcation. Hopf bifurcation alsocan occurs not only on the symmetric eqilibrium but

on the asymmetric ones which lead to rich bifurcationstructure. The numerical simulation is carried outfor various parameter values to show that di↵usion-induced instability can lead to multiple steady statesand oscillatory states.

Global Stability of the Predator-Prey System with Prey-Taxis

Zhian WangHong Kong Polytechnic University, Hong KongHaiyang Jin

We consider the global boundedness and stability tothe predator-prey system with prey-taxis. By Moseriteration and Lp-estimates, we show that the intrin-sic interaction between predators and preys in thepredator-prey system is su�cient to prevent the pop-ulation overcrowding without any technical assump-tion on prey-taxis imposed in the existing results.Furthermore, by constructing appropriate Lyapuno-val functional, we show that prey-only steady stateis globally asymptotically stable if the predation isweak, and the co-existence steady state is globallyasymptotically stable under some conditions (like theprey-taxis is weak or prey di↵use fast) if the preda-tion is strong. The exponential decay rates of so-lutions to the steady states are also derived in ourwork.

Spatiotemporal Patterns of a Ho-mogeneous Di↵usive Predator-PreySystem with Holling Type III Func-tional Response

Jinfeng WangHarbin Normal University, Peoples Rep of China

The dynamics of a di↵usive predator-prey systemwith Holling type-III functional response subject toNeumann boundary conditions is investigated. Theparameter region for the stability and instability ofthe unique constant steady state solution is derived,and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifur-cation method and Leray-Schauder degree theory.The e↵ect of various parameters on the existence andnonexistence of spatiotemporal patterns is analyzed.These results show that the impact of Holling type-IIIresponse essentially increases the system spatiotem-poral complexity.


The Numerical Method for Time-Fractional Convection-Di↵usionProblems with High-Order Accu-racy

Boying WuHarbin Institute of Technology, Peoples Rep ofChinaWenjuan Yao, Jiebao Sun

In this paper, we consider the numerical methodfor solving the two-dimensional fractional convection-di↵usion equation with a time fractional derivative ofhigh order accuracy.

Global Existence of Solutions andUniform Persistence of a Di↵usivePredator-Prey Model with Prey-Taxis

Sainan WuHarbin Institute of Technology, Peoples Rep ofChinaJunping Shi, Boying Wu

We prove the global existence and boundedness of so-lutions to a general reaction-di↵usion predator-preysystem with prey-taxis defined on a smooth boundeddomain with no-flux boundary condition. The re-sult holds for domains in arbitrary spatial dimensionand small prey-taxis sensitivity coe�cient. We alsoprove the existence of a global attractor and the uni-form persistence of the system under some additionalconditions. Applications to models from ecology andchemotaxis are discussed.

Traveling Pulses in a Lateral Inhibi-tion Neural Network

Aijun ZhangDrexel University, USAYixin Guo

We study the spatial propagating dynamics in a neu-ral network of excitatory and inhibitory populations.Our study demonstrates the existence and nonexis-tence of traveling pulse solutions with a nonsaturat-ing piecewise linear gain function. We prove thattraveling pulse solutions do not exist for such neuralfield models with even(symmetric) couplings. Theneural field models only support traveling pulse solu-tions with asymmetric couplings. We also show thatsuch neural field models with asymmetric couplingswill lead to a system of delay di↵erential equations.We further compute traveling 1-bump solutions usingthe system of delay di↵erential equations. Finally, wedevelop Evans functions to assess the stability of 1-bump traveling pulse solutions.

Invariant Manifolds for ImpulsiveEquations and Nonuniform Polyno-mial Dichotomies

Jimin ZhangHeilongjiang University, Peoples Rep of ChinaLuis Barreira, Meng Fan, Claudia Valls

For impulsive di↵erential equations in Banach spaces,we construct stable and unstable invariant manifoldsfor su�ciently small perturbations of a polynomialdichotomy. We also consider the general case ofnonuniform polynomial dichotomies. Moreover, weintroduce the notions of polynomial Lyapunov expo-nent and of regularity coe�cient for a linear impul-sive di↵erential equation, and we show that when theLyapunov exponent never vanishes the linear equa-tion admits a nonuniform polynomial dichotomy.

Stability, Bifurcation and PatternFormation of a Di↵usive PopulationModel for Pioneer-Climax Species

Xingfu ZouUniversity of Western Ontario, CanadaYing Su

We consider a general di↵usive model that describesthe interaction of pioneer and climax species in abounded domain subject to the zero-flux boundarycondition is considered. Among the results are lo-cal and global steady state bifurcations, Hopf bifur-cations, as well as conditions for Turing instabilityleading to the formation of spatial patters. We willalso present some numerical simulation results thatcan visualize and extend the analytic results. Theseresults suggest that di↵usion may make the climaxspecies more dominant.

Traveling Wave Solutions of aDi↵usive Ratio-Dependent Holling-Tanner System with DistributedDelay

Wenjie ZuoChina University of Petroleum, Peoples Rep ofChinaJunping Shi

The paper is devoted to the existence of travel-ing wave solutions of a di↵usive ratio-dependentpredator-prey system with distributed delay. For thecase without distributed delay, the existence of theminimal wave speed and traveling wave front solu-tions are established by using the comparison prin-ciple and upper/lower solutions method. Secondlythe existence of periodic traveling wave train solu-tions are shown using Hopf bifurcation theory withthe di↵usive coe�cient as a bifurcation parameter.For the case with distributed delay, the existencesof traveling wave solutions and traveling wave trainsare proved via geometric singular perturbation the-ory when the mean delay is su�ciently small.


Special Session 50: Transition Dynamics of Parabolic Type Equations

Wan-Tong Li, Lanzhou University, Peoples Rep of ChinaGuo Lin, Lanzhou University, Peoples Rep of China

Zhi-Cheng Wang, Lanzhou University, Peoples Rep of China

This special session focuses on the transition dynamics of parabolic type equations including reaction-di↵usion equations, lattice di↵erential equations, integrodi↵erence equations, nonlocal dispersal equations,etc. Due to the backgrounds of these equations in life sciences and physics as well as the importance inmathematical literature, we pay attention to the dynamics formulated by traveling wave solutions, entiresolutions, asymptotic spreading and some generalized transition modes. This session will try to present asmuch information as possible and o↵er opportunities for potential cooperation.

Solvability and Numerical Schemesfor Non-Local Kolmogorov ForwardEquations in L1 Space

Linghua ChenNorwegian University of Science and Technology,NorwayEspen Robstad Jakobsen, Arvid Naess

In this talk we present new results for non-local Kol-mogorov Forward equations with unbounded coe�-cients: solvability and the strong convergence of nu-merical schemes in L1 space. The solution of suchequations corresponds to the evolution of probabil-ity density functions of non-linear stochastic di↵eren-tial equations (SDE) driven by non-Gaussian noise.The latter has a large number of applications in vari-ous areas - including physics, economics, and finance.Existence and uniqueness of a mild solution is de-rived. On the numerical side, we study the so-calleddiscrete path integration method which produces ap-proximate probability density functions for the so-lutions of corresponding SDEs. We prove that thisscheme strongly converges in L1 norm, uniformly inany finite time horizon. Specifically, we use the con-cept of dissipative operators, combined techniquesfrom semigroup and PDE theory, as well as meth-ods from stochastic analysis.

Entire Solutions in a Nonlocal Dis-persal Epidemic Model

Wan-Tong LiLabzhou University, Peoples Rep of China

This talk is concerned with entire solutions of a non-local dispersal epidemic model. Unlike local dispersalproblems, a nonlocal dispersal operator is not com-pact and the solutions of nonlocal dispersal systemstudied here lack regularity in suitable spaces, whicha↵ects the uniform convergence of the solution se-quences. A key idea is to characterize the asymptoticbehaviors of the traveling wave solutions at infinite.This is the joint work with Li Zhang and Shi-LiangWu.

Persistence and Spreading Speeds ofa Spatial Model with an ExpandingOr Contracting Habitat

Bingtuan LiUniversity of Louisville, USASharon Bewic, Michael R. Barnard, WilliamF. Fagan

We discuss a spatial model that describes the spatialdynamics of a species in an expanding or contractinghabitat. We give conditions under which the speciesdisperses to a region of poor quality where the specieseventually becomes extinct. We show that when thespecies persists in the habitat, the rightward spread-ing speed and leftward spreading speeds are deter-mined by c, the speed at which the habitat qualityincreases or decreases in time, as well as the disper-sal kernel and species growth rates in both directions.We demonstrate how c a↵ects the spreading speeds.We also show that it is possible for a solution to forma two-layer wave, with the propagation speeds of thetwo layers analytically determined.

Persistence and Failure of CompleteSpreading in Delayed Reaction-Di↵usion Equations

Guo LinLanzhou University, Peoples Rep of ChinaShigui Ruan

This talk deals with the long time behavior in termsof complete spreading for a population model de-scribed by a reaction-di↵usion equation with de-lay, of which the corresponding reaction equation isbistable. When a complete spreading occurs in thecorresponding undelayed equation with initial valueadmitting compact support, it is proved that the in-vasion can also be successful in the delayed equationif the time delay is small. To spur on a completespreading, the choice of the initial value would bevery technical due to the combination of delay andAllee e↵ects. In addition, we show the possible failureof complete spreading in a quasimonotone delayedequation to illustrate the complexity of the problem.


Existence and Continuous Depen-dence of Random Impulsive NeutralStochastic Functional Integro Dif-ferential Evolution systems

Tongyi MaHexi University, Zhangye, Gansu, Peoples Rep ofChina

In this talk, we establish the results by using a fixedpoint analysis approach. The results improve somerecent results.

Traveling Wave Solutions of Non-monotone Delayed Systems withDispersal

Shuxia PanLanzhou University of Technology, Peoples Rep ofChina

In this talk, I shall present some results of travelingwave solutions for nonlocal dispersal systems withtime delay. When the system is not monotone andtime delay is not small, we investigate the existence

of traveling wave solutions by generalized upper andlower solutions. To establish the asymptotic behav-ior, we combine the contracting rectangle with theasymptotic spreading. These results are applied tocompetitive and predator-prey systems to obtain theminimal wave speed.

Time Periodic Traveling Waves fora Periodic and Di↵usive SIR Epi-demic Model

Zhi-Cheng WangLanzhou University, Peoples Rep of China

In this paper, we study the time periodic travelingwave solutions for a periodic SIR epidemic modelwith di↵usion and standard incidence. We establishthe existence of periodic traveling waves by investi-gating the fixed points of a nonlinear operator definedon an appropriate set of periodic functions. Then weprove the nonexistence of periodic traveling via thecomparison arguments combined with the propertiesof the spreading speed of an associated subsystem.This is a joint work with Dr. Liang Zhang and Pro-fessor Xiao-Qiang Zhao.


Special Session 51: Advances in Population Dynamics and Epidemiology

Necibe Tuncer, Florida Atlantic University, USAMaia Martcheva, University of Florida, USA

Hayriye Gulbudak, Georgia Institute of Technology, USA

This special session will bring together researchers, who are experts and at the heart of the areas of mathemat-ical biology, and students to share ideas and methods for mathematical modeling, analysis, and simulationsrelated to biological systems. It will allow for discussion of pressing topics and exchange of novel ideas. Itis expected that the session will lead to the development of mathematical theory for biological dynamics.Potential topics of interest include population dynamics, pathogen-immune system dynamics, and infectiousdisease epidemiology. The purpose of the special session is to address recent advances of mathematicalmethods used to study these focal areas and discuss a wide range of topics: computational, mathematicaland statistical.

Modelling Insect Population Dy-namics: the Physiologically-Structured Approach Applied toLaboratory Data on DrosophilaSuzukii

Souvik BhattacharyaUniversity of Florida, India

Developmental and demographic rates of an Italianpopulation of Drosophila suzukii were measured inthe laboratory in three separate experiments: un-der optimal temperature and humidity conditions;at low temperatures, following or not a freeze expo-sure; at low humidity levels. The results were similarbut not identical to those already obtained for Ore-gon populations (Dalton et al., 2011, Tochen et al.,2014, 2015). On the basis of these data as well ofthose already published, we developed a physiolog-ically structured population model; parameter esti-mation yielded a most parsimonious model where alltraits are a↵ected by temperature according to sim-ple polynomial functions, while humidity a↵ects onlyadult mortality and fruit-diet only fecundity. Themodel produces a reasonably good fit of availablelaboratory data, while di↵erences among populationsand experiments prevent a perfect fit. The model hasalso been tested with actual field temperatures in twodi↵erent years in three nearby sites; this test high-lights the relevance of small temperature di↵erencesfor the potential population growth. Use of the modelwith field data will require adequate modelling of re-source availability, and of insect movement, in orderto build a useful tool for predicting and managing D.suzukii infestations

Virus-Immune Dynamics in Age-Structured HIV Model

Cameron BrowneUniversity of Louisiana at Lafayette, USA

Mathematical modeling of viruses, such as HIV, hasbeen an extensive area of research over the past twodecades. For HIV, some important factors that af-fect within-host dynamics include: the CTL (Cyto-toxic T Lymphocyte) immune response, intra-hostdiversity, and heterogeneities of the infected cell life-cycle. Motivated by these factors, I consider exten-

sions of a standard within-host virus model. In acell infection-age structured PDE model with multi-ple virus strains, I show that competitive exclusionoccurs. I also investigate the e↵ect of CTL immuneresponse acting at di↵erent times in the infected-celllifecycle based on recent studies demonstrating supe-rior viral clearance e�cacy of certain CTL clones thatrecognize infected cells early in their lifecycle. Inter-estingly, explicit inclusion of early recognition CTLscan induce oscillatory dynamics and promote coexis-tence of multiple distinct CTL populations. Finally,I discuss several directions of ongoing modeling workattempting to capture complex HIV-immune systeminteractions suggested by experimental data.

Containing Emerging Epidemics: aQuantitative Comparison of Quar-antine and Symptom Monitoring

Lauren ChildsHarvard Chan School of Public Health, VirginiaTech, USACorey Peak, Yonatan Grad, Caroline Buckee

The containment of emerging epidemics of infectiousdiseases like Ebola or SARS requires the rapid de-ployment of interventions, which may require non-pharmaceutical strategies such as isolation becausedrugs or vaccines do not yet exist. Two strategieshave historically been considered to accelerate theisolation of a potentially infected contact: completequarantine of potentially infected contacts, or inter-mittent monitoring of contacts, followed by quaran-tine if and when symptoms occur. Complete quar-antine is a highly conservative approach to epidemiccontainment, but there are diverse associated costs,ranging from implementation expenses and restric-tion of personal liberties, to less obvious impacts likethe stigmatization of health workers, a reluctance tocomply with health orders, and interruption of finan-cial and trade markets. Here, we develop a mathe-matical model to compare the performance of symp-tom monitoring and quarantine containing an emerg-ing infectious disease. We consider case studies ofseven known pathogens with a wide range of naturalhistories that have the potential for causing sudden,severe epidemics and even potentially pandemics. We


identify which disease characteristics and interven-tion attributes are most critical for judging betweenquarantine and symptom monitoring, and provide aclear, general framework for understanding the con-sequences of isolation policies during an epidemic.

Global Stability of Age-StructuredPopulation Models

Daniel FrancoUniv. Nacional de Educaci‘on a Distancia (UNED),SpainJuan Peran

During this talk, we will discuss recent su�cientconditions for the existence of a globally asymptoti-cally stable non-zero equilibrium of certain nonlineardiscrete-time population models, describing the dy-namics of age-structured species.

From Ecology to Evolution ofHost Ajd Vector-Bornepathogenin a Structured Immuno-Epidemiological Model

Hayriye GulbudakGeorgia Institute of Technology, USAVincent Cannataro, Necibe Tuncer, MaiaMartcheva

The interaction of a pathogen and a host’s immunesystem governs the pathogen’s transmission poten-tial between hosts. After a host is infected, thepathogen population grows inside the host, triggeringan immune response of pathogen-specific antibodies,which help clear the infection. Pathogen and an-tibody dynamics are often monitored in laboratoryexperiments and modeling their interaction may in-form our understanding of disease spread. In thisstudy, we formulate a novel immunological model tocapture the within-host dynamics of vector-borne dis-eases and link it to a vector-host age-since-infectionstructured epidemiological model. By incorporatingwithin-host pathogen and immune response dynam-ics, we are able to capture the het erogeneity in in-fectivity of infected individuals and analyze how thisheterogeneity scales up and influences population-level dynamics. On the evolutionary scale, we an-alytically derive host and parasite fitness functionsdepending on intra-host pathogen-immune responseantibody dynamics. By using numerical schemes,we study host and pathogen evolutionary trajectory,along with the e↵ect of tradeo↵ functions and vec-tor inoculum load on the coevolutionary attractor.This nested modeling approach provides a tool tostudy the e↵ect of immune-pathogen interactions onthe epidemiological threshold and disease prevalence,which may give important insights on control strate-gies such as drug treatment and vaccination. In ad-dition, it o↵ers a compelling method for investigat-ing virus-host evolution, which is an important topicin emergence of new viruses, virulence and emergingoutbreaks.

Backward Bifurcations in Modelsfor Disease Dynamics

Abba GumelArizona State University, USA

Backward bifurcation, a dynamic phenomenon asso-ciated with the coexistence of a stable disease-freeequilibrium and a stable endemic equilibrium whenthe reproduction threshold of the model is less thanunity, has been shown to arise in a number of epi-demiological settings. The presence of such bifurca-tion in the transmission dynamics of a pathogen in acommunity makes its e↵ective community-wide con-trol di�cult (since, in such a backward bifurcationsituation, bringing the reproduction number of themodel to a value less than unity, while necessary, isno longer su�cient for such e↵ective control). Thispresentation discusses some of the common and newcauses of backward bifurcations, as well as their gov-erning mechanisms, in disease transmission dynam-ics.

Dynamical Models of Task Organi-zation in Social Insect Colonies

Yun KangArizona State University, USAGuy Theraulaz

The organizations of insect societies, such as divisionof labor, task allocation, collective regulation, massaction responses, have been considered as main rea-sons for the ecological success. In this article, we pro-pose and study a general modeling framework thatincludes the following three features: (a) the averageinternal response threshold for each task (the inter-nal factor); (b) social network communications thatcould lead to task switching (the environmental fac-tor); and (c) dynamical changes of task demands (theexternal factor). Since workers in many social in-sect species exhibit age polyethism, we also extendour model to incorporate age polyethism in whichworker task preferences change with age. We applyour general modeling framework to the cases of twotask groups: the inside colony task versus the out-side colony task. Our analytical study of the modelsprovides important insights and predictions on thee↵ects of colony size, social communication, and agerelated task preferences on task allocation and divi-sion of labor in the adaptive dynamical environment.Our study implies that the smaller size colony investsits resource for the colony growth and allocates moreworkers in the risky tasks such as foraging while thelarger colony shifts more workers to perform the safertasks inside the colony. Social interactions among dif-ferent task groups play an important role in shapingtask allocation depending on the relative cost anddemands of the tasks.


Structural and Practical Identifia-bility Issues of Outbreak Models

Trang LeUniversity of Tulsa, USANecibe Tuncer

Structural identifiability in epidemic models is a cru-cial issue because unreliable estimates of parame-ters could result in inaccurate estimates of importantepidemiological values such as the basic reproduc-tion number. We performed structural identifiabilityanalysis on various epidemic models using a di↵eren-tial algebra approach to investigate the characteris-tics of these models. It is necessary to note that amodel which is structurally identifiable may not bepractically so. Furthermore, we carried out practicalidentifiability analysis on these models using MonteCarlo simulations and sensitivity based analysis.

Estimating the Risk of Disease Out-break

Evan MillikenUniversity of Florida, USASergei S. Pilyugin

Infectious Salmon Anemia (ISA) virus (ISAV) is anOrthomyxovirus which causes ISA, a disease whichhas caused increased mortality in Atlantic salmon,Salmon salar L. ISA has had a substantial impacton the aquaculture industry in all major salmon pro-ducing countries. The disease can be transmitteddirectly from fish to fish and indirectly from contami-nated sea water. In a salmon farm, salmon movementis restricted within individual net cages, but contam-inated sea water can move from cage to cage. De-terministic and stochastic models are presented forthe disease in a single patch, in two patches whenfish occupy only one patch and in two patches whenfish occupy both patches. Analytical and numeri-cal methods are combined to approximate the risk ofoutbreak in the stochastic models. Results of approx-imation techniques are compared to numerical simu-lations. In each case, the threshold for the extinctionof the virus in the stochastic model is compared tothe threshold for the associated deterministic model.

Generation of Plasmodium Falci-parum Parasite Diversity WithinMosquito

Olivia ProsperUniversity of Kentucky, USALauren Childs

Plasmodium falciparum, the malaria parasite caus-ing the most severe disease in humans, undergoes anasexual stage within the human host, and a sexualstage within the vector host, Anopheles mosquitoes.Because mosquitoes may be superinfected with par-asites of di↵erent genotypes, this sexual stage of theparasite life-cycle presents the opportunity to cre-

ate genetically novel parasites. To investigate therole that mosquitoes‘ biology plays on the genera-tion of parasite diversity, which introduces bottle-necks in the parasites‘ development, we first con-structed a stochastic model of parasite developmentwithin-mosquito, generating a distribution of para-site densities at five parasite life-cycle stages: ga-mete, zygote, ookinete, oocyst, and sporozoite, overthe lifespan of a mosquito. We then coupled a modelof sequence diversity generation via recombinationbetween genotypes to the stochastic parasite pop-ulation model. With this modeling framework, wedemonstrate that parasite diversity increases at thepopulation level as a consequence of the stage tran-sitions within-mosquito, which has important impli-cations for disease-transmission and control.

Computing Human to HumanAvian Influenza R0 Via Trans-mission Chains and ParameterEstimation

Omar SaucedoUniversity of Florida, USAMaia Martcheva, Juliet Pulliam

The transmission of avian influenza between humansis extremely rare, and it mostly a↵ects individualswho are in contact with infected poultry. Althoughthis scenario is uncommon, there have been multi-ple outbreaks that occur in small infection clustersin Asia with relatively low transmissibility, and thusare too weak to cause an epidemic. Still, subcriticaltransmission from chain data is vital for determin-ing whether avian influenza is close to the thresholdof R

0

> 1. In this talk, we will explore two meth-ods of computing R

0

using transmission chains andparameter estimation via data fitting.

Coupled Infectious Disease ModelVia Human Movement

Zhisheng ShuaiUniversity of Central Florida, USA

Many recent outbreaks and spatial spread of infec-tious diseases have been influenced by human move-ment over air, sea and land transport networks.Mathematical models can be used to investigatethe e↵ects of human movement on disease dynam-ics. Models that explicitly incorporate human move-ment are commonly called multi-patch models or La-grangian models. A classic multi-patch SIR modelwill be revisited incorporating nonlinear incidencetransmission which can be used to model large classesof infectious diseases. Dynamics of the model sys-tem will be investigated both analytically and nu-merically, providing interesting dynamic phenomenaand biological insights.


Structural and Practical Iden-tifiability Issues of Immuno-Epidemiological Vector-Host Mod-els with Application to Rift ValleyFever

Necibe TuncerFlorida Atlantic University, USAMaia Martcheva, Hayriye Gulbudak, VincentCannataro

In this talk, I will discuss the structural and practi-cal identifiability of a nested immuno-epidemiologicalmodel of arbovirus diseases, where host-vector trans-mission rate, host recovery, and disease induceddeath rates are governed by the within-host immunesystem. We fit multi-scale models to multi-scale data.For an immunological model we use Rift Valley FeverVirus (RVFV) time-series data obtained from live-stock under laboratory experiments and for an epi-demiological model we incorporate the human com-partment to the nested model and use the numberof human RVF cases reported by the CDC duringthe 2006 � 2007 Kenya outbreak. We show thatthe immunological model is not structurally identi-fiable for the measurements of time-series viremiaconcentrations in the host. Thus, we study the non-dimensionalized and scaled versions of the immuno-logical model and prove that both are structurallyglobally identifiable. After fixing estimated parame-ter values for the immunological model derived fromthe scaled model, we develop numerical methods tofit observable RVF epidemiological data to the nestedmodel for the remaining parameter values of themulti-scale system. For the given (CDC) data set,numerical studies indicate that only two parametersof the epidemiological model are identifiable whenthe immune model parameters are fixed. Alterna-tively, we fit multi-scale data to multi-scale modelsimultaneously. Numerical results for simultaneous

fitting suggest that the parameters of the immuno-logical model and the parameters of the immuno-epidemiological model are not identifiable. We sug-gest that analytic approaches for studying the struc-tural identifiability of nested models is a necessity, sothat identifiable parameter combinations of the pa-rameters can be derived to reparameterize the nestedmodel to obtain an identifiable one. This is a crucialstep in developing multi-scale models which explainmulti-scale data.

Coexistence and Competitive Exclu-sion in an SIS Model with StandardIncidence and Di↵usion

Yixiang WuUniversity of Western Ontario, CanadaNecibe Tuncer, Maia Martcheva

In this talk, we present a two strain SIS model withdi↵usion, spatially heterogeneous coe�cients of thereaction part and distinct di↵usion rates of the sep-arate epidemiological classes. First, it is shown thatthe model has bounded classical solutions. Next, itis established that the model with spatially homoge-neous coe�cients leads to competitive exclusion andno coexistence is possible in this case. Furthermore,it is proved that if the invasion number of strain jis larger than one, then the equilibrium of strain i isunstable; if, on the other hand, the invasion numberof strain j is smaller than one, then the equilibriumof strain i is neutrally stable. In the case when alldi↵usion rates are equal, global results on competi-tive exclusion and coexistence of the strains are es-tablished. Finally, evolution of dispersal scenario isconsidered and it is shown that the equilibrium ofthe strain with the larger di↵usion rate is unstable.Simulations suggest that in this case the equilibriumof the strain with the smaller di↵usion rate is stable.


Special Session 52: Function Spaces and Inequalities

Annamaria Barbagallo, Naples University “Federico II”, ItalySorina Barza, Karlstad University, Sweden

Maria Alessandra Ragusa, University of Catania, Italy

The aim of this Section is to study the relations between some of the main used spaces in MathematicalAnalysis: Lebesgue Spaces, Orlicz Spaces, Lorentz Spaces, Sobolev Spaces, Morrey Spaces. These classesand the their relations can be used to study regularity properties of solutions of partial di↵erential equationsand variational inequalities.

Semigroup C* - Crossed Productsof Lattice-Orderedgroups and TheirIdeal Structure

Mamoon AhmedPrincess Sumaya University for Technology, Jordan

Let (G,G+

) be a quasi-lattice-ordered group withpositive cone G

+

and H+

a hereditary subsemigroupof G

+

. Adji and Raeburn have shown a result aboutthe structure of the primitive ideal space of the C⇤-algebra BG+ ⇥↵ G

+

for a totally ordered abeliangroup. In this paper we extend their result to thismore general setting. In particular we show that ifP

(G) is the set of subgroups H := H+

� H+

par-tially ordered by inclusion, then there exists a well-defined map F from the disjoint union

F{ bH : H 2P

(G)} to the primitive ideals of the Toeplitz alge-bra BG+ ⇥↵ G

+

. This allows us to deduce infor-mation about the irreducible representations of theC⇤-crossed product BG+ ⇥↵ G

+

.

A General Quasi-Variational In-equality for Cournot-Nash Principleand Inverse Formulation

Annamaria BarbagalloUnuversity of Naples Federico II, ItalyPaolo Mauro

The talk deals with the study of an evolution-ary quasi-variational inequality, which expresses theequilibrium conditions of a general oligopolistic mar-ket equilibrium model, and to present its inverse for-mulation. More precisely, the model allows the pres-ence of capacity constraints, production and demandexcesses and, moreover, both the production and de-mand functions are considered depending on the fore-casted equilibrium distribution. As a consequence,the capacity constraint set is defined by a multivaluedfunction. The equivalence between the dynamic elas-tic Cournot-Nash equilibrium principle and an evo-lutionary quasi-variational inequality is proved. Fur-thermore, we provide the analysis of existence, regu-larity and sensitivity of solution. Finally, the behav-ior of control policies for our problem, whose aim isto regulate the exportation through the adjustmentof taxes on the firms, is analyzed.

Factorizations of Weighted Cesaroand Copson Spaces

Sorina BarzaKarlstad University, Sweden

Any inequality expressing the boundedness of a lin-ear operator on a Banach space may be interpretedas an inclusion between two spaces. One way ofimproving the inequality is to replace the smallerspace by a larger space such that the best constant inthe original inequality remains the same. The prod-uct between the original space and the space of allmultipliers from the smaller space to the bigger onewill satisfy the above requirements. In some con-crete cases the space of multipliers can be describedand the inclusion becomes actually an indentity. Thefirst inequality, produced in this way, is an enhance-ment of the original operator inequality, while the re-versed inclusion replaces the operator inequality byan equality. Based upon this factorization we getalso a renorming of the bigger space. We presentsuch factorizations for some weighted spaces of func-tions, namely Cesaro and Copson, generalizing in thisway previous results proved by Graham Bennet forclassical spaces of sequences.

Gradient Regularity for Solutions ofa Class of Nonlinear Elliptic Equa-tions with a Degenerating LowerOrder Term.

Salvatore D’AseroUniversita di Catania, ItalyP. Cianci, G.R. Cirmi, S. Leonardi

We consider the following prototype problem:

8<

:� �u+M

|ru|2|u|✓ = f in⌦

u = 0 on @⌦

,

and we study the regularity of the gradient of a solu-tion both in Morrey spaces and in fractional Sobolevspaces, in correspondence of the regularity of theright-hand side.


Duality for A1 Weights on the RealLine

Luigi D’OnofrioNapoli Parthenope University, Italy

The aim of the talk is to describe the relation betweenthe notion of Bi-Sobolev function of the real line andthe Gehring and Muckenhoupt classes of weight. Todo this we give explicit formulas for the sharp transi-tion exponents and sharp improvement exponents ofthese classes

The Brezis-Nirenberg Problem:Recent Results on Sign-ChangingSolutions

Alessandro IacopettiUniversity of Torino, Italy

In this talk we show some recent results about sign-changing solutions for the Brezis-Nirenberg problem,which is the following semilinear elliptic problem:

⇢��u = �u+ |u|p�1u in⌦u = 0, on @⌦,

(1)

where ⌦ is a bounded smooth domain of RN , N �3, � is a positive parameter, while p + 1 = 2N

N�2

is the critical Sobolev exponent for the embeddingof H1

0

(⌦) into Lp+1(⌦). In 1990 Atkinson-Brezis-Peletier pointed out some interesting phenomena re-lated to the existence or nonexistence of radial sign-changing solutions for the Brezis-Nirenberg problemin the ball, in the low dimensions N = 4, 5, 6. Fromtheir results several questions arose, in particularconcerning the presence of analogous situations inother domains. A related question about the exis-tence or not, in general domains, of sign-changing so-lutions whose positive and negative part concentrateat the same point, raised from later papers of BenAyed-El Mehdi-Pacella. We will show recent con-tributions on these questions, proving, in particular,some existence and nonexistence results in generalbounded domains, possibly with some symmetry.

Pointwise Convergence of FourierSeries Near L1

Victor LiePurdue University, USA

In our talk we will discuss the old and celebratedquestion regarding the pointwise behavior of FourierSeries near L1. This presentation will include: 1) theresolution of Konyagin’s conjecture (ICM, Madrid2006) on the pointwise convergence of Fourier Se-ries along lacunary subsequences; 2) the L1-strongconvergence of Fourier Series along lacunary subse-quences. We end with several considerations on therelevance/impact of the above two items on the sub-ject of the pointwise convergence of Fourier Series.

Sublinear Operators in WeightedGrand Morrey Spaces

Alexander MeskhiA. Razmadze Mathematical Institute of I.Javakhishvili Tbilisi State University, Rep of Georgia

The boundedness of sublinear integral operators ingrand Morrey spaces defined by means of measuresgenerated by the Muckenhoupt weights is derived.The operators under consideration involve operatorsof Harmonic Analysis such as Hardy-Littlewood andfractional maximal operators, Calderon-Zygmundoperators, potential operators etc. The same prob-lem for commutators of singular and fractional in-tegral operators is also studied. This research is acontinuation of the investigation carried out by thespeaker jointly with V. Kokilashvili and H. Rafeiroin the unweighted case (see e.g., Chapter 16 of theforthcoming monograph V. Kokilashvili, A. Meskhi,S. Samko, and H. Rafeiro, Integral operators innon-standard function spaces. Volumes I and II,Birkhauser, 2016 ).

Di↵erentiability Results Concerningthe Minima of Nondi↵erentiableFunctionals

Maria Alessandra RagusaCatania University, Italy

The author shows some results obtained in coop-eration with Atsushi Tachikawa. We study Holderregularity for minimizers of functionals having morethan quadratic growth and discontinuous coe�cients.Starting with the well-known results by Giaquinta,Giusti and Modica , the direct approach was in-troduced. Later the study made by Giaquinta andGiusti and through the work of many authors, amongothers Huang, Danecek and Viszus was developed atheory of partial regularity of solutions of minimizersof variational integrals in vector valued case, usingthe direct method.

A Regularity Result for Solutionsof Some P.D.E. with DiscontinuousCoe�cients

Andrea ScapellatoUniversity of Catania, ItalyMaria Alessandra Ragusa

The author shows some conditions to obtain esti-mates for singular and fractional integral operatorson Generalized Morrey Spaces. The boundedness ofthe operators in this kind of spaces is finalized toobtain regularity results for solutions of partial dif-ferential equations of elliptic type.


Lusin (N) Condition and The Dis-tributional Determinant

Roberta SchiattarellaUniversita degli Studi di Napoli Federico II, ItalyLuigi D’Onofrio- Stanislav Hencl-Jan Maly

Let⌦ ⇢ Rn be an open set. We show that for acontinuous mapping f 2 W 1,n�1(⌦,Rn) with Jf 2L1(⌦) the validity of the Lusin (N) condition impliesthat the distributional Jacobian equals to the point-wise Jacobian.

Approximation of Smooth Func-tions on [1,1) and R \ (�1, 1) byEntire Functions of ExponentialType

Alexander TovstolisUniversity of Central Florida, USA

In 1946, Sergey M. Nikolskıi discovered an e↵ect ofbetter pointwise approximation of a smooth functionby algebraic polynomials. Namely, for a functionfrom Sobolev class W 1 [�1, 1], there is a sequenceof algebraic polynomials {pn}1n=0

, pn 2 Rn [x], suchthat

|f (x)� pn (x)| ⇡2

p1� x2

n+ 1+O

✓ln (n+ 2)

(n+ 1)2

◆.

The constant ⇡/2 in the first term cannot be im-proved.There are several generalizations of this result. Themost recent one is due to Roald M. Trigub (1993),where the asymptotically sharp estimate was ob-tained for the W r [�1, 1] class.

We will focus on pointwise approximation of a func-tion from the Sobolev class W r (R \ (�1, 1)) by entirefunctions of exponential type at most �. Known esti-mates of such approximation is due to Ju. A. Brudnyı(1959). Our goal is to obtain an asymptotically sharpestimate. As in the Trigub’s article, we deduce ourestimates from corresponding result on the uniformapproximation. In our case, this is the Akhiezer’stheorem on uniform approximation (on R) of a func-tion from W r (R) class by entire functions of expo-nential type �.

Mean Field Equations with Proba-bility Measure in 2D-Turbulence

Gabriella ZeccaUniversity of Naples Federico II, Italy

We study the mean field equation derived by Neri[C. Neri, Ann. Inst. H. Poincare Anal. Non Lineaire(2004)] in the context of the statistical mechanicsdescription of 2D-turbulence, under a stochastic as-sumption on the vortex circulations. The correspond-ing mathematical problem is a nonlocal semilinearelliptic equation with exponential type nonlinearity,containing a probability measure P 2 M([�1, 1])which describes the distribution of the vortex circu-lations. Unlike the more investigated “deterministicversion, we prove that Neri’s equation may be viewedas a perturbation of the widely analyzed standardmean field equation, obtained by taking P = �

1

. Inparticular, in the physically relevant case where P isnon-negatively supported and P({1}) > 0, we provethe mass quantization for blow-up sequences. We ap-ply this result to construct minimax type solutions onbounded domains in Rd and on compact 2-manifoldswithout boundary. Those results are obtained in col-laboration with T. Ricciardi


Special Session 53: Interface Dynamics and Transport Phenomena

Tatiana Savina, Ohio University, USAAlexander Nepomnyashchy, Technion - Israel Institute of Technology, Israel

The goal of this session is to bring together mathematicians and physicists studying di↵erent processesrelated to the surface dynamics and transport phenomena, including free boundary problems and anomalousdi↵usion.

Non-Analytic Reconstruction of theVortex Core in Bosonic Superfluids,and Its Implications on the VortexDynamics and the instability ofAbrikosov’s lattice

Oded AgamThe Hebrew University, Israel

We analyze the motion of quantum vortices in a two-dimensional spinless superfluid within Popov’s hy-drodynamic description. In the long healing lengthlimit (where a large number of particles are inside thevortex core) the superfluid dynamics is shown to bedescribed by weak solutions of the Gross-Pitaevskiiequation. We solve the resulting equations of motionfor a vortex moving with respect to the superfluidand find the reconstruction of the vortex core to bea non-analytic function of the force applied on thevortex. As a result, the spectrum associated withthe vortex motion exhibits narrow resonances lyingwithin the phonon part of the spectrum, contrary totraditional view. The implication of this nonanalyticcore reconstruction to the disordered limited motionof vortices and the instability of Abrikosov’s latticeis discussed.

Free Boundary Problems in Com-pressible Vortex Flows

Darren CrowdyImperial College London, EnglandVikas Krishnamurthy

The usefulness of conformal mapping techniques insolving Laplacian-growth problems (e.g. the Hele-Shaw problem) where the governing field equation isharmonic is well-known. It has led to the discoveryof broad classes of analytical solutions to such freeboundary problems. But conformal mapping meth-ods can also be useful in solving free boundary prob-lems that are not conformally invariant, or wherethe governing field is not harmonic. These mathe-matical ideas will be showcased in the application tovortex dynamics for weakly compressible flows wherenew analytical solutions for physically important flowscenarios, such as compressible von Karman vortexstreets, have recently been uncovered.

Gibbs-Thomson Equation for theRapidly Moving Interfaces

Peter GalenkoUniversity of Jena, Germany

Using a phase-field model for fast phase transfor-mations an interface condition for the rapidly mov-ing solid-liquid interface is obtained. The modelis described by equations for the hyperbolic trans-port and fast interface dynamics, which are reducedto a sole equation of the phase field with the driv-ing force given by deviations of temperature andconcentration from their equilibrium values withinthe di↵use interface. It is shown that the obtainedinterface condition presents the acceleration- andvelocity-dependent Gibbs-Thomson interfacial con-dition. This condition is identical to the advancedBorn-Infeld equation for the hyperbolic motion bymean curvature with the driving force. As a limit-ing case, the interface condition presents “velocity-driving force relationships found earlier as travelingwave solutions for slow and fast phase field profiles.Predictions of the analytical solutions are qualita-tively compared with literature data of atomistic sim-ulations on crystal growth kinetics.

Continuum Mean-Field Theory forTransport in Highly ConcentratedElectrolytes

Nir GavishTechnion - IIT, IsraelDoron Elad, Arik Yochelis

The Poisson-Nernst-Planck (PNP) theory is one ofthe most widely used analytical methods to de-scribe electrokinetic phenomena for electrolytes. Themodel, however, considers isolated charges and thusis valid only for dilute ion concentrations. The keyimportance of concentrated electrolytes in applica-tions has led to the development of a large familyof generalized PNP models. In particular, the Bik-erman model that takes into account the finite sizeof the ions has been one of the most commonly usedextension to PNP. In this talk, we derive a thermo-dynamically consistent mean-field model for concen-trated solutions. Our model recovers the Bikermanterm, but shows that it is inconsistent in the sensethat additional terms of equal magnitude should betaken into account. Furthermore, our study showsthat the Bikerman approach inherently fails to de-scribe finite-size e↵ects at the highly concentratedregime, and presents a supplementary approach inthis regime. The result is a modelling framework


that is valid over the whole range of concentrations- from dilute electrolyte solutions to highly concen-trated solution, such as ionic liquids. Importantly,the new model predicts distinct transport propertieswhich are not governed by Einstein-Stokes relations,but are rather e↵ected by inter-di↵usion and emer-gence of nano-structure.

Global Aspects of a Free BoundaryProblem for the Laplace Operator

Lavi KarpORT Braude College, Israel

The talk will consider free boundary problems thatarise from Hele-Shaw flows and unbounded quadra-ture domains. We will show how to construct solu-tions with arbitrary growth near infinity and discussconditions which guarantee a quadratic growth. Thelast one is crucial to the instigation of the geometricproperties of the free boundaries. Another aspect isthe asymptotic behavior of the free boundaries, whilein the two dimensional case there is a satisfactory de-scription of it, in higher dimensions the problem iswide open.

Anomalous Di↵usion in a Systemwith a Thin Membrane

Tadeusz KosztolowiczJan Kochanowski University, Kielce, PolandKatarzyna D. Lewandowska

We consider anomalous di↵usion in a system whichconsists of two media separated by a thin partiallypermeable membrane. In each part of the system,bounded by the membrane, there may be di↵er-ent kinds of di↵usion (normal di↵usion, subdi↵usion,slow subdi↵usion). We derive the Green’s functionsusing a simple random walk model with both dis-crete time and spatial variables. Next we move tothe continuous variables. The obtained Green’s func-tions are used to derive a boundary condition at themembrane. It is shown that the boundary conditioncontains a specific term which can be interpreted asa ‘memory term‘ depending on kinds of di↵usion oc-curring in the system.

Discontinuity in the Behavior ofMother Body and 2D OrthogonalPolynomial

Seung-Yeop LeeUniversity of South Florida, USAMeng Yang

We consider the two-dimensional orthogonal polyno-mials whose zeros are conjectured to converge on themother body, a certain potential theoretic skeleton.We take the simplest case where the mother body be-haves discontinuously over the continuous variationof the underlying domain. We studied the behaviorof the zeros in detail at the discontinuity.

Anomalous Di↵usion and ErgodicViolation in Intermembrane andMembrane-Mediated Transport

Ralf MetzlerPotsdam University, Germany

It will be shown that di↵usion of lipid moleculesand embedded proteins in bilayer membranes istransiently anomalous. The temporal range of theanomalous regime is significantly extended when dis-order in the form of cholesterol or proteins is added.In protein crowded membranes long range anomalousdi↵usion is observed, combining antipersistent long-ranged noise with non-Gaussian fluctuations. Pass-ing to in vivo membranes, the transport is againanomalous, but dominated rather by jump like dif-fusion with scale free, power-law waiting time den-sity, causing the di↵usion to be non-ergodic in theBoltzmann sense that time and ensemble averagesof physical observables no longer converge to eachother. Adding another spatial dimension, it will bediscussed how surface-bulk exchange due to reactiveboundary conditions leads to e↵ective, Levy flight-type anomalous di↵usion on the membrane surfacemediated by transient bulk excursions. Again, thisprocess is non-ergodic in the above sense.

Interfacial Instability During PhaseChange

Ranga NarayananUniversity of Florida, USAK.E. Uguz

The physics of evaporative convection for binary sys-tems is presented. Two results are of importance.The first is that a binary system, in the absence ofgravity, can generate an instability only when heatedfrom the vapor side. This is to be contrasted withthe case of a single component where instability canoccur only when heated from the liquid side. Thesecond result is that a binary system, in the presenceof gravity, will generate an instability when heatedfrom either the vapor or the liquid side provided theheating is strong enough. In addition to these resultswe show the conditions at which interfacial patternscan occur.

Dynamics of Subdi↵usion-ReactionFronts

Alexander NepomnyashchyTechnion, IsraelMohammad Abu Hamed, Vladimir Volpert

The dynamics of a reaction front is considered inthe framework of the Seki-Lindenberg subdi↵usion-reaction model, where the time derivative in thedi↵usion-reaction equation is replaced by a Caputofractional derivative. Using a piecewise linear reac-tion function, we obtain exact analytical solutionsfor plane fronts (i) between two stable uniform statesand (ii) between a stable state and an unstable state


(‘pulled‘ and ‘pushed‘ fronts). In the case of equalpotentials of stable phases, where the plane front ismotionless, the dynamics of fronts is determined (i)by the front curvature and (ii) by interaction of mul-tiple fronts. We obtain a closed equation governinga slow motion of a small-curvature front and find itsapproximate solutions. Also, we derive and solve, an-alytically and numerically, integro-di↵erential equa-tions that describe the interaction of distant fronts.

On a Muskat Problem with LineDistributions of Sinks and Sources

Tatiana SavinaOhio University, USAA.A. Nepomnyashchy, L. Akinyemi

We consider a Muskat problem with zero surface ten-sion. We give examples of exact solutions when thedistribution of sinks and sources is defined by theinitial shape of the interface separating two fluids.

Stationary Boundary Points fora Laplacian Growth Problem inHigher Dimensions

Tomas SjoedinLinkoeping University, SwedenStephen J. Gardiner

This talk will concern the behaviour of corners forcertain Laplacian growth processes driven by sourceterms in higher dimensions. In two dimensions thisprocess corresponds to Hele-Shaw flow, and it isknown that corners of interior angle less than ⇡/2in the boundary of a plane domain are initially sta-tionary for such growth processes. The aim here isto present analogous results in higher dimensions.

Weak Resolution of Singularities inIntegrable Free-Boundary Dynam-ics

Razvan TeodorescuUniversity of South Florida, USA

In two dimensions, free-boundary dynamics featur-ing an infinite set of conserved quantities is an uni-versal model rich in applications and also in mathe-matical properties. In many cases, such models are

characterized by the formation of boundary singular-ities, which sometimes does not allow continuationof strong solutions past a finite evolution time. Theweak resolution of singularities is obtained by a de-formation of the original dynamical system, whichintroduces an associated class of weak solutions. Weprove that the deformation is universal and that itleads to weak solutions in the family of modulated el-liptic functions. For the case of generic singularities,we provide explicit weak solutions and we indicatean interesting connection to complex Burgers-typeequations and convex nonlinear optimization.

FRAP Dynamics in the RandomComb Model

Santos YusteDpt. Fisica. U.Extremadura, SpainE. Abad, A. Baumgaertner

The problem of computing concentration recoverycurves mimicking FRAP experiments in comb-like ge-ometries is addressed. Our approach relies on thee↵ective mean-field CTRW description of the ran-dom comb model, which means that the di↵usionof particles can be described by means of a frac-tional di↵usion equation. We first obtain analyti-cally the value of the anomalous di↵usion coe�cientand, then, we solve analytically the fractional dif-fusion equation in the Laplace space with boundaryconditions typical of FRAP experiments. We showthat, although the recovery curves cannot be fitted bya standard di↵usion equation with a time-dependentdi↵usion coe�cient (scaled Brownian motion model),the di↵erences between the exact curves and such fitsare small. This provides support for the generalizedpractical use of scaled Brownian motion for describ-ing di↵usion experiments in comb-like systems. Ourtheoretical results are confirmed by numerical simu-lations.


Special Session 54: Nonlinear PDEs and Variational Methods

Olimpio Hiroshi Miyagaki, Universidade Federal de Juiz de Fora-UFJF, BrazilDjairo Guedes de Figueiredo, IME-UNICAMP, Brazil

Joao Marcos Bezerra do O, Universidade Federal da Paraiba-UFPB, Brazil

This session will focus on some recent developments in the theory of Nonlinear Partial Di↵erential Equationsof the elliptic type. Questions like uniqueness, multiplicity, geometric properties and behaviour of solutionswill be discussed. Nonlinear elliptic problems have been studied in recent years,also motivated by theirapplicability to other areas of PDE, as well as in many areas of Physics, Biology, Economy.

On the Persistence of the Eigen-values of a Perturbed FredholmOperator

Pierluigi BenevieriUniversity of Sao Paulo, BrazilAlessandro Calamai, Massimo Furi, MariaPatrizia Pera

Let H be a real Hilbert space and denote by S itsunit sphere. Consider the nonlinear eigenvalue prob-lem Lx + "N(x) = �x, where ",� 2 R, L : H ! His a bounded self-adjoint, linear operator with non-trivial kernel and closed image, and N : H ! H isa nonlinear perturbation term. A unit eigenvector~x 2 S \ mKer, L of L (corresponding to the eigen-value � = 0) is said to be persistent if it is closeto solutions x 2 S of the above equation for smallvalues of the parameters " � 0 and �. We give ana�rmative answer to a conjecture formulated by R.Chiappinelli, M. Furi and M.P. Pera in 2008. Namely,we prove that, if N is Lipschitz continuous and theeigenvalue � = 0 has odd multiplicity, then the sphereS\mKer, L contains at least one persistent eigenvec-tor. We provide examples in which our results apply,as well as examples showing that if the dimension ofmKer, L is even, then the persistence phenomenonmay not occur.

On a Class of Superlinear EllipticProblems

David CostaUniversity of Nevada Las Vegas, USASiegfried Carl, Hossein Tehrani

We consider a class of superlinear slliptic problems inRN ((N � 3). By means of variational methods, wepresent a new approach to finding multiple solutions.

Some Classical Inequalities Revis-ited for the Fractional Laplacian

Olivaine de QueirozUNICAMP, Brazil

We are interested in the study, in the context of thefractional Laplacian in bounded domains, of someclassical inequalities such as Sobolev-Trudinger-Moser and also the Faber-Khran. We apply our re-sults in the study of some free boundary problemsand also in some nonlinear PDEs from ConformalGeometry.

An Inverse Iteration Method forObtaining Q-Eigenpairs of the P-Laplacian

Grey ErcoleUniversidade Federal de Minas Gerais, Brazil

We consider a Lane-Emden type problem for theDirichlet p-Laplacian operator in an N-dimensionalLipschitz domain. The nonlinearity contains anonlocal factor that makes the problem (p-1)-homogeneous, so that it can be considered as aneigenvalue problem. We show that eigenpairs can beobtained as limits of sequences suitably built from aninverse iteration scheme.

Ground States of Elliptic ProblemsInvolving Non Homogeneous Oper-ators

Giovany FigueiredoUniversidade Federal do Para, BrazilHumberto Ramos

We investigate the existence of ground states for func-tionals with nonhomogenous principal part. Roughlyspeaking, we show that the Nehari manifold methodrequires no homogeinity on the principal part of afunctional. This result is motivated by some ellipticproblems involving nonhomogeneous operators. Asan application, we prove the existence of a groundstate and infinitely many solutions for three classesof boundary value problems.


Self-Similar Solutions for the HeatEquation

Marcelo FurtadoUniverisity of Brasilia, Brazil

We are concerned with the existence of solutions forthe equation

��u� 12(x ·ru) = f(u), x 2 ⌦,

where ⌦= RN or ⌦= RN+

, with suitable nonlin-ear boundary condition on the second case. It canbe showed that this problem arises in finding self-similar solutions for the nonlinear heat equation. Byusing variational methods we obtain solutions whichrapidly decay to zero at infinity. We consider severalcases depending on the special profile of the functionf and the dimension of the space.

Convex Solutions of the FractionalHeat Equation

Antonio IannizzottoUniversity of Cagliari, ItalyAntonio Greco

We prove several properties, involving sign, convex-ity and uniqueness results, for unbounded solutions ofstationary problems driven by the fractional Lapla-cian on the whole space. Exploiting such proper-ties, we prove an existence/uniqueness result for thefractional heat equation provided the (possibly un-bounded) initial datum has a moderate growth, andmoreover that the fractional heat flow preserves con-vexity of the initial datum. Our result complementsthe Widder’s type theorem of Barrios, Peral, Soriaand Valdinoci (Arch. Rational Mech. Anal. 213(2014) 629-650).

Nonradial Positive Solutions of theP-Laplace Emden-Fowler Equation

Ryuji KajikiyaSaga University, Japan

In this lecture, we study the p-Laplace Emden-Fowlerequation with a radial and sign-changing weight un-der the Dirichlet boundary condition. We show thatif the weight function is negative in the unit ball ex-cept for a small neighborhood of the boundary andpositive at somewhere in this neighborhood, then noleast energy solution is radially symmetric. More-over, in the one dimensional case, we prove that ifthe neighborhood is large, then a positive solution isunique.

Singularly Perturbed PDEs andPatterns with Periodic Profiles

Fethi MahmoudiUniversidad de Chile, Chile

We consider a class of singularly perturbed equationsin planar domains: as the singular perturbation pa-rameter tends to zero, we exhibit a family of solutionsconcentrating at the boundary with asymptoticallyperiodic profile. As solutions with uniform profile atthe boundary were known to exist, the result here re-flects the phenomenon of Turing‘s instability, whichtriggers formation of inhomogeneous structures frommore homogeneous ones.

A Critical Nonlinear FractionalElliptic Equation with Saddle-LikePotentical in RN

Olimpio MiyagakiUFJF-Universidade Federal de Juiz de Fora, BrazilClaudianor O. Alves

In this work, we study the existence of positive solu-tion for the following class of fractional elliptic equa-tion

✏2s(��)su+V (z)u = �|u|q�2u+|u|2⇤s

�2u, , , in, , ,RN ,

where ✏,�> 0 are positive parameters, q 2(2, 2⇤s), 2

⇤s = 2N

N�2s, N > 2s, s 2 (0, 1), (��)su is

the fractional laplacian, and V is a saddle-like po-tential. The result is proved by using minimizingmethod constrained to the Nehari manifold. A spe-cial minimax level is obtained by using an argumentmade by Benci and Cerami.

Some Results about Singular El-liptic Problems in Orlicz-SobolevSpaces

Carlos SantosUniversity of Brasilia, BrazilClaudianor Alves, Je↵erson Santos, JoseValdo Goncalves, Marcos Leandro Carvalho,Jiazheng Zhou

In this talk, I will present some new results about sin-gular elliptic problems. We will do this for di↵erentsettings of function spaces and non-linearities.


Special Session 56: Junior Session on Nonlinear Hyperbolic Equationsand Related Topics

Laura Spinolo, IMATI-CNR, Pavia, ItalyFabio Cavalletti, University of Pavia, Italy

Hyperbolic systems of conservation laws are a class of nonlinear partial di↵erential equations with severalapplications coming from both physics and engineering. In particular, the archetype are the Euler equationsof fluid dynamics. The mathematical understanding of this class of equations is complicated by the presenceof highly nonlinear phenomena, like the fact that, in general, classical solution can breakdown in finite timeowing to the formation of shocks. In recent years, the analysis of system of conservation laws has takengreat advantage from the interplay with di↵erent but closely related fields, like geometric measure theory,convex integration, and others. This session aims at gathering young mathematicians working on nonlinearhyperbolic equations and related topics, in order to present some of the most recent developments of thetheory and favour scientific interactions.

Finite Energy Weak Solutions of theQuantum Navier-Stokes Equations- Part II

Paolo AntonelliGran Sasso Science Institute, ItalyStefano Spirito

In this talk I will present an existence result for finiteenergy weak solutions for the quantum Navier-Stokessystem. By considering a suitable regularization ofthe system, which maintains the compactness prop-erties of the original system, I will show the existenceof a sequence of regular approximate solutions. Thea priori bounds obtained will then guarantee the con-vergence of the sequence to a finite energy weak so-lution to the quantum Navier-Stokes system. This isa joint work with S. Spirito from Gran Sasso ScienceInstitute, L‘Aquila.

Surprise and Predictability inBounded Sources of Non-ConvexBalance Laws

Laura CaravennaUniversita di Padova, Italy

In the talk I will show surprising and predictable as-pects of bounded source terms in a non-convex bal-ance law, with smooth flux, when it admits a contin-uous solution. Namely, I will discuss to what extentthe conservation law can be reduced to an (infinitelydimensional) system of ODEs along the characteristiccurves. This correspondence is evident in the classi-cal setting but it is surprising in this context withlack of regularity. Part of the correspondence justrequires suitable definitions and smart technicality,but concerning part of it new odd unexpected behav-iors show up. The presentation is mostly based ona joint work with G. Alberti (Pisa) and S. Bianchini(SISSA) , and it extends previous works by severalauthors relative to the case of the quadratic flux.

Recent Progress on the LargeSolution of Compressible EulerEquations

Geng ChenGeorgia Institute of Technology, USA

Compressible Euler equations (introduced by Eulerin 1757) model the motion of compressible inviscidfluids such as gases. It is well-known that solutionsof compressible Euler equations often develop discon-tinuities, i.e. shock waves. Successful theories havebeen established in the past 150 years for small so-lutions in one space dimension. The theory on largesolutions is widely open for a long time, even in onespace dimension. In this talk, I will discuss some re-cent exciting progresses in this direction. In the firstpart of this talk, I will discuss our complete resolu-tion of shock formation problem, which extends thecelebrated work of Peter Lax on small solutions in1964. Our breakthrough relies on the discovery of asharp time-dependent lower bound on density, whensolutions approach vacuum in infinite time. In thesecond part, I will show our recent negative exam-ple concerning the failure of current available frame-works on approximate solutions in order to establishlarge BV (bounded total variation) theory. The talkis based on my joint works with A. Bressan, H.K.Jenssen, R. Pan, R. Young, Q. Zhang, and S. Zhu.

Surprising Solutions to the Isen-tropic System of Gas Dynamics

Elisabetta ChiodaroliEPFL, Lausanne, SwitzerlandC. De Lellis, O. Kreml

In this talk we discuss some applications of themethod of convex integration to the compressible Eu-ler system of gas dynamics in several space dimen-sions. This leads to the construction of non-standardsolutions which disprove the e�ciency of di↵erent ad-missibility criteria proposed in the literature.


On the Global Structure of the Setof 2D Stationary Euler Flows

Antoine Cho↵rutUniversity of Warwick, EnglandHerbert Koch

The incompressible Euler equations govern the evo-lution of an ideal fluid. They enjoy a very ele-gant geometric Hamiltonian formulation and as suchtheir steady-states are of particular interest. Beingan infinite-dimensional system, the analysis becomesconsiderably more challenging and the usual toolboxis not adequate for the Euler equations. In this talkI will present some recent work with Herbert Kochon the existence of steady-states with prescribed vor-ticity distribution. It is a global version of previouswork in collaboration with Vladimir Sverak. One cru-cial ingredient is to derive su�ciently strong a prioriestimates in order to repeat the local, perturbativeresult. I will also discuss other interesting aspects ofthe proof.

Finite Time Singularities of a Hy-perbolic System of Elasticity

Tao HuangNew York University Shanghai, Peoples Rep ofChinaGeng Chen, Chun Liu

We study the formation of finite time singularities inthe form of super norm blowup for a spatially hy-perbolic system modeling inhomogeneous elasticity,which is related to the variational wave equations.The system possesses a unique C1 solution beforethe emergence of vacuum in finite time, for giveninitial data that are smooth enough, bounded anduniformly away from vacuum. At the occurrence ofblowup, the density becomes zero, while the momen-tum stays finite, however the velocity and the densityof the energy are both infinite.

Holder Continuous Euler Flows

Philip IsettMIT, USA

Motivated by the theory of hydrodynamic turbu-lence, L. Onsager conjectured in 1949 that solutionsto the incompressible Euler equations with Holderregularity less than 1/3 may fail to conserve energy.C. De Lellis and L. Szekelyhidi, Jr. have pioneered anapproach to constructing such irregular flows basedon an iteration scheme known as convex integration.This approach involves correcting “approximate solu-tions“ by adding rapid oscillations that are designedto reduce the error term in solving the equation. Inthis talk, I will discuss an improved convex integra-tion framework, which yields solutions with Holderregularity as much as 1/5-.

On Measure Valued Solutions to theCompressible Euler Equations

Ondrej KremlCzech Academy of Sciences, Czech RepElisabetta Chiodaroli, Eduard Feireisl, EmilWiedemann

In a very interesting paper, Szekelyhidi and Wiede-mann (2012) proved that every measure valued so-lution to the incompressible Euler equations can beapproximated by a sequence of weak solutions, im-plying that there is no significant di↵erence betweenweak and measure valued solutions to the incom-pressible Euler system. In this talk we prove thatsuch a property does not hold for the compressiblecase and we show the construction of a measure val-ued solution which can not be generated by weaksolutions. Moreover we show an abstract neccesarycondition for measure valued solutions to be gener-ated by sequences of weak solutions. The proof isbased on a generalization of a rigidity result by Balland James, the necessary condition is obtained as aconsequence of the works of Fonseca and Muller.

E↵ective Monte Carlo Methods forComputing Measure Valued Solu-tions and Statistical Solutions

Kjetil Olsen LyeETH Zurich, SwitzerlandSiddhartha Mishra

Recent developments have indicated the correct no-tion of solution for system of non-linear conservationlaws, especially in the presence of turbulence, is thenotion measure valued solutions. We briefly reviewthe theory of entropy measure valued solutions andstatistical solutions and show how we can computestatistics of multidimensional systems of conservationlaws using entropy preserving schemes and stochasticsampling. We assess the applicability of Multi-levelMonte Carlo methods for both scalar conservationlaws and system of equations.

Lower Bound on the Tempera-ture Background Field Method forRayleigh-Benard Convection

Camilla NobiliUniversity of Basel, Switzerland

By the background field method, finding bounds onthe average upward heat transport for the Rayleigh-Benard convection reduces to a variational problem:minimize over background profiles that satisfy a cer-tain stability condition. After recalling the method(introduced in 1996 by Doering and Constantin inthis context) we characterize the admissible profilesand then we argue the non-optimality of this method,by exhibiting a lower bound on its solution. This isa joint work with Felix Otto.


A Time Discretization Scheme forthe Multi-Dimensional Compress-ible Euler Equations.

Marc SedjroKAUST, Saudi ArabiaFabio Cavalletti, Michael Westdickenberg

We construct a time discretization scheme for themulti-dimensional compressible Euler equations, in-spired by minimizing movements for curves of maxi-mal slope. Each timestep requires the minimizationof a functional measuring the acceleration of fluid el-ements, over the cone of monotone transport maps.We prove convergence to measure-valued solutions.

Finite Energy Weak Solutions of theQuantum Navier-Stokes Equations

Stefano SpiritoGran Sasso Science Institute, ItalyPaolo Antonelli

In this talk we focus on a new compactness resultabout finite energy weak solutions of the quantumNavier-Stokes equations. The novelty of the result isthat we are able to consider the vacuum in the def-inition of weak solutions. The main tool is a newformulation of the equations which allows us to getadditional a priori estimates to prove compactness.Some remarks concerning the choice of the approxi-mation system to get global existence will be made.This is a joint work with Paolo Antonelli (GSSI -Gran Sasso Science Institute).


Special Session 57: Lie Symmetries, Conservation Laws and OtherApproaches in Solving Nonlinear Di↵erential Equations

Chaudry Masood Khalique, North-West University, So AfricaMufid Abudiab, Texas A&M University-Corpus Christi, USA

Maria Gandarias, Universidad de Cadiz, Spain

This session is devoted to research areas that are related to nonlinear di↵erential equations and their appli-cations in science and engineering. The main focus of this special session is on the Lie symmetry analysis,conservation laws and their applications to ordinary and partial di↵erential equations. Other approachesin finding exact solutions to nonlinear di↵erential equations will also be discussed. This includes, but notlimited to, asymptotic analysis methodologies, the simplest equation method, the multiple exp-functionmethod, inverse scattering transform techniques, the upper-lower solutions method, the Hirota method, andothers.

Solutions and Conservation Lawsfor a Two Plus One DimensionalModified Korteweg De Vries

Mufid AbudaibTexas A&M University-Corpus Christi, USAChaudry Masood Khalique

In this talk we study a two plus one dimensional mod-ified Korteweg de Vries equation which has applica-tions in many scientific fields. We obtain exact solu-tions and conservation laws for this equation.

Symbolic Computation of Conserva-tion Laws and Exact Solutions of aCoupled Variable-Coe�cient Modi-fied Korteweg-De Vries System

Abdullahi AdemNorth-West University, So Africa

In this talk we study a generalized coupled variable-coe�cient modified Korteweg-de Vries (CVCmKdV)system that models a two-layer fluid, which is appliedto investigate the atmospheric and oceanic phenom-ena such as the atmospheric blockings, interactionsbetween the atmosphere and ocean, oceanic circula-tions and hurricanes. The conservation laws of theCVCmKdV system are derived using the multiplierapproach and a new conservation theorem. In addi-tion to this, a similarity reduction and exact solutionswith the aid of symbolic computation are computed.

A Predator-Prey Model with Star-vation Driven Di↵usion in Hetero-geneous Environment

Inkyung AhnKorea University, KoreaWonhyung Choi, Kwangjoong Kim

Recently, population models with starvation drivendi↵usions is introduced to represent the fitness prop-erty of certain species, which describes a random dis-persal strategy with a motility increase on starvation.In this talk, we discuss predator-prey models withstarvation driven di↵usion(SDD) in a heterogeneousenvironment. The Lotka-Volterra type is considered

to understand the e↵ect of SDD for the stability ofsemi-trivial solutions of models with SDD under no-flux boundary conditions. In addition, the positivecoexistence is investigated in terms of the eigenvalueanalysis for the linearized operator derived from themodel. The methods employed is the fixed-point the-ory applied to a positive cone on a Banach space. Wepresent the biological interpretation and simulationsbased on the result.

Cattanneo Christov Heat FluxModel Study for Water BasedCNT Suspended Nanofluid Past aStretching Sheet

Noreen AkbarNational University of Sciences and Technology,Pakistan

In this talk I will discussed the magnetic field ef-fects on the flow of Cattanneo–Christov heat fluxmodel for water based CNT suspended nanofluidover stretching sheet. Cattanneo–Christov heat fluxmodel for water based CNT suspended nanofluid isnot explored so far for stretching sheet. The flowequations will be presented first time in literaturetransformed into ordinary di↵erential equations us-ing similarity transformations. The numerical solu-tions using shooting technique will be presented andComparison will be presented with existing litera-ture. Graphical results will be presented to illustratethe e↵ects of various fluid flow parameters on veloc-ity, heat transfer, Nusselt number, Sherwood num-ber and skin friction coe�cient for di↵erent type ofnanoparticles.

Existence Result for a Class ofStrongly Nonlinear StochasticParabolic Initial Boundary ValueProblem

Zakaria AliUniversity of Pretoria, So AfricaAli Zakaria Idriss, Mamadou Sango

This paper treats a very important class of stochas-tic partial di↵erential equations. our main purposein this paper is to prove an existence result for suchtype of problems under Dirichlet boundary condi-


tions. The main obstacle in solving the present prob-lem is that the existence result cannot be easily re-trieved from the well known results of Krylov - Ro-zovskii and Etienne Pardoux illustrating the exten-sion of many papers including some in the referencestherein. Strong solutions are obtained for stochasticevolution equations driven byWiener processes underLipchitz assumptions of the random forces G(t, u).

Computation of Conservation Laws

Stephen AncoBrock University, Canada

Some recent work on general methods for computa-tion of conservation laws of di↵erential equations willbe reviewed.

Exact Solutions for Stokes FlowModel of an Incompressible Third-Grade Nanofluid

Asim AzizNational University of Sciences and Technology,PakistanTaha Aziz

The fully developed time-dependent flow and heattransfer of an incompressible, thermodynamicallycompatible non-Newtonian third-grade nanofluid isinvestigated. Classical Stokes model is considered inwhich the flow is generated due to the impulsive mo-tion of the plate in its own plane. Lie symmetryapproach is utilized to convert the governing nonlin-ear partial di↵erential equation to a system of linearand nonlinear ordinary di↵erential equations. Exactsolutions for the model equation are deduced in theform of closed-form exponential functions which arenot available in the literature before. The physicalfeatures of the pertinent parameters are discussed indetail through several graphs

Group Theoretical Analysis andInvariant Solutions for Time-Dependent Flow Model of a Non-Newtonian Fluid

Taha AzizDST/NRF Centre of Excellence in Mathematicaland Statistical Sciences, University of the Witwater-srand, So AfricaFazal M. Mahomed

The present work deals with the modelling and so-lution of the unsteady flow of an incompressiblethird grade fluid over a porous plate within a porousmedium. The flow is generated due to an arbitraryvelocity of the porous plate. The fluid is electricallyconducting in the presence of a uniform magneticfield applied transversely to the flow. Lie group the-ory is employed to find symmetries of the model equa-tion and these symmetries are used to transform theoriginal third order partial di↵erential equation to athird order ordinary di↵erential equations. The third

order ordinary di↵erential equations are then solvedanalytically and numerically. The manner in whichvarious emerging parameters have an e↵ect on thestructure of the velocity is discussed with the help ofseveral graphs.

Nonclassical Symmetries and Con-servation Laws for a Generalizationof the Korteweg-De Vries Equation

Maria BruzonUniversity of Cadiz, SpainTamara Garrido, Rafael de la Rosa

In this work, a class of sixth-order nonlinearwave equations, which leads to the well-knownSawada-Kotera-Caudrey-Dodd-Gibbons, Kaup-Kupershmidt, and Drinfeld-Sokolov-Satsuma-Hirotasystem of coupled Korteweg-de Vries equations, hasbeen considered to determine nonclassical symme-tries. Furthermore, nonlinear self-adjointness isproved and conservation laws are obtained.

Survival Strategy on a Predator-Prey Interacting System with SDD

Wonhyung ChoiKorea University, KoreaInkyung Ahn, Seunghyeon Baek

In this talk, we present a Lotka-Volterra typepredator-prey model with starvation driven di↵usionunder Neumann boundary condition. A starvationdriven di↵usion(SDD) is a dispersal strategy thatincrease their motility when food or other resourceis not su�cient. We investigate whether the star-vation driven di↵usion is a better survival strategythan constant di↵usion by using stability analysis forsemi-trivial solutions to the system. Furthermore,the extinction and coexistence of predator and preyare discussed.

On a Homogeneous Evolution Equa-tion: Lax Pair and Peakon Solutions

Priscila da SilvaUniversidade Federal do ABC, BrazilIgor Leite Freire

In this talk we discuss integrability and peakon solu-tions of a family of homogeneous evolution equations.Recursion operators are obtained and two membersrelated to KdV-type equations are shown to be com-pletely integrable using a Lax representation and theexistence of an infinite number of conserved quanti-ties. Conditions for the existence of peakon solutionsare then given.


Conditional Symmetries for Or-dinary Di↵erential Equations andApplications

Aeeman FatimaDST/NRF Centre of Excellence in Mathematicaland Statistical Sciences, University of the Witwater-srand, So AfricaFazal M. Mahomed

We refine the definition of conditional symmetries ofordinary di↵erential equations and provide an algo-rithm to compute such symmetries. A proposition isproved which provides criteria as to when the sym-metries of the root system of ODEs are inheritedby the derived higher-order system.We provide ex-amples and then investigate the conditional symme-try properties of linear nth-order equations subjectto root linear second-order equations. First this isconsidered for simple linear equations and then forarbitrary linear systems. We prove criteria when thesymmetries of the root linear ODEs are inherited bythe derived scalar linear ODEs and even order lin-ear system of ODEs. Furthermore, we show that ifa system of ODEs has exact solutions, then it ad-mits a conditional symmetry subject to the first orderODEs related to the invariant curve conditions whicharises from the known solution curves. Moreover, wegive examples of the conditional symmetries of non-linear third-order equations which are linearizable bysecond-order Lie linearizable equations. Applicationsto classical and fluid mechanics are presented.

A Multicomponent System ofCamassa-Holm-Novikov Equations

Igor FreireUFABC, BrazilDiego Catalano Ferraioli

In this talk we discuss a system of equations gen-eralising the Camassa-Holm and Novikov equations.Point symmetries are shown, as well as some classi-cal solutions. Peakons and multipeakon solutions arealso considered.

On Conservation Laws for a Boussi-nesq Equation with a DampingTerm

Maria Luz GandariasUniversity of Cadiz, SpainMaria Rosa

In this talk we present the classical Lie symmetriesadmitted by a Boussinesq equation with a damp-ing term as well as the reduced ordinary di↵erentialequations. We find a classification of the low-orderconservation laws for this equation. By using somesymmetry-invariant conservation laws we apply thedouble reduction method.

Interconnectedness of Group The-ory and Dynamical Systems Analy-sis

Kesh GovinderUKZN, So Africa

There are a number of di↵erent approaches to findinganalytic solutions to, as well as determining prop-erties of, solutions of di↵erential equations. Groupanalysis (which exploits the invariance of equationsunder transformations) has been very successfully ap-plied in the classification and explicit solution of dif-ferential equations. Singularity analysis (which re-quires solutions of equations to have, at worst, move-able poles) has also been used to classify equations.The long term evolution of systems of equations canbe determined by application of dynamical systemsanalysis (where the behaviour of equilibrium pointsis paramount).In this talk we will show how, in some cases, the re-sults obtained via these methods coincide (especiallyin parameter determination) while in other cases, thedi↵erent approaches are complementary. We will alsocomment on the role of transformations in these syn-chronous/complementary results and show that well–known results may not be as accurate as previouslythought. Examples from a variety of applications willbe used to illustrate our findings.

Noether and Lie Group Classifica-tion of a Radial Form of a CoupledHyperbolic System

Chaudry Masood KhaliqueNorth-West University, So Africa

In this talk we present a complete Noether symmetryclassification of the radial form of a coupled systemof hyperbolic equations and construct conservationlaws corresponding to the Noether operators. Liegroup classification of the system is then performed.

On a Di↵usive Predator-Prey Sys-tem with an Infected Prey withRatio-Dependent Functional Re-sponse

Kwangjoong KimKookmin University, KoreaInkyung Ahn

We examine a di↵usive ratio-dependent predator-prey system with disease in the prey under homo-geneous Dirichlet boundary conditions with a hos-tile environment at its boundary. We investigate thepositive coexistence of three interacting species (sus-ceptible prey, infected prey, and predator) and pro-


vide the asymptotical behavior of semi-trivial solu-tion with disease free. The methods are employedfrom a comparison argument for the elliptic problemas well as the fixed-point theory as applied to a pos-itive cone on a Banach space.

Di↵erentiation of Solutions to aParameter Dependent BVP withIntegral Boundary Conditions

Je↵rey LyonsNova Southeastern University, USAKaitlyn Seabrook

We discuss derivatives of the solution of a parame-ter dependent boundary value problem with an in-tegral boundary condition with certain assumptionsand its relationship to a nonhomogeneous di↵eren-tial equation closely associated with the traditionalvariational equation.

Lambda-Symmetries and Mu-Symmetries by Integrable Couplings

Wen-Xiu MaUniversity of South Florida, USA

We will talk about connections of lambda-symmetriesfor ODEs and mu-symmetries for PDEs with Lie-Baecklund symmetries of integrable couplings. Spe-cial integrable couplings are used to construct novellambda-symmetries and mu-symmetries. The clas-sification problem of integrable couplings of soli-ton equations and symmetries of integro-di↵erentialequations will be discussed.

Relativistic Stars and Symmetries

Sunil MaharajKwaZulu-Natal University, So Africa

A systematic analysis of the stellar surface junctioncondition is undertaken. This is a highly nonlinearpartial di↵erential equation in general. We obtainthe Lie point symmetries that leave the boundarycondition invariant. Using a linear combination ofthe symmetries, we transform the junction conditioninto simpler form. We present several new exact so-lutions to the junction condition. In each case we canidentify the exact solution with a Lie point generator.Some of the solutions obtained satisfy a barotropicequation of state. As a special case we regain pre-viously known models. Our analysis highlights theinterplay between Lie algebras, nonlinear di↵erentialequations and application to relativistic astrophysics.

Hamiltonians: Symmetries and In-tegrals

Fazal MahomedWits University, So Africa

We give explicit criteria when symmetries for aHamiltonian system correspond to their first inte-grals. It is shown what conditions need to be im-posed on the Hamiltonian symmetry so that it con-structively yields a first integral. We show that boththe classical and recent Noether based approachesare in fact equivalent. It is proved that when theHamiltonian symmetries provide first integrals theyform a Lie algebra. We prove that the Hamilton firstintegral is invariant under the Hamilton symmetry.Examples are provided.

Lie Symmetry Classification of aChemotaxis Model

Motlatsi MolatiNational University of Lesotho, Lesotho

We perform Lie symmetry classification of a chemo-taxis model. The model comprises a system of non-linear di↵erential equation with an arbitrary functionof the dependent variable whose functional form isspecified via the direct method of group classifica-tion.

Symmetry Analysis of a PotentialKadomtsev-Petviashvili Equationwith Power Nonlinearity

Dimpho MothibiNorth-West University, Mafikeng Campus, So Africa

This talk aims to study the potential Kadomtsev-Petviashvili equation with power nonlinearity, whicharises in many nonlinear problems of mathematicalphysics. We perform Noether symmetry classifica-tion and determine exact solutions for this equation.

Conservaton Laws for a GeneralizedHyperbolic Lane-Emden System

Ben MuatjetjejaNorth-West University, So Africa

In this talk, we aim to perform Noether symmetryclassification of a coupled (1+ 1)-dimensional hyper-bolic Lane-Emden system. Several cases arise forwhich Noether symmetries exist. In addtion, con-servation laws are constructed for each case.


Lie Ymmetries of Functional Di↵er-ential Equations of Hopf Type

Martin OberlackTU Darmstadt, GermanyDaniel Janocha, Marta Waclawczyk

Functional di↵erential equations are partial di↵eren-tial equations with an infinite set of independent vari-ables in a continuous sense. They appear in quantummechanics and turbulence theory, where it is calledHopf functional equation. In turbulence theory theHopf functional describes the full turbulence statis-tics as its n-th functional derivative defines the n-thstatistical moment. The Hopf equation contains bothfunctional as well as regular derivatives. As they areessentially no methods to derive solutions for thistype of equations we have extended Lie symmetrymethods. In a first example we show how to deriveLie symmetries of the Hopf functional equations de-rive from the Burgers equation.

Non-Modal Hydrodynamic StabilityTheory Based on an Extended Setof Lie Symmetries

Martin OberlackTU Darmstadt, GermanyAndreas Nold, Alexei Cheviakov, CedricSanjon

Classical hydrodynamic stability is, to a large part,based on the modal ansatz, which, implemented intothe linearized Navier-Stokes equation, leads to thefamous Orr-Sommerfeld equation for the eigenfunc-tions. Analyzing linearized Navier-Stokes equationwith respect to its Lie symmetries it turned out, thatthe modal ansatz is based on translation in space andtime as well as on scaling of the dependent variable- independent of the base flow, which is analysed onits stability. However, various canonical flows suchas linear shear, Couette, Poiseuille, pipe or Taylor-Couette flows exhibit one or more additional symme-tries, which in turn lead to invariant solutions thatare very di↵erent from the modal ansatz. Hence, it al-lows for a new and, in fact, non-modal type of hydro-dynamic stability theory. In turn, the resulting equa-tion for the respective eigenfunctions are derived.

On the General Solution Based onthe Time-Independent Integral, theLagrangian and The HamiltonianFunctions for Fin Equation

Ozlem OrhanIstanbul Technical University, TurkeyTeoman Ozer

We derive the time-independent integral for a nonlin-ear equation, namely fin equation in which the ther-mal conductivity and heat transfer coe�cient are as-sumed to be functions of the temperature. Using themodified Prelle-Singer approach, we point out thatexplicit the time-independent first integral and gen-

eral solution of the equation corresponding to theseintegrals can be identified for the fin equation fordi↵erent thermal conductivity and heat transfer co-e�cient functions. Then using this approach, an ap-propriate the Lagrangian and the Hamiltonian formsare obtained. Finally, we discuss on these solutionsby their graphics.

A Novel Method for Solving a Classof Nonlinear Integro-Di↵erentialEquations

Suares Clovis Oukouomi NoutchieNorth-West University, So AfricaRichard Guiem

The global solvability of a class of nonlinear non-autonomous integro- di↵erential equations describingcoagulation-fragmentation processes with growth isinvestigated using a modified monotone method. Ex-istence and uniqueness results are obtained thanks toGronwall’s inequality. In particular, a new conceptof upper-lower solution is introduced and the com-parison principle established.

Design, Analysis and Simulation ofa Robust Numerical Technique toSolve Fractional Partial Di↵erentialEquations Arising in ComputationalFinance

Kailash C. PatidarUniversity of the Western Cape, So AfricaF. Gideon, S.M. Nuugulu

Conventional partial di↵erential equation approacheshave been explored extensively over past threedecades to solve option pricing problems. With theprogress in computational methodologies, it is nowpossible to consider solving many complex problems,for example, fractional partial di↵erential equations(FPDEs) that are used to model more realistic phe-nomena. In this talk, we will discuss pricing of op-tions using such FPDEs. Apart from the construc-tion of the numerical method, we will also discussits thorough analysis. Finally, we will present somereliable numerical results for a class of problems forwhich the proposed approach is applicable.

Conservation Laws and Exact So-lutions for a Nonlinear p-LaplacianEvolution Equation

Elena RecioBrock University, CanadaStephen Anco, Maria S Bruzon, Maria LuzGandarias

In this work, we consider a nonlinear p-Laplacian evo-lution equation that in the initial value problem, ad-mits solutions exhibiting an interesting extinction be-havior. A complete classification of point and contactsymmetries and conservation laws is presented for the


corresponding n-dimensional radial p-Laplacian dif-fusion equation. The structure of a gradient flow ofthis equation is studied, leading to an useful energyidentity. In addition, exact group-invariant solutionsare obtained.

Exact Lie Symmetric Solutions ofNonlinear Partial Di↵erential Equa-tions

Barbara ShraunerWashington University, USA

Barbara Abraham-Shrauner, Washington University,St. Louis, MO A balance of nonlinearity of nonlinearpartial di↵erential equations (NLPDEs) terms iden-tifies if Jacobian elliptic functions are probably trav-eling wave solutions. Another approach of identifica-tion is to use NLPDEs solved by hyperbolic functions.Search for superposition of Jacobian elliptic functionsolutions of NLPDEs is simplified by a change of basisof sn, cn and dn functions. Examples include general-ized KdV equations, nonlinear Schrodinger equationsand Zakharov equations. The extended tanh methodfor the Zakharov equations is contrasted to the directmethod discussed here.

Nonlinear Di↵erential Equationsin Exterior Domains from GeneralRelativity

Nicolae TarfuleaPurdue University Northwest, USA

We study a class of nonlinear elliptic equations withnonlinear Neumann boundary conditions in exteriordomains. Problems of this type arise in many diversecontexts. An application related to the initial dataproblem in general relativity is presented.

On Kolmogorov’s Energy Spectrumfor MHD Turbulence

Tesfalem Abate TegegnUniversity of Pretoria, So AfricaMamadou Sango

A Fourier Analysis method has been used rigorouslyto investigate the spectral behaviour Magnetohydro-dynamics flow governed by:8>><

>>:

@tu+ (u ·r)u+r⇧ � (b ·r)b� ⌫�u = f1

in R3 ⇥ (0, T )@tb+ (u ·r)b� (b ·r)u� ⌘�b = f

2

in R3 ⇥ (0, T )r · u = r · b = r · f

1

= r · f2

= 0 in R3 ⇥ (0, T )u|t=0

= u0

, b|t=0

= b0

in R3

where u(x, t) is the flow velocity, b(x, t) is the mag-netic field, ⇧= p+ 1

2

|b|2 is the total pressure, ⌫ > 0is the kinetic viscosity of the fluid, ⌘ > 0 is the resis-tivity of the fluid, and f

1

and f2

are external forceson the system.

Denoting a Fourier transforms of u and b by u(⇠, t)and u(⇠, t) respectively, we define the energy spectralfunction by the surface integral

E(k, t) =

Z

|⇠|=k

|u(⇠, t)|2 + |b(⇠, t)|2dS(⇠),

where ⇠ 2 R3 is a Fourier space variable. Thecelebrated 1941 works of Kolmogorov predict thatthe Navier-Stokes equation in three dimensions withlarge Reynolds number should obey

E(k) ⇠ C0

✏3/2k�5/3,

for an inertial range k1

k k2

in some sense. Moti-vated by the 2012 work of Andrei Biryuk and WalterCraig, we have investigated bounds on the inertialrange for the weak solution of the MHD equation byusing Fourier transform, rate of energy dissipation.

Lie Symmetry Classification of aQuantum Hydrodynamical Model

Rita TracinaUniversity of Catania, Italy

In this talk we present the symmetry classificationobtained by the infinitesimal method of a quantumhydrodynamical model. The model comprises a sys-tem of partial di↵erential equations with arbitraryelements.

Bifurcations in Traveling Wave So-lutions of the Kuramoto SivashinskyType Perturbed Forced Equation

Muhammad UsmanUniversity of Dayton, USAChi Zhang

In this work a perturbation analysis is used to studythe Kuramoto Sivashinsky type perturbed forcedequation. Using traveling coordinates the PDE isconverted into an ODE. We drive the operating curvefor the relation between the bifurcation parameters.A condition is derived for the stability of the solutionsusing eigenvalue analysis. This model also includesthe delays.

Symmetry Reductions, Exact Solu-tions and Conservation Laws for aNew (3+1)-Dimensional NonlinearEvolution Equation with MKdVEquation Constituting Its MainPart

Emrullah YasarUludag University, Turkey

In this talk, we perform the Lie group method toa new (3+1)-dimensional nonlinear evolution equa-tion with mKdV equation constituting its mainpart. This new (3+1)-dimensional equation models


shallow-water waves and short waves in nonlinear dis-persive models with a more realistic way. The sym-metry reductions and exact solutions which includethe 1-soliton and 2-soliton solutions are obtained. Inaddition, the conservation laws of the equation arealso derived using the invariance-multiplier approach.

Local and Nonlocal Constants ofMotion in Conservative and Dissi-pative Dynamics

Gaetano ZampieriUniversity of Verona, ItalyGianluca Gorni

We give a recipe to generate nonlocal constants ofmotion for ODE Lagrangian systems and we applythe method to find useful constants of motion whichpermit to prove global existence and estimates of so-lutions to dissipative mechanical systems. We showexamples where our recipe can be used to find gen-uine first integrals too. Our applications are the me-chanical systems with homogeneous potential of de-gree �2, and the conservative Maxwell-Bloch systemwith RWA, where in particular we can separate one

of the variables. Finally, we show a generalizationof our recipe to some non-variational systems. As anapplication we find another (time-dependent) first in-tegral for the dissipative Maxwell-Bloch equations forsome special values of the parameters.

Traveling Wave Solutions andInfinite-Dimensional Linear Spacesof Multiwave Solutions to Jimbo-Miwa Equation

Lijun ZhangZhejiang Sci-Tech University, Peoples Rep of China

In this talk, the traveling wave solutions and multi-wave solutions to (3 + 1)-dimensional Jimbo-Miwaequation are investigated. We obtain the existenceof two families of bounded periodic traveling wavesolutions and their implicit formulas by analysis ofphase portrait of the corresponding traveling wavesystem.We derive the exact 2-wave solutions and twofamilies of arbitrary finite N-wave solutions by study-ing the linear space of its Hirota bilinear equation,which confirms that the (3 + 1)-dimensional Jimbo-Miwa equation admits multiwave solutions of any or-der and is completely integrable.


Special Session 58: Qualitative Properties of Nonlinear Di↵erentialEquations of Elliptic and Parabolic Type

Raul Manasevich, University of Chile, ChileMarta Garcia Huidobro, Catholic University, Chile

The aim of this session is to discuss new properties of solutions to the various types of nonlinear di↵erentialequations of elliptic and parabolic PDEs and systems as well as applications to importante applied problems.Topics will include nonlinear analysis methods, existence, uniqueness and stability of solutions, Liouville-typetheorems, bifurcation of solutions, pattern recognition, blow up of solutions, and related methods.

Precompactness of MinimizingSequences Under Multiple Indepen-dent Constraints

Santosh BhattaraiTrocaire College, USA

The variational technique based on the concentrationcompactness principle has been adapted by many dif-ferent authors to study existence and stability of soli-tary waves for a variety of nonlinear dispersive equa-tions. A key to proving the existence of minimizers(and hence the stability of solitary waves) using thismethod is to exclude possibilities of vanishing anddichotomy alternatives. The more delicate part is toprove that the dichotomy alternative cannot occur.A crucial technical obstacle in order to rule out thedichotomy alternative is to establish the strict sub-additivity condition. For one constraint problems,several techniques have been developed to establishthe strict subadditivity condition. To prove the strictsubadditivity condition for multiconstraint problems,however, seems to be more delicate. As a matter offact only a few papers address the issue of compact-ness of minimizing sequences for coupled systems ofdispersive equations. Moreover, in most of these pa-pers, the variational problem has consisted of eitheronly one constraint or two constraints were not in-dependently chosen. In this talk we consider somecoupled nonlinear Schrodinger systems and prove theprecompactness of the minimizing sequences undermultiple independent constraints. To exclude the di-chotomy alternative, we show the subadditivity con-dition using a technique based on certain rearrange-ment inequality. As a consequence of the concen-tration compactness, we also obtain the stability ofsolitary waves associated to the set of minimizers.

Multiplicity Results for Bound StateSolutions of a Semilinear Equation

Marta Garcia-HuidobroPontificia Universidad Catolica de Chile, ChileCarmen Cortazar, Pilar Herreros

In this talk we give conditions on the nonlineary f sothat the problem

�u+ f(u) = 0, x 2 RN , N � 2, lim|x|!1

u(x) = 0,

has at least two solutions having a prescribed numberof nodal regions and for which u(0) > 0.

Any nonconstant solution to this problem is called abound state solution. Bound state solutions that arenontrivial and nonnegative for all x 2 RN , are re-ferred to as a first bound state solution, or a groundstate solution.

Multiplicity of Solutions for anElliptic Equation with a SingularNonlinearity and A Gradient Term

Ignacio GuerraUniversidad de Santiago de Chile, Chile

We consider the problem

��u = �(1 + |ru|q)/(1� u), u 2 (0, 1) in B,

u = 0 on @B,

where B is the unit ball in RN , q � 0 and � � 0. Theproblem with q = 0 is well known. In fact, Joseph &Lundgren found that for 2 < N < 4 + 2

p2 there

are infinitely many solutions for some � = �⇤ > 0.On the other hand, they also found that for N >4 + 2

p2 there exists �⇤ such that there exists a

unique solution for each 0 < � < �⇤.Here we study the existence of solutions for this prob-lem when q > 0. In particular, we found a rangeof q and N where there exists �⇤ > 0 such thatthere are infinitely many solutions for � = �⇤.

Klein-Gordon-Maxwell Systems inBounded Spatial Domains

Monica LazzoUniversity of Bari, ItalyLorenzo Pisani

I will talk about a system of elliptic equations, ob-tained in the framework of Klein-Gordon-Maxwellsystems when looking for standing waves in equilib-rium with a purely electrostatic field. I will presentsome results about nonexistence, existence, and mul-tiplicity of solutions, under di↵erent boundary con-ditions and assumptions on the data. In view of thevariational structure of the system, most results arebased on global variational methods.


Obtaining Solutions with PrescribedNumber of Zeros for a NonlinearEquation Containing the P-LaplaceOperator with Weights

Raul ManasevichUniversity of Chile, ChileMarta Garcia-Huidobro, Carmen Cortazar,Jean Dolbeault

We consider radial sign-changing solutions of

div�a |ru|p�2ru

�+ b f(u) = 0 , lim

|x|!+1u(x) = 0,

where a and b are two positive, radial, smooth func-tions defined on Rd\{0}, f 2 C(R) satisfying someadditional conditions, and div

�a |ru|p�2ru

�de-

notes a p�Laplace operator with weight, p > 1.For radial solutions this problem becomes

�a(r)�p(u‘)

�‘ + b(r) f(u) = 0 , r > 0,

limr!0+

a(r)�p(u‘(r)) = 0 , limr!+1

u(r) = 0

where r = |x|, and a(r) = rd�1 a(x), b(r) =rd�1 b(x).Our problem is to find solutions of this last prob-lem changing sign exactly k times, for any arbitraryk 2 N.

Interior W 1,1 Estimates for So-lutions of Nonlinear DegenerateParabolic Systems

Truyen NguyenUniversity of Akron, USA

We study interior W 1,1 regularity for weak solutionsof nonlinear degenerate parabolic systems of the formut = divA(x, t, u,ru)+B(x, t, u,ru), which includethose of p-Laplacian type. We derive interior L1 es-timate for u by employing Moser’s iteration. Usingsome recent regularity results together with a suit-able modification of the arguments by DiBenedettoand Friedman, we also establish the boundedness ofru.

Moderate Solutions of SemilinearElliptic Equations with Hardy Po-tential

Phuoc Tai NguyenPUC, ChileMoshe Marcus

Let ⌦ be a bounded smooth domain in RN . We studypositive solutions of equation (E) �Lµu + uq = 0in ⌦ where Lµ = � + µ

�2, µ > 0, q > 1 and

�(x) = dist (x,@ ⌦). A positive solution of (E) is mod-erate if it is dominated by an Lµ-harmonic function.If µ is small then every positive Lµ- harmonic func-tions can be represented in terms of a finite measure

on @⌦ via the Martin representation theorem. How-ever the classical measure boundary trace of any suchsolution is zero. We introduce a notion of normalizedboundary trace by which we obtain a complete clas-sification of the positive moderate solutions of (E).

Travelling Waves for an Inter-nal Water Waves Model of theBenjamin-Ono Type

Jose QuinteroUniversidad del Valle, ColombiaJuan C. Munoz

In this talk we discuss the existence of solitary wavesolutions for the regularized Benjamin-Ono system8><

>:

⇣t � ((1� ↵⇠)u)x = ✏2

6

⇣xxt,

ut + ↵uux +⇣1� ⇢2

⇢1

⌘⇣x = ⇢2

⇢1✏H(uxt) + ✏2

6

uxxt.

(1)System (1) describes the propagation of a weaklynonlinear internal wave propagating at the interfaceof two immiscible fluids with constant densities ⇢

1

, ⇢2

with ⇢2

/⇢1

> 1, which are contained at rest in a longchannel with a horizontal rigid top and bottom, andthe thickness of the lower layer is assumed to be e↵ec-tively infinite (deep water limit). Constants a,✏ aresmall positive numbers such that ↵ = O(✏2) definedas ↵ = a

h1and ✏ = h1

Lthat measure the intensity

of nonlinear and the dispersive e↵ects, respectively.h1

denotes the thickness of the upper fluid layer andthe parameters L and a correspond to the character-istic wavelength and characteristic wave amplitude,respectively. The function u = u(x, t) is the veloc-

ity monitored at the normalized depth z = 1 �q

2

3

,

and ⇠ = ⇠(x, t) is the wave amplitude, measured withrespect to the rest level of the two-fluid interface.The existence of travelling waves is a consequence ofsome results for positive operators in a cone due toT. Benjamin, J. Bona and D. Bose , which is usefulin the case of not having an appropriate variationalstructure, as is the case

Boundary Blow-Up in PolyharmonicEquations with Power Nonlineari-ties

Paul SchmidtAuburn University, USA

We study explosive radial solutions of semilinear el-liptic PDEs involving an integer power of the Lapla-cian and a power-type nonlinearity. Depending onthe sign and monotonicity of the nonlinearity, twovery di↵erent types of blow-up behavior are observed.Explosive solutions of the first type diverge to infin-ity or negative-infinity, and their blow-up profile isby now fairly well understood. Explosive solutionsof the second type, which do not occur in the clas-sical second-order case, are unbounded from aboveand from below, oscillating wildly with increasing


amplitude and frequency; details of their asymptoticbehavior are only beginning to emerge. Our analy-sis employs dynamical-systems methods, applied toan associated system of asymptotically autonomousODEs.

Boundedness of Weak Solutionsof a Class of Non-Linear EllipticSystems with Morrey Data

Lyoubomira SoftovaSecond University of Naples, Italy

Let u : ⌦ ⇢ Rn ! RN , n,N � 2, be a weak solutionof the non-linear elliptic system

divA(x,u(x), Du(x)) = b(x,u(x), Du(x)), x 2 ⌦(2)

where ⌦ is a Reifenberg-flat domain and A(x, z,⌅)and b(x, z,⌅) are Caratheodory maps satisfying con-trolled growth conditions. Precisely, for n > 2 thereexist a constant⇤ > 0 such that for i = 1, . . . , n andj = 1, . . . , N it holds

(|aji (x, z,⌅)| ⇤

�'(x) + |z|

n

n�2 + |⌅|�

|bi(x, z,⌅)| ⇤� (x) + |z|

n+2n�2 + |⌅|

n+2n

�

for a.a. x 2 ⌦, all (z,⌅) 2 RN ⇥ RN⇥n and' 2 Lp,�(⌦), 2 Lq,µ(⌦) being suitable Morreyfunctions.We cannot expect boundedness of the solution u of(2) unless we impose some restrictions on the struc-ture of the operator A. For this goal, we suppose thatfor large values of uj , the operator A is component-wise coercive.We show that if u 2 W 1,2

0

(⌦,RN ) is a weak solutionof the system (2), than u is essentially bounded in⌦in terms of known quantities.

Bifurcation of Positive Solutions forthe One-Dimensional (p, q)-LaplaceEquation

Satoshi TanakaOkayama University of Science, Japan

In this talk, we study the bifurcation of positive solu-tions for the one-dimensional (p, q)-Laplace equation('p(u‘))‘ + ('q(u‘))‘ + �('p(u) + 'q(u)) = 0 underthe Dirichlet boundary condition u(�L) = u(L) = 0,where 'p(u) = |u|p�2u, p > 1. We investigate theshape of the bifurcation diagram and prove that thereexist five di↵erent types of bifurcation diagrams. As aconsequence, we prove the existence of multiple posi-tive solutions and show the uniqueness of positive so-lutions for a bifurcation parameter in a certain range.This is a joint work with Ryuji Kajikiya and MiekoTanaka.

Global Bifurcation Diagrams forProblems with Sign-Changing Non-linearities with High Multiplicity ofSolutions

Andrea TelliniEHESS, Paris, FranceJulian Lopez-Gomez, Marcela Molina-Meyer,Fabio Zanolin

In this talk I will present some results of high mul-tiplicity of positive steady states for the one dimen-sional logistic equation with di↵usion and a weight infront of the nonlinearity that changes sign. Moreover,I will show the structure of the bifurcation diagramsof such positive steady states, obtained by varying aparameter which modulates the positive part of theweight. In particular, I will focus on situations wheresuch diagrams are connected or present an arbitrar-ily high number of isolated components, according tothe properties of the weight. The results have beenobtained in collaboration with Julian Lopez-Gomez(Univ. Complutense de Madrid, Spain), MarcelaMolina-Meyer (Univ. Carlos III, Madrid, Spain) andFabio Zanolin (Univ. Udine, Italy).


Special Session 59: Mathematical Models of Cell Motility and CancerProgression in Microenvironment: Design, Experiments, Mathematical

Framework, and Hypothesis Test

Yangjin Kim, Konkuk University, KoreaYi Jiang, Georgia State University, USA

Cancer is a complex, multi-scale process, in which genetic mutations occurring at a sub-cellular level manifestthemselves as functional changes at the cellular and tissue scale. The main aim of this session is to discusscurrent stages and challenges in modeling tumor growth and developing therapeutic strategies. Specific goalsof the session include: (i) to analyze both computational and analytical solutions to mathematical modelsfrom tumor modeling (ii) to discuss creative ways of laboratory experimentation for better clinical diagnosis(iii) to improve our biochemical/biomechanical understanding of fundamental mechanism of tumor growthsuch as analysis of signaling pathways in relative balances between oncogenes and suppressors. Both theimmediate microenvironment (cell-cell or cell-matrix interactions) and the extended microenvironment (e.g.vascular bed) are considered to play crucial roles in tumour progression as well as suppression. Microenvi-ronment is known to control tumor growth and cancer cell invasion to surrounding stromal tissue. However,it also prohibits therapeutics from accessing the tumor cells, thus causing drug resistance. Therefore, athorough understanding of the microenvironment would provide a foundation to generate new strategiesin therapeutic drug development. At the cellular level, cancer cell migration is a main step for metastasisand further progression of cancer and metastasis in a given microenvironment. Thus, understanding of cellmotility under the control of signal transduction pathways would improve technical and specific advances incancer therapy by targeting the specific pathways that are associated with the diseases. Analysis of mathe-matical models would identify fundamental (abstract) structure of the model system and shed a light on ourunderstanding of tumor growth in the specific host tissue environment and biochemical and biomechanicalinteractions between players in cancer progression. More comprehensive multi-scale (hybrid) models can beused to meet the needs of designing patient-specific agents. The focus of this session is threefold: (a) topresent mathematical models of tumor growth and analysis of the models (b) to discuss therapeutic strate-gies for curing the disease and to showcase mathematical models incorporating mechanical aspects of cancercell movement and clinical implications (c) see recent development of cell-mechanical aspect of cell-ECMinteractions.

Exploiting Evolutionary Trade-O↵sAs a Novel Cancer Therapy

Alexander AndersonIMO, Mo�tt Cancer Centre, USAArig Ibrahim Hashim, Mark Robertson-Tessi,Robert Gillies, Robert Gatenby

Solid tumours export metabolically derived acid intosurrounding stroma. We view this microenviron-mental acidosis as a niche construction evolutionarystrategy in which acid-producing/acid-adapted can-cer cell phenotypes benefit by decreasing the fitnessof non-adapted stromal competitors, promoting localinvasion. These phenotypic properties, in turn, pro-mote transition from in-situ to invasive cancer anda progressive expansion of primary or metastatic tu-mours. However, there is a significant cost for main-taining an acid-producing/adapted phenotype due toreduced e�ciency in energy production and increasedenergy demand for adaptations to the acidic environ-ment. We hypothesize that this cost may be cancersAchilles heel and a novel route for therapeutic inter-vention. We have generated a multiscale mathemat-ical model that predicts that even subtle perturba-tions in pH can dramatically alter the progression ofinvasive and non-invasive tumour populations.

We investigated these model predictions in theTRAMP mouse model, our experiments demon-strated that an increase of intratumoural pHe by only0.2 pH units prevented transition from in situ to inva-sive cancer during carcinogenesis and significantly re-duced growth and invasion in primary and metastaticcancers. Taken together, our results demonstrate anovel strategy that exploits evolutionary trade-o↵sto steer the tumour population towards less invasivephenotypes.

A Model for Direction Sensing inDictyostelium Discoideum: RasActivity and Symmetry BreakingDriven by a Gbetagamma- Medi-ated, Galpha2-Ric8 – DependentSignal Transduction Network

Yougan ChengUniversity of Minnesota, USAHans Othmer

Many eukaryotic cells, including Dictyostelium dis-coideum (Dicty), neutrophils and other cells of theimmune system, can detect and reliably orient them-selves in chemoattractant gradients. In Dicty, sig-nal detection and transduction involves a G-protein-coupled receptor (GPCR) through which extracellu-lar cAMP signals are transduced into Ras activationvia an intermediate heterotrimeric G-protein (G2).Ras activation is the first polarized response to cAMPgradients in Dicty. Recent work has revealed mu-


tiple new characteristics of Ras activation in Dicty,thereby providing new insights into direction sensingmechanisms and pointing to the need for new modelsof chemotaxis. Here we propose a novel reaction-di↵usion model of Ras activation based on three ma-jor components: one involving the GPCR, one cen-tered on G2, and one involving the monomeric Gprotein Ras. In contrast to existing local excitation,global inhibition (LEGI) models of direction sensing,in which a fast-responding but slowly-di↵using ac-tivator and a slow-acting rapidly di↵using inhibitorset up an internal gradient of activity, our model isbased on equal di↵usion coe�cients for all cytoso-lic species, and the unbalanced local sequestrationof some species leads to gradient sensing and ampli-fication. We show that Ric8-modulated G2 cyclingbetween the cytosol and membrane can account formany of the observed responses in Dicty. includingimperfect adaptation, multiple phases of Ras activ-ity in a cAMP gradient, rectified directional sensing,and cellular memory.

Identifiability of Multistage ClonalExpansion Models in Cancer

Marisa EisenbergUniversity of Michigan, Ann Arbor, USAAndrew Brouwer, Rafael Meza

Multistage clonal expansion (MSCE) models of car-cinogenesis and other continuous-time Markov pro-cess models are often used to relate cancer and otherdisease incidence to biological mechanism. However,successful parameter estimation depends on the iden-tifiability of the underlying models. In this talk Iwill discuss recent work using a di↵erential algebra-based approach to examine the identifiability of aclass of clonal expansion models for cancer, deter-mining identifiable combinations that can be success-fully estimated from data and evaluating how di↵er-ent mechanisms can a↵ect parameter estimation.

Biomarker for Cancer

Avner FriedmanOhio State University, USA

Tumor microenvironment includes proteins, mRNAs,and microRNAs (miRs) expressed at abnormal lev-els. Some of these concentration levels are highlypositively correlated to their concentration in bloodserum, and could therefore potentially be used asnon-invasive biomarkers for early diagnosis of can-cer, as well as for monitoring the stage of cancer. Inthis talk I give two examples of potential biomark-ers. The first example is uPAR as a biomarker forearly detection of breast cancer recurrence. The sec-ond example involved expression of miRs in pancre-atic cancer. The corresponding mathematical modelsare represented by systems of PDEs in a growing tu-mor. Assuming radial symmetry, the expression levelof the potential biomarkers are quantitatively associ-ated with the growing radius of the tumor. This isjoint work with Wenrui Hao.

Modeling Cell-ECM Interactions inCancer Invasion

Yi JiangGeorgia Sate University, USA

The extracellular matrix (ECM) is important inmany cellular processes, from development to woundhealing and cancer invasion. While in vivo and invitro data have established the strong correlation be-tween the extracellular matrix remodeling and can-cer invasion, it is not clear how much biomechanicsmatter in cancer invasion or it is all genetics andbiochemistry. We have designed a series of mathe-matical models to understand how structure, density,alignment and mechanics of the ECM in regulatingcell migration. I will present our latest ongoing workon quantifying ECM structure during cancer invasionand on modeling of biomechanical cell-ECM interac-tions.

Mathematical Modeling of EnzymeCluster Formation in Cancer

Hye-Won KangUniversity of Maryland, Baltimore County, USA

Glycolysis is a metabolic pathway involving enzyme-catalyzed reactions. Previous studies have observedlarge clusters of the enzymes involved in glycolysisand increased production of serine in cancer cells.We hypothesize that the production of large clus-ters of the enzymes will change the glucose flux intoserine production in cancer cells, and constructed amathematical model to describe enzyme clusteringin normal and cancer cells using a system of ordinarydi↵erential equations. Using the method of the par-tial rank correlation coe�cient, a subset of sensitivemodeling parameters is identified. Enzyme turnovernumbers, Michaelis constant, and concentrations ofenzymes and metabolites were estimated based onthe literature. Using this model, we analyzed vari-ous cases with di↵erent-sized enzyme clusters in bothcancer and normal cells. We simulated these casesand were able to see the increased serine productionin cancer cells when there are large-sized enzyme clus-ters. The simulation result is qualitatively consistentwith the experimental result. This is joint work withSongon An and Jane Pan.

Cancer-Immune Dynamics of On-colytic Virotherapy and DendriticCell Vaccines

Peter KimUniversity of Sydney, AustraliaAdrianne Jenner, Il-Kyu Choi, Jana Gevertz,Joanna Wares, Arum Yoon, Chae-Ok Yun

Recent experiments with engineered oncolytic aden-ovirus have caused substantial reduction in growthrates of tumors in mice. We develop ordinary dif-ferential equation (ODE) models based on the datafrom five di↵erent treatments. By fitting time se-


ries data of tumor growth to our ODE models, weattempt to elucidate the underlying cancer-virus andcancer-immune dynamics to clarify the strengths andlimitations of oncolytic virotherapy combined withdendritic cell vaccines. Using modeling, we considerhow di↵erent treatment strategies can be used to(1) rapidly kill the tumor with a goal of completeelimination or (2) maintain the tumor long-term atlow levels. We also describe the problem of improv-ing the delivery of oncolytic virus into tumors. Im-ages show that viruses seem to penetrate hardly morethan a few millimeters from the site of injection andonly infect isolated and sparse clusters of cells, ratherthan dispersing comprehensively throughout the tu-mor. Understanding the kinetics of virus deliveryinto a tissue and the extracellular matrix poses auseful problem that could require the formulation ofpartial di↵erential equation or other spatial models.

Androgen Resistance Prediction inProstate Cancer Patients UnderAndrogen Suppression Therapy

Yang KuangArizona State University, USAJavier Baez

Predicting castrate resistant prostate cancer is criti-cal to optimize treatment and improve the quality oflife of advanced prostate cancer patients. We com-pare and analyze two plausible mathematical modelsthat aim to forecast future levels of prostate-specificantigen (PSA) with the help of clinical data of locallyadvanced prostate cancer patients undergoing andro-gen deprivation therapy (ADT). While these modelsare simplifications of a previously published model,they fit data with the same level of accuracy and im-prove forecasting results. Both models can predictthe progression of resistance. Model 1 models can-cer cells in a single population while model 2 dividescancer cells into two types. Model 1 is simpler butcan fit clinical data at the same or greater precision.However, we found the biologically more plausiblemodel 2 can forecast future PSA levels more accu-rately. These findings suggest that including morerealistic mechanisms of resistance may help predictlong-term androgen resistance.

Computational Simulations ofGlioma Invasion Using the Im-mersed Boundary Method

Wanho LeeNational Institute for Mathematical Sciences, KoreaSookkyung Lim, Yangjin Kim

Gliomas are malignant tumors that are commonlyobserved in primary brain cancer. Glioma cells mi-grate through a dense network of normal cells inmicroenvironment and spread long distances withinbrain. In this poster we present a two-dimensionalhybrid model in which a glioma cell is surrounded bynormal cells and its migration is controlled by cell-

mechanical components in the harsh microenviron-ment via the regulation of myosin II in response tochemoattractants. Our simulations reveal that themyosin II plays a key role in deformation of the cellnucleus as the glioma cell passes through the narrowintercellular space smaller than its nuclear diameter.In addition, our results suggest that in the presence ofmyosin II the strong signal of chemoattractants mayretract invasive glioma cells back to the resection siteso that they can be removed completely. This studysheds lights on the understanding of glioma infiltra-tion through the narrow intercellular spaces and a po-tential approach for the development of anti-cancerinvasion strategies.

Image-Guided Genomics RevealsPhenotypic Heterogeneity Support-ing a Symbiotic Model of CollectiveCancer Invasion

Adam MarcusEmory University, USAJ. Konen, E. Summerbell, B. Dwivedi, K.Galior, Y. Hou, L. Rusnak, A. Chen, J.Saltz, P. Vertino, L. Cooper, K. Salaita, J.Kowalski, A.I. Marcus

To probe the phenotypic heterogeneity found in cellpopulations, we developed an image-guided genomicstechnique termed spatiotemporal genomic and cellu-lar analysis (SaGA) that allows for precise selectionand amplification of living and rare cells. SaGA wasused on collectively invading 3-D cancer cell packsto create unique purified leader and follower cell cul-tures. The leader cell cultures are phenotypically sta-ble and highly invasive in contrast to follower cul-tures, which show phenotypic plasticity over timeand do not invade. Genomic and molecular inter-rogation reveals a VEGF-based vascular sproutingmimicry that facilitates recruitment of follower cellsbut not for leader cell motility itself, which insteadutilizes focal adhesion kinase- fibronectin signaling.While leader cells provide an escape mechanism forfollowers, follower cells in turn provide leaders withincreased proliferation and survival. These data sup-port a symbiotic model of collective invasion wheredi↵erent phenotypic cell types cooperate to promotetheir escape.

Decoding of Intracellular SignalTransfer from FRET Imaging: Dis-tinct Functions of Rac1 and Cdc42in Cell Migration

Honda NaokiKyoto Univ., JapanYamao Masataka, Shin Ishii

We propose a computational approach for decodinghow intracellular molecular signals are transferredduring cell migration. In this approach, we per-formed FRET time-lapse imaging of Rho GTPases,Rac1 and Cdc42, and quantitatively identified theresponse functions that describe how molecular sig-


nals are transferred into the motile morphodynam-ics. Based on the identified response functions, wefound that Rac1 and Cdc42 activation triggers later-ally propagating membrane protrusion, and that themembrane protrusion is driven by temporal deriva-tives of Rac1 and Cdc42 activities. Using the re-sponse function, we could predict the morphologi-cal change from molecular activity, and its predictiveperformance provides a new quantitative measure ofhow much the Rho GTPases participate in the cellmigration. Interestingly, we discovered distinct pre-dictive performance of Rac1 and Cdc42 dependingon the migration modes, indicating that Rac1 andCdc42 distinctly contribute to persistent and randommigration, respectively. Thus, our approach enabledus to uncover the hidden information processing rulesof Rho GTPases in the cell migration.

Role of Cellular and Microenvi-romental Heterogeneities in Mul-tidrug Resistance in Cancer: aMultiscale Approach

Gibin PowathilSwansea University, WalesMark Chaplain

Although multidrug combination chemotherapy hasbeen widely used in cancer treatment, the develop-ment of drug resistance by cancer cells continuesto be a major impediment to the successful deliv-ery of these multi-drug therapies. Studies have in-dicated that intra-tumoural heterogeneities have asignificant role in driving resistance to chemother-apy in many human malignancies. Multiple factorsincluding the internal cellular phenotypes, cell-cycledynamics and the external microenvironment con-tribute to the intra-tumoural heterogeneity. In thistalk, I will present a hybrid, multiscale, individual-based mathematical model, incorporating intracellu-lar dynamics and changes in oxygen concentration,to study the e↵ects of multidrug chemotherapy on aheterogeneous tumour. We will analyse various fac-tors that may contribute to the potential multidrugresistance within the heterogeneous cancer cell mass.

A Systems Biology Approach toUnderstanding Cell-Substrate In-teraction During Cell Spreading

Magdalena StolarskaUniversity of St. Thomas, USAAravind Rammohan

The mechanical interaction of a cell with its environ-ment a↵ects various intracellular processes. For ex-ample, stem cell fate is determined by substrate sti↵-ness, and extracellular matrix sti↵ness determinesmalignant phenotypes in cancer cells. We developa two-dimensional mathematical model and compu-tational tool that allows us to investigate how vari-ous mechanical processes interact during cell spread-ing over a deformable substrate. The cell is mod-

eled as a deformable hypoelastic continuum, and thesubstrate is treated as linear elastic. Focal adhe-sions, the macromolecular assemblies through whicha cell attaches to the substrate, are modeled as col-lections of linear springs that can dynamically growand shrink in a stress-dependent manner. The com-plex biochemical interactions governing the forma-tion of the focal adhesions are reduced to a singlereaction, and actin polymerization and actomyosincontractions are incorporated into the model via anactive rate of deformation tensor. We use this modelof the cell-substrate system to explain the mechanicalmechanisms required to obtain increased cell spreadareas on sti↵er substrates and to investigate the ef-fects of focal adhesion size on intracellular forces thatdevelop during spreading.

Multiscale Computational Mod-elling of Cancer Growth and Spread:a Novel Three-Scale MathematicalApproach

Dumitru TrucuUniversity of Dundee, Scotland

Cancer invasion is a complex multi-scale phenomenoninvolving many inter-related genetic, biochemical,and cellular processes at many di↵erent spatial andtemporal scales that play a crucial role in the overallcancer development. The process of invasion consistsof cancer cells secreting various matrix-degrading en-zymes, which degrade the surrounding tissue or theextracellular matrix. Combining abnormal prolifer-ation with favourable migratory conditions enabledby altered cell adhesion characteristics, the cancercells actively spread locally into the surrounding tis-sue. As these multiscale phenomena lead naturallyto a question concerning the establishment of a fun-damental framework that would enable a rigorousanalysis and modelling of cancer invasion, in thistalk we will present a new integrated multiscale mod-elling framework involving two biological scales, celland tissue, as well as the links between these cell-and tissue-scale processes. This novel two-scale ap-proach will explore the dynamics of cell adhesion in-side the tumour in conjunction with the activity ofvarious proteolytic processes occurring along the in-vasive edge of the tumour. Finally, we will presentcomputational simulations of the resulting multiscalemoving boundary model and discuss a number of im-portant fundamental properties that follow.

Quantitative Studies Reveal a RelayMechanism for TGF-Beta Signaling

Jianhua XingUniversity of Pittsburgh, USAJingyu Zhang, Xiao-Jun Tian, YiJiun Chen,Weikang Wang

Epithelial to mesenchymal transition (EMT), a pro-cess of transforming polygon-shaped epithelial cellswith tight cell-to-cell attachment to spindle-like mes-enchymal cells with loose or rare cell-to-cell attach-


ment, has been suggested to play a key role in manypathological processes such as fibrosis and cancermetastasis. Previous studies showed that exogene-sis signals such as TGF-� can induce EMT in manymammalian epithelial cell lines. In the canonicalTGF-� signal transduction pathway, transmembraneTGF-� receptors (TGFBR) receive the extracellu-lar signal, pass downstream via the Smad transcrip-tion factor family, and activate multiple genes suchas Snail1, a key regulator of EMT. However, ourmeasurements reveal a temporal gap between theearly pulsed upregulation of Smads and late sus-tained Snail1 expression. After careful examina-tion of the Smad dependent and independent TGF-�pathways, we hypothesize that sustained Snail1 acti-vation is achieved through a network motif composedof Smads, Gli1/2 (a main component of the SHHpathway), and GSK3�/� � catenin (main compo-nents of the WNT pathway). Our combined math-ematical modeling and quantitative measurementsconfirmed this hypothesis. Currently we are usingthe CRISPR-Cas9 technique to fuse fluorescence pro-teins to selected players in the network, and will uselive cell imaging to monitor the cell dynamics in realtime.

Controlling the Heterogeneous Cel-lular Quiescent State by an Rb-E2FNetwork Switch

Guang YaoUniversity of Arizona, USASarah Jungeun Kwon, Xia Wang, Nick Ev-eretts, Kimiko Della Croce

Quiescence is a critical non-proliferative cellularstate. Reactivating quiescent cells (e.g., fibroblasts,lymphocytes, and stem cells) to proliferation is fun-damental to tissue repair and regeneration. Oftendescribed as the “G0 phase“, quiescence is in fact nota homogeneous state. As cells remain quiescent forlonger durations, they move progressively “deeper“into quiescence, exiting from which requires pro-longed and stronger growth stimulation. Neverthe-less, underlying mechanisms of deep vs. shallow qui-escence remain an enigma, and represent a currentlyunderappreciated layer of complexity in growth con-trol. Previously, we have shown that the retinoblas-toma (Rb)-E2F gene network functions as a bistableswitch, converting graded and transient growth sig-nals into an all-or-none E2F activity, which underliesthe all-or-none transition from quiescence to prolif-eration. Here by coupling modeling and single-cellmeasurements, we show that quiescent depth is con-trolled by the activation threshold of the Rb-E2Fswitch, and that di↵erent network components havedi↵erent e�cacies in modulating quiescence depth.We also show that by a↵ecting the Rb-E2F activa-tion threshold, Notch pathway, circadian clock, aswell as metabolic state modulate quiescence depth.Further elucidating the control of quiescence depthmay help develop novel strategies to correct abnor-mal quiescent states of diseased cells.


Special Session 60: Infinite-dimensional Dynamical Systems fromDi↵erential Equations under Singular Perturbations

Alexandre Nolasco de Carvalho, Universidade de Sao Paulo, BrazilMarcone Corra Pereira, Universidade de Sao Paulo, Brazil

This special session will discuss recent results concerned with qualitative behavior of infinite-dimensionaldynamical systems subjected to singular perturbations which are originated by approximations in the pa-rameters of the model. Such dynamical systems are often associated to autonomous and non-autonomousdi↵erential equations which possess a maximal compact and invariant set that attracts bounded subsets un-der the action of the semi-flow and contains all relevant information about the asymptotics of the system andis an important object of study (the global attractors). There are many researchers working to understandits existence, characterisation, robustness, as well as upper and lower semicontinuity under perturbations.Observe that the study of elementary invariant sets such as steady state solutions, periodic orbits and theirunstable (stable) manifolds and their behavior under singular perturbation play an important role in thestudy of the characterisation and robustness of attractors.

Transverse Stability of NonlinearKlein-Gordon Periodic Wavetrains

Jaime Angulo PavaState University of Sao Paulo, BrazilRamon G. Plaza

In this talk we present the orbital stability of a classof spatially periodic wavetrain solutions to multi-dimensional nonlinear Klein-Gordon equations withperiodic potential. We show that the orbit gener-ated by the one-dimensional wavetrain is stable un-der the flow of the multi-dimensional equation underperturbations which are, on one hand, co-periodicwith respect to the translation (or Galilean) variableof propagation, and, on the other hand, periodic withrespect to the transverse directions. That is, we showtheir transverse orbital stability. The class of periodicwavetrains under consideration is the family of sub-luminal rotational waves, which are periodic in themomentum but unbounded in their position.

Limit of Nonlinear Elliptic Equa-tions with Concentrated Terms andVarying Domains

Simone BruschiUniversity of Brasilia, BrazilGleiciane S. Aragao

In this talk we discuss the limit of the solutions of asemilinear elliptic equations with nonlinear Neumannboundary conditions when the boundary presents ahighly oscillatory behavior and the nonlinear term isconcentrated in a region which neighbors the bound-ary of the domain. We consider two cases: the uni-formly and the non uniformly Lipschitz oscillatorybehavior in the boundary.

Non-Autonomous Dynamical Sys-tems and Their Attractors UnderPerturbations

Alexandre CarvalhoUniversidade de Sao Paulo, BrazilE. R. Aragao-Costa, M. C. Bortolan, T.Caraballo, J. A. Langa

We present a careful description of the relationshipbetween pullback and uniform attractors, leading toa detailed description of the uniform attractor andproviding the understanding of its dynamical struc-tures. That will enable us to talk about upperand lower semicontinuity, topological and geometri-cal structural stability of uniform attractors, at leastfor a non-autonomous perturbation of a semigroup.

A Di↵usion Equation with LocalizedChemical Reactions

Cesar Hernandez MeloMarina State University, BrazilEdgar Mayorga

In this work, we study the existence and stability ofequilibrium solutions of a reaction-di↵usion equationwith a Dirac delta distribution, Namely

ut = uxx + Z�(x)u+ wu+ aup + bu2p�1,

where u : R ⇥ [0,1) ! R denotes the unknown,� : H1(R) ! R defined by �(g) = g(0) is the Diracdelta distribution localized at zero, and a, b, p, Z de-note real numbers with p > 1.

The equilibrium solutions are obtained explicitly bydirect integration of the equation after removing thedistribution, as well as, from certain conditions whichappear as an e↵ect of the Dirac delta distribution.The stability/unstability of these solutions is ob-tained in the classical way: by analysing the spec-trum of the linear operator that results of the linear


approximation of the vector field associated to theequation at the equilibrium solution. In this part,the spectral analysis of these operators is based onthe analytic perturbation theory of linear operators.

Periodic Orbits of Strongly Indef-inite Ill-Posed PDEs Via RigorousNumerics

Jean-Philippe LessardLaval University, CanadaMarcio Gameiro,Roberto Castelli

In this talk, we introduce a computer-assisted tech-nique for the analysis of periodic orbits of ill-posedPDEs. As a case study, our proposed method is ap-plied to the Boussinesq equation, which has been in-vestigated extensively because of its role in the theoryof shallow water waves. The idea is to use the symme-try of the solutions and a Newton-Kantorovich typeargument (the radii polynomial approach), to obtainrigorous proofs of existence of the periodic orbits in aweighted ell-one Banach space of space-time Fouriercoe�cients with geometric decay. We present sev-eral computer-assisted proofs of existence of periodicorbits at di↵erent parameter values.

Stability of Partially DampedNonuniform Timoshenko System

To Fu MaUniversity of Sao Paulo, BrazilM. A. Jorge Silva, J. E. Munoz Rivera

This talk is concerned with the Timoshenko systemwith non-homogeneous coe�cients. The main re-sult shows the exponential stability of the partiallydamped system by requiring only locally, the well-known equal wave speeds assumption. Otherwise,we prove that the system is polynomially stable.

Bifurcation and Stability of SteadyStates of Parabolic Equations UnderIndefinite Logistic Flux Boundaryconditions

Gustavo MadeiraFederal University of Sao Carlos, BrazilA. S. do Nascimento

We consider the heat equation on a bounded smoothdomain supplied with a flux boundary conditiondriven by a logistic function, an indefinite weight anda positive parameter. Our aim is to completely de-scribe the bifurcation and stability properties of thesteady states for the dynamical system associated tothe problem and draw the corresponding diagrams.The techniques for establishing our results are de-manded according to the vanishing or not of the av-erage of the weight over the boundary of the domain.

Orbital Stability of Periodic Travel-ing Wave Solution for a Kawahara-Type Equation

Fabio NataliState University of Maringa, Brazil

In this talk, we investigate the orbital stability of pe-riodic traveling waves for a Kawahara-Type equation.We prove that the periodic traveling wave, under cer-tain conditions, minimizes a convenient functional byusing an adaptation of the well knownmethod developed by Grillakis, Shatah e Strauss.The required spectral property for the orbital sta-bility was obtained by knowing the positiveness ofFourier transform associated with the periodic wave.


On the A Priori Bounds and TheExistence of Positive Solutions forHamiltonian Elliptic Systems.

Rosa PardoUniversidad Complutense de Madrid, SpainNsoki Mavinga

In [1] the authors prove the existence of a-prioribounds for positive solutions of elliptic equations��u = f(u) with Dirichlet homogeneous bound-

ary conditions, when f(u) = uN+2N�2 / ln(e+ u)↵, with

↵ > 2/(N �2), see [1, Corollary 2.2]. Previous resultdo not cover this nonlinearity. In view of this recentadvance, it is natural to ask whether it is possibleto obtain the corresponding results for systems. Weprove a priori L1-bounds for classical positive solu-tions of Hamiltonian elliptic systems in bounded con-vex domains⌦ ⇢ RN , with a type of subcritical non-linearity. By subcritical we understand a nonlinear-

ity, say f = f(s, t), such that f(s, t)/|(s, t)|N+2N�2 ! 0

as |(s, t)| ! 1. Our analysis provides a new classof nonliterary depending on both components, forwhich classical positive solutions of Hamiltonian el-liptic systems are a priori bounded. We use a versionfor systems of moving planes arguments and an ex-tension of Rellich-Pohozaev type identity, combinedwith some estimates lying on Morrey’s Theorem.

References

[1] A. Castro and R. Pardo, A priori bounds forPositive Solutions of Subcritical Elliptic Equa-tions, Rev. Mat. Complut. 28 (2015), 715-731.

[2] N. Mavinga and R. Pardo, A priori bounds forpositive solutions of Hamiltonian elliptic sys-tems, Preprint.

Parabolic Problems in OscillatingThin Domains

Marcone PereiraUniversidade de Sao Paulo, Brazil

In this talk we discuss some recent results ob-tained about the asymptotic behavior of the solutionsof semilinear parabolic problems with homogeneousNeumann boundary conditions posed on two dimen-sional thin domains with locally periodic structureon the boundary. We first obtain the limit problemassuming the thin domain degenerates to the unit in-terval also analyzing its dependence with respect tothe geometry of the thin channel. Next we study theconvergence of the nonlinear semigroup investigatingthe upper and lower semicontinuity of the family ofglobal attractors taking dissipative assumptions tothe system.

Unbounded Attractors for ParabolicEquations

Juliana PimentelUniversidade Federal do ABC, BrazilA. Carvalho, C. Rocha

We consider non-dissipative semilinear parabolicequations on a bounded interval. We review the re-cently developed theory for the autonomous versionof this class of problems and present a characteri-zation for the associated noncompact global attrac-tor. We also examine a non-autonomous variationand take into account distinct regimes for the non-autonomous linear term; in this setting we investigatethe dynamics on the related unbounded pullback at-tractor.

On Quasiconvergence of Solutionsof Parabolic Equations on the RealLine

Peter PolacikUniversity of Minnesota, USA

We examine bounded solutions of semilinearparabolic equations on the real line. Such a so-lution is quasiconvergent, if it approaches a set ofsteady states in a localized topology. We start byshowing examples of bounded solutions which are notquasiconvergent. Then we identify classes of initialdata which give quasiconvergent solutions. Some ofthese results are based on joint papers with HiroshiMatano.

Dynamic and Continuity for aNon-Classical Non-AutonomousDi↵usion Problem

Felipe RiveroUniversidade Federal Fluminense, BrazilMatheus C. Bortolan

In this talk I will show our study of the exis-tence and continuity of four di↵erent notions ofnon-autonomous attractors for a family of non-autonomous non-classical parabolic equations givenby

⇢ut � �(t)�ut � �u = g✏(t, u), in⌦

u = 0, on @⌦.

in a smooth bounded domain⌦ ⇢ Rn, n > 3, wherethe terms g✏ are a small perturbation, in some sense,of a function f that depends only on u.


Behavior of the P-Laplacian on ThinDomains

Ricardo SilvaUniversity of Brasilia, BrazilMarcone C. Pereira

In this talk we present the limiting behavior of so-lutions of quasilinear elliptic equations on thin do-mains. As we will see the boundary conditions playan important role. If one considers homogeneousDirichlet boundary conditions the sequence of solu-tions will converges to the null function, whereas, ifone considers Neumann boundary conditions there isa non trivial equation which determines the limitingbehavior.

Reaction-Di↵usion Problems withLarge-Di↵usion Up to the Boundary

Alejandro Vidal-LopezXi‘an Jiaotong-Liverpool University, Peoples Rep ofChinaAnibal Rodriguez-Bernal

We will discuss some results regarding the continuityof reaction-di↵usion problems when the di↵usion insome region of the domain becomes larger and larger.This induces a homogenisation process which deter-mines the type of the limit problem. We will considerthe case in which the large-di↵usion area is either aneighbourhood of the boundary or it has non-emptycontact with parts of the boundary having Robin andDirichlet conditions.


Special Session 61: Recent Trends in Navier-Stokes Equations andRelated Problems

Sarka Necasova, Institute of Mathematics, Academy of Sciences of the Czech Republic, Czech RepReimund Rautmann, University of Paderborn, Germany

Werner Varnhorn, University of Kassel, Germany

Due to the active research on this field in many places around the world, there are important new resultsin diverse directions: e.g. concerning local smoothness criteria for 3D-solutions, regularity by properties ofthe vorticity, singular solutions, lower bounds to blowing up solutions, norm inflation, analyticity in generalspaces, Lagrangean representation of flows, fluid flow with chemotaxis. The aim of our special session is tobring together reseachers working in di↵erent directions and to initiate fruitfull discussions.

Global Well-Posedness of the Ax-isymmetric Navier-Stokes Equationsin the Exterior of an Infinite Cylin-der

Ken AbeKyoto University, Japan

We consider the initial-boundary value problem ofthe Navier-Stokes equations for axisymmetric initialdata with swirl in the exterior of an infinite cylinder,subject to the slip boundary condition. We constructglobal solutions and estimate potential singularitiesof axisymmetric flows in the whole space by using thesize of the cylinder. The proof is based on the Boussi-nesq system. We show that the system is globallywell-posed in the exterior domain for axisymmetricinitial data without swirl.

Local Regularity Condition Involv-ing Two Velocity Components forthe Navier-Stokes Equations

Hyeong-Ohk BaeAjou University, KoreaJorg Wolf

The present paper deals with the problem of lo-cal regularity of weak solutions to the Navier-Stokesequation in⌦ ⇥ (0, T ) with forcing term f in L2.We prove that u is strong in a sub-cylinder Qr ⇢⌦ ⇥ (0, T ) if two velocity components u1, u2 satisfy-ing a Serrin-type condition.

Regularity of the 3D Navier-StokesEquations and Related Problems

Mimi DaiUniversity of Illinois Chicago, USAAlexey Cheskidov

As one of the most significant problems in the studyof partial di↵erential equations arising in fluid dy-namics, Leray’s conjecture in 1930s regarding theappearance of singularities for the 3-dimensional(3D) Navier-Stokes equations (NSE) has been neitherproved nor disproved. The problems of blow-up havebeen extensively studied for decades using di↵erenttechniques. By using a method of wavenumber split-ting which originated from Kolmogorov’s theory of

turbulence, we obtained a new regularity criterion forthe 3D NSE. The new criterion improves the classi-cal Prodi-Serrin, Beale-Kato-Majda criteria and theirextensions. Related problems, such as the well/ill-posedness, will be discussed as well.

On Measure Valued Solutions to theEuler and Navier-Stokes System

Eduard FeireislCzech Academy of Sciences, Prague, Czech RepE. Chiodaroli, P. Gwiazda, O. Kreml, A.Swierczewska-Gwiazda, E. Wiedemann

We introduce a new concept of dissipative measurevalued solution to the Euler and Navier-Stokes sys-tem. We show several properties of these solutionsincluding the weak-strong uniqueness principle withapplications to certain stability problems. We alsoshow the existence of a measure-valued solution of thecompressible Euler system that cannot be obtainedas a limit of a sequence of weak solutions. This is insharp contrast with the measure-valued solutions ofthe incompressible Euler system when this is alwayspossible.

Onsager’s Conjecture for the Com-pressible Models

Piotr GwiazdaPolish Academy of Sciences, PolandEduard Feireisl, Agnieszka Swierczewska-Gwiazda, Emil Wiedemann

We give su�cient conditions on the regularity of solu-tions to the inhomogeneous incompressible Euler andthe compressible isentropic Euler systems in order forthe energy to be conserved. Our strategy relies oncommutator estimates similar to those employed byP. Constantin W. E and E. Titi for the homogeneousincompressible Euler equations.


Robustness of Strong Solutions ofthe Navier-Stokes Equations withVarious Types of Boundary Condi-tions

Petr KuceraCzech Technical University, Czech Rep

We study qualitative properties of solutions of a sys-tem of the Navier-Stokes equations with various typesof boundary conditions. Some properties of thesesolutions, e.g. local in time existence of strong so-lutions, are presented. Further, consequences of per-turbations of initial conditions of strong solutions areinvestigated. We prove that the corresponding solu-tions are also strong for su�ciently small perturba-tions in some norms.

Existence Results for Equations De-scribing Multicomponent ReactiveFlows

Martine MarionEcole Centrale de Lyon, FranceR. Temam

We investigate some mathematical issues arising inthe context of the coupling of multi-species exother-mic chemical reactions to fluid motion. The incom-pressible Navier-Stokes equations are coupled withequations for the temperature and the concentrationsof the chemical species. The equations for the chem-istry involve non-linear di↵usion coe�cients thatare obtained by resolution of the so-called Stefan-Maxwell equations.

The Motion of Incompressible Vis-cous Fluid Around a Moving RigidBody

Sarka NecasovaAcademy of Sciences, Institute of Math., Czech RepS.Kracmar, P. Deuring

The dynamics of fluids, i.e. liquids and gases, is animportant part of the continuum mechanics. Thislecture is devoted to the qualitative analysis of math-ematical models of motion of a viscous incompress-ible fluid around a compact body B, translating androtating in the fluid with given time–independenttranslational and angular velocities u1 and !. Thetranslation can be considered, without the loss of gen-erality, to be parallel to the x

3

–axis. We shall discussthe fundamental solution of the Oseen rotating sys-tem and the asymptotic decay for the Oseen case andalso for nonlinear case,

Stability of a Quasi-Steady Flow ofa Viscous Incompressible Fluid Pasta Rotating Obstacle

Jiri NeustupaCzech Academy of Sciences, Czech Rep

We deal with stability of a solution of the mathemat-ical model, describing the flow of a viscous incom-pressible fluid past a rotating body. The consideredsolution is steady in the body-fixed frame. We showthat the stability follows from a “su�ciently fast” de-cay (in time) of a finite number of suitable functionsin a norm restricted to a “su�ciently large” boundedregion around the body. We also discuss the relationto the spectrum of an associated linearized operator.No assumption on the smallness of the steady solu-tion is required.

Navier-Stokes Flow in the WeightedHardy Spaces

Takahiro OkabeHirosaki University, JapanYohei Tsutsui

The asymptotic expansions of the Navier-Stokes flowin the whole spaces and the rates of decay are dis-cussed with aid of weighted Hardy spaces. Fuji-gaki and Miyakawa, Miyakawa proved the n-th or-der asymptotic expansion of the Navier-Stokes flowif initial data has suitable pointwise decay in spaceand n-th moment is finite. In this presentation, it isclarified that the moment condition for initial data isessential in order to obtain higher order asymptoticexpansion of the flow and to consider the rapid timedecay problem. Firstly we derived the existence the-orem in the weighted Hardy spaces with smallnessonly in Ln as a refinement of the previous work ofthe second author, Tsutsui. As an application, therapid time decay of the flow are investigated withaid of asymptotic expansions and of the symmetryconditions introduced by Brandolese.

Heat-Conducting, CompressibleMixtures with Multicomponent Dif-fusion

Milan PokornyCharles University in Prague, Czech RepVincent Giovangigli, Piotr B. Mucha, TomaszPiasecki, Ewelina Zatorska

We study a model for heat conducting compress-ible chemically reacting gaseous mixture, basedon the coupling between the compressible Navier–Stokes–Fourier system and the full Maxwell–Stefanequations. The viscosity coe�cients are density-dependent functions vanishing on vacuum and theinternal pressure depends on species concentrations.We consider the question of existence of a solutionto this system and based on several levels of approx-imations we construct a weak solution without any


restriction on the size of the data. Furthermore, wealso consider (for a mixture of isomers) the steadyversion of the system above. The viscosity coe�-cients are considered to be temperature-dependent.We construct weak solutions for di↵erent formula-tions of the problem (weak solution or variationalentropy solution).

Monotonicity in Singular VolterraIntegral Equations and Application.

Reimund RautmannPaderborn University, Germany

The wellknown bounds to the resolvent of linearsingular Volterra integral equations imply a usefulmonotonicity in the prescibed data. The applicationto a (seemingly new) approximation scheme to the3D- initial-boundary value problem of the Navier-Stokes equations leads to the proof of convergenceand error estimates in a scale of Banach spaces oneach compact time interval in case of su�cientlysmall initial values.

News on the Helmholtz Decomposi-tion and Very Weak Solutions

Juergen SaalUniversity Dusseldorf, Germany

The aim of the talk is to construct very weak so-lutions to the Navier-Stokes equations for a gen-eral class of boundary conditions including partialslip type. We essentially follow the approach givenby Herbert Amann which relies on interpolation-extrapolation scales generated by the correspondingStokes operator. In this regard also a suitable exten-sion of the Helmholtz projector to function spaces ofnegative order plays a crucial role. Related to thisfact, Amann’s approach includes a misinterpretationof the properties of this Helmholtz extension opera-tor. A second aim of the talk therefore is to clarifythis misinterpretation and to demonstrate that it hasno influence on the correctness of Amann’s results onthe existence of very weak solutions.

Optimal Control in Blood FlowSimulations

Adelia SequeiraUniversity of Lisbon, PortugalJorge Tiago, Telma Guerra

Blood flow simulations can be improved by integrat-ing known data into the numerical simulations. DataAssimilation techniques based on a variational ap-proach play an important role in this issue. We pro-pose a non-linear optimal control problem to recon-struct the blood flow profile from partial observationsof known data in di↵erent geometries. To simplify,blood flow is assumed to behave as a Navier-Stokesfluid. Using a Discretize then Optimize (DO) ap-proach, we solve a non-linear optimal control prob-

lem and present numerical results that indicate itsrobustness with respect to di↵erent idealized geome-tries and measured data. Blood flow in real vesselswill also be considered, including the discussion of aparticular clinical case.

The Regularity of Solutions to theNavier-Stokes Solutions Based onItems of the Velocity Gradient

Zdenek SkalakCzech Technical University, Czech Rep

We will present a survey of recent criteria for theregularity of the solutions to the 3D Navier-Stokesequations based on information on several items ofthe velocity gradient.

About Questions of Stability forMultiple Gasballs

Gerhard StrohmerUniversity of Iowa, USA

We are considering barotropic gasballs in space. Weshow that an energy stability condition implies thelinear stability of such balls and discuss the conse-quences for non-linear stability.

On Ill-Posedness of Euler Systemwith Non-Local Terms

Agnieszka Swierczewska-GwiazdaUniversity of Warsaw, PolandJ. A. Carrillo, E. Feireisl, P. Gwiazda

The talk will concern the issue of existence of weaksolutions to the Euler equations with pairwise at-tractive or repulsive interaction forces and non-localalignment forces in velocity appearing in collectivebehavior patterns. We consider several modificationsof the Euler system of fluid dynamics including itspressureless variant driven by non-local interactionrepulsive-attractive and alignment forces in the spacedimension N = 2, 3. These models arise in the studyof self-organisation in collective behavior modeling ofanimals and crowds. We adapt the method of con-vex integration, adapted to the incompressible Eu-ler system by De Lellis and Szekelyhidi, to show theexistence of infinitely many global-in-time weak so-lutions for any bounded initial data. Then we con-sider the class of dissipative solutions satisfying, inaddition, the associated global energy balance (in-equality). The discussed result is in a certain sensenegative result concerning stability of particular solu-tions. It turns out that the solutions must be soughtin a stronger class than that of weak and/or dissi-pative solutions. We essentially show that there areinfinitely many weak solutions for any initial dataand that there is a vast class of velocity fields thatgives rise to infinitely many admissible (dissipative)weak solutions.


Local in Time Regularity Regionfor General Weak Solutions of theNavier-Stokes Equations

Werner VarnhornKassel University, GermanyReinhard Farwig, Hermann Sohr

We consider a general weak solution of the Navier-Stokes equations concerning the unsteady motion ofa viscous incompressible fluid. Our main result con-cerns Leray’s structure theorem (Leray 1934) , ex-tending some well-known results (Galdi 2000) to sev-eral directions. In particular, we do not assume thevalidity of the strong energy inequality for the under-lying weak solution.


Special Session 62: Imaging Methods in Coupled Physics Models

Alexandru C. Tamasan, University of Central Florida, USA

A current trend in Inverse Problems assume models where interacting fields are model by two or more physicalphenomena. Examples include but are not llimited to Elastography, Thermo/Photoacoustic, AcoustoOptics,Magnetic Resonance Electrical Impedance Tomography (MREIT), Current Density Impedance Imaging,Lorentz force based Electrical Impedance Tomography. This minisymposium will bring together researchersin the above mentioned areas to present their advances made both in the mathematics, and the modelingemployed in acquiring the data in these imaging methods. The organizers believe that time passed since thelast minisymposium within the AIMS meeting (the 9th) in these areas is optimal for reporting in the newfindings.

Photoacoustic Imaging and Ther-modynamic Attenuation

Sebastian AcostaBaylor College of Medicine, USACarlos Montalto

We consider a mathematical model for photoacousticimaging to take into account attenuation due to ther-modynamic dissipation. The propagation of acousticwaves is governed by a scalar wave equation coupledto the heat equation for the excess temperature. Weseek to recover the initial acoustic profile from knowl-edge of boundary measurements. This inverse prob-lem is a special case of boundary observability for athermoelastic system. This leads to the use of con-trol/observability tools to prove the unique and sta-ble recovery of the initial acoustic profile in the weakthermoelastic coupling regime. We propose and im-plement (numerically) a reconstruction algorithm.

Generalized Radon TransformsArising in Single Scattering OpticalTomography

Gaik AmbartsoumianUniversity of Texas at Arlington, USA

Single scattering optical tomography (SSOT) useslight photons that scatter once in the body to re-cover its interior features. The mathematical modelof 2D SSOT is based on the broken ray (or V-lineRadon) transform, which puts into correspondence toan image function its integrals along V-shaped piece-wise linear trajectories in a plane. The related con-ical Radon transform appears in some 3D imagingtechniques based on Compton scattering e↵ect. Theprocess of image reconstruction in these modalitiesrequires inversion of the corresponding transforms.The talk will discuss the known results and recentdevelopments in the study of these integrals trans-forms, their applications and the open problems.

On the Stability of ReconstructingConvection Terms from BoundaryMeasurements

Toufic el ArwadiBeirut Arab University, LebanonHoda Malak, Alexandru Tamasan

This talk concerns the inverse convection problem.A. Tamasan introduced a direct method to recon-struct the convection terms. However, the inverseconvection problem is ill posed. In this talk, a reg-ularization technique is presented. Some technicallemmas and estimates for the forward and inverseproblem are presented, moreover, error estimates areobtained from this regularization will be discussed .

Direct Computation of the Radia-tive Transfer Equation for Near-Infrared Light Propagation in Bio-logical Tissue

Hiroshi FujiwaraKyoto University, JapanNaoya Oishi

Near-infrared light is a new modality to measurebrain activities noninvasively. To realize it, the anal-ysis of the radiative transfer equation (RTE), whichis a mathematical model of light propagation in tis-sue, is required. Since the discretization of RTE isa large-scale problem, Monte Carlo method has beenconventionally used. In this talk we show some nu-merical examples using the direct discretization ofthe stationary RTE in the three dimensions for accu-rate and reliable imaging based on the recent progressof computer resources.

Fourier Expansion of Disk Automor-phisms and Scattering in LayeredMedia

Peter GibsonYork University, Canada

An exotic family of orthogonal polynomials on thedisk serves to formulate a remarkable Fourier expan-sion of the composition of a sequence of Poincaredisk automorphisms. The resulting identity is inti-


mately connected with the scattering of plane wavesin piecewise constant layered media. Indeed, a re-cently established combinatorial analysis of scatter-ing sequences provides a key ingredient of the proof.At the same time, the polynomial obtained by trun-cation of the Fourier expansion elegantly encodes thestructure of the nonlinear measurement operator as-sociated with the finite time duration scattering ex-periment.

Solvability of Discrete Inverse Prob-lems on Networks

Fernando Guevara VasquezUniversity of Utah, USA

The complex geometric optics approach has beenused to show uniqueness for several inverse problems,including the electrical impedance tomography prob-lem. We show how to use ideas from complex geomet-ric optics to show uniqueness for inverse problems onnetworks, where the goal is to recover possibly com-plex node or edge based quantities. Our results showthat if the linearized problem is injective then thenon-linear problem admits a unique solution, excepton a zero measure set.

Photoacoustic Tomography withCircular Integrating Detectors

Yulia HristovaUniversity of Michigan - Dearborn, USASunghwan Moon, Dustin Steinhauer

We propose a detector geometry for photoacoustic to-mography that o↵ers the practical advantage that itcould be implemented with a single rotating circulardetector. The Radon type transform that arises canbe decomposed into the spherical Radon transformand the Funk transform. An inversion formula andnumerical simulations demonstrating the proposedalgorithm will be presented.

Schwartz Theorem for the RadonTransform and Metal Artifact Re-duction

Sungwhan KimHanbat University, KoreaChi Young Ahn, Kiwan Jeon

In computed tomography (CT), X-ray photons pass-ing through metallic objects in a X-ray scanned ob-ject are highly attenuated and the number of photonsdetected at the detector pixels behind them is actu-ally zero. This photon starvation causes the metalshadow area in the sinogram domain in which theprojection data are considered to be missing. CTimages reconstructed by the filtered back projection(FBP) from such sinogrms su↵er from severe streakartifacts which deteriorate the image quality andmake it di�cult to interpret valuable details closeto metallic objects for medical diagnosis. During thelast couple of decades, it has become a challenging

problem to reduce streak artifacts caused by metallicobjects. One of important methods for metal arti-fact reduction is the projection completion approach.In this method, the missing projections in the sono-gram domain are synthesized from metal-free pro-jections neighboring to the metal shadow area usingvarious inpainting techniques such as polynomial in-terpolation, wavelet interpolation, adaptive filtering,total variation, etc. In this talk, we show that if asinogram synthesized by the projection completiontechnique is in the range of the X-ray Radon trans-form, its reconstructed CT image is same to the trueimage outside the convex hull of metal regions im-planted in the scanned object and introduce a novelprojection completion algorithm which exploits theSchwartz theorem for the X-ray Radon transform inorder to reduce metal artifacts in CT.

Nonlinear Imaging with Waves ViaReduced Order Model Backprojec-tion

Alexander MamonovUniversity of Houston, USAVladimir Druskin, Andrew Thaler, MikhailZaslavsky

We introduce a novel nonlinear method for imagingreflectors with waves. The method is based on re-duced order models (ROMs). The ROMs are projec-tions of the wave equation propagator on the sub-space of certain time domain wavefield snapshots.These projections can be computed entirely fromthe time domain data measured on the boundary.The image is a backprojection of the ROM usingthe subspace basis for a known kinematic velocitymodel. Nonlinear implicit orthogonalization of wave-field snapshots di↵erentiates our approach from con-ventional linear methods. It automatically removesthe multiple reflection artifacts. Moreover, it dou-bles the resolution in range direction compared toconventional time reversal. Both seismic explorationand medical ultrasound imaging applications are con-sidered.

Multiwave Imaging in an Enclosurewith Variable Sound Speed

Carlos Montalto CruzUniversity of Washington, USASebastian Acosta

This talk considers the problem of photoacoustic andthermoacoustic tomography in the presence of phys-ical boundaries such as reflectors or interfaces, whichreflect some wave energy back into the domain. Tomodel the physical boundaries we consider the waveproblem where an impedance Robin boundary condi-tion is imposed. We relate the inverse problem witha statement in boundary observability and stabiliza-tion of waves. We present uniqueness and stability


of the inverse problem and propose two di↵erent re-construction methods. In both cases, if well-knowngeometrical conditions are satisfied, the approachesare naturally suited for variable wave speed and formeasurements on a subset of the boundary.

Photoacoustic Tomography Modelwith Varying Material Density andVariable Bulk Modulus.

Kamran SadiqThe Johann Radon Institute for Computational andApplied Mathematics (RICAM), AustriaA. Beigl, P. Elbau , O. Scherzer

We study the Photoacoustic tomography model tak-ing into consideration that the material has variabledensity and bulk modulus is spatially varying. Wepropose an approach to simultaneously reconstructabsorption density, bulk modulus and material den-sity from photoacoustic measurements using photoa-coustic sectional imaging.

Acoustic CT for Concrete Struc-tures

Takashi TakiguchiNational Defense Academy of Japan, Japan

For the time being, there exist no non-destructiveinspection method for concrete structures which en-ables us to reconstruct their interior structure, con-cretely and completely. In this talk, we shall dis-cuss how to develop such a non-destructive inspec-tion method for concrete structures, for which weshall apply ultrasonic waves. We shall pose a mathe-matical problem in order to establish a tomographicnon-destructive inspection method and discuss howto solve this problem from the viewpoint of practicalapplications.


Special Session 63: Topological Methods for Nonlinear Boundary andInitial Value Problems

John R. Graef, University of Tennessee at Chattanooga, USAMiroslawa Zima, University of Rzeszow, Poland

Topological methods have proved to be an important technique in the study of boundary and initial valueproblems and related topics for ordinary and partial di↵erential equations. There has been a rapidly growinginterest in applying these techniques to such problems in recent years, and this session is devoted to the useof such methods in the study of boundary and initial value problems including singular problems and thosewith multi-point conditions.

On the Solution to the ConservedKuramoto-Sivashinski Equation

Gabriella BognarUniversity of Miskolc, Hungary

The study of the growth processes of surfaces hasturned out to be source of a variety of nonlinear dy-namics. The growth is intrinsically two-dimensionaland the variable describing the growth dynamics isthe local height h(x, y, t) of the surface. Our aim isto examine the solutions to the conserved Kuramoto-Sivashinsky equation (CKS)

ht + ��h+ �h+ � |rh|2

�= 0, (x, y) 2 (a, b)2, t > 0

and � is a positive parameter. The CKS equation dis-plays coarsening in the growth processes of surfaces.

Asymptotically Linear System ofThree Equations Near Resonance

Maya ChhetriUNC Greensboro, USAPetr Girg

We will consider an asymptotically linear system ofthree semilinear equations satisfying Dirichlet bound-ary conditions. The nonlinear perturbations areCaratheodory functions that are bounded by someappropriate nonnegtaive function. There are onlytwo simple eigenvalues, associated to the linear part,whose corresponding eigenfunctions are component-wise nonnegative. We will discuss bifurcation of pos-itive solutions from infinity from these simple eigen-values. In particular, we will provide su�cient condi-tions under which the system has bifurcation of posi-tive solutions (from infinity) from both, one, or noneof the simple eigenvalues.

A Survey on Recent Results Relatedto the Pumping E↵ect

Jose Angel CidUniversity of Vigo, Spain

We shall review some recent results about the ex-istence of positive periodic solutions for a singularsecond order equation. This boundary value prob-lem was presented in 2006 by G. Propst as a simplemodel for the “pumping e↵ect”.

Generalized Third Order CoupledSystems with Full Nonlinearities

Infeliz CoxeESP- Malanje, AngolaFeliz Minhos

This work studies the solvability of a nonlinear thirdorder coupled system composed by coupled third or-der fully di↵erential equations with L1-Caratheodorynonlinearities, and two-point boundary conditions.An adequate truncature together with Nagumo-typeconditions allow the dependence of the nonlinearitieson the second derivatives. By lower and upper solu-tions method is obtained some strips where the un-known functions and their derivatives must lie, whichprovides some qualitative data on the solutions.

On the Existence of Three PositiveSolutions for a Two-Point Second-Order BVP

Abdulkadir DoganAbdullah Gul University, Turkey

In this paper, we consider the existence of threepositive solutions for the boundary value problem,with using the new fixed-point theorem and impos-ing growth conditions on the nonlinearity. The inter-esting point is that the nonlinear term f explicitlyinvolves a first-order derivative.

Green’s Functions of Di↵erentialSystems with Reflection

F. Adrian F. TojoUniversity of Santiago de Compostela, SpainAlberto Cabada

In this work we generalize the previous theory of dif-ferential equations with reflection to the case of sys-tems. This is achieved by remaking the classical ap-proach to ordinary di↵erential systems through fun-damental matrices. In this process we can appreciatesome important similarities and di↵erences betweenboth cases.


Modelling Idiopathic Scoliosis - aTest Case Using ATT(Antalgic-TrakTechnology) Device

Joao FialhoAmerican University of the Middle East, Kuwait

In this paper the author studies and develops themathematical model for idiopathic scoliosis in humanspine and its therapy using the ATT (Antalgic TrackTechnology) device. The therapy for idiopathic scol-iosis using this device relies on the application of ver-tical traction to the spine. It is a relatively new tech-nique that di↵ers from the commonly used orthoticstabilization, making the existent models inaccurateor insu�cient.The model is composed by the fourth order di↵eren-tial equation

EIy(iv)1

(x) + Py001

(x) = �Py000

(x) + Py0

(x) (1)

for x 2⇥L2

,�L2

⇤and P, EI, constants to be defined,

along with the boundary conditions

y1

✓�L

2

◆= y

1

✓L2

◆= 0, (2)

y001

✓�L

2

◆= 0, (3)

�EIy001

✓L2

◆+ kiy1

✓L2

◆= 0. (4)

The arguments applied are based on the upper andlower solution method and a test case is studied asthe main result. In this test case, based on the infor-mation provided by the upper and lower solution, atherapy plan using the ATT device is outlined.

Existence Results for Initial ValueProblems of First-Order Systems ofStieltjes Di↵erential Equations

Marlene FrigonUniversity of Montreal, CanadaR. Lopez Pouso

We present the basic theory of existence and unique-ness of solutions for systems of di↵erential equa-tions with the usual derivative replaced by a Stieltjesderivative. This derivative, called g-derivative, wasintroduced by Lopez Pouso and Rodrıguez [1]. Theproblems that we consider contain as particular casesdynamic equations on time scales and impulsive or-dinary di↵erential equations.

References

[1] R. Lopez Pouso and A. Rodrıguez, A new uni-fication of continuous, discrete, and impulsivecalculus through Stieltjes derivatives, Real Anal.Exchange 40 (2014/15), no. 2, 1–35.

[2] M. Frigon and R. Lopez Pouso, Theory and ap-plications of first–order systems of Stieltjes dif-ferential equations, Adv. Nonlinear Anal. (to ap-pear).

An Application of a New Cone toBVPs with Nonlocal, NonlinearBoundary Conditions

Christopher GoodrichCreighton Preparatory School, USA

We demonstrate that by using a new order cone,existence results for boundary value problems withnonlocal, nonlinear boundary conditions can be im-proved. In addition, the cone is flexible enough to beuseful in the case of vanishing or nonpositive Green’sfunctions. Finally, together with the use of a particu-lar open set, existence results utilizing only pointwise(rather than limit- or interval-type) conditions can begiven.

A Nonlinear Fractional BoundaryValue Problem Involving Riemann-Liouville Fractional Derivatives ofTwo Di↵erent Orders

John GraefUniversity of Tennessee at Chattanooga, USAMin Wang, Grant Yost

The authors consider the nonlinear boundary valueproblem consisting of the fractional di↵erential equa-tion

�D↵0+

z + aD�0+

z = w(t)f(t, z), 0 < t < 1,

and the boundary conditions


z(0) = z‘(0) = · · · = z(n�2)(0) = 0, z(1) = k,

where k 2 R, a 2 R, n 2 N, 0 � n � 2, n � 1 <↵ < n, w 2 L[0, 1] with w(t) � 0, f 2 C([0, 1]⇥R,R),and D�

0+

is the �-th left Riemann-Liouville fractionalderivative. They give su�cient conditions for the ex-istence and uniqueness of solutions to this problem.In so doing, they introduce a new approach that maybe useful in analyzing such problems with other typesof boundary conditions.

Impulsive Boundary Value Prob-lems on Unbounded Domains

Nickolai KosmatovUniversity of Arkansas at Little Rock, USA

We apply cone-theoretic methods to study positivesolutions of impulsive di↵erential equations of secondorder on an unbounded domain. The formulation ofthe problem involves infinitely many impulsive con-ditions.

Positive Solutions of Some Bound-ary Value Problems Via the FixedPoint Index Theory for NowhereNormal-Outward Compact Maps

Kunquan LanRyerson University, Canada

In this talk, I shall briefly present recent joint workon the fixed point index theory for nowhere normal-outward compact maps. Applications to the exis-tence of positive solutions for some boundary valueproblems will be given.This is joint work with Professor Guangchong Yang

Existence of Solutions to a DiscreteFourth Order Periodic BoundaryValue Problem

Xueyan LiuUniversity of Tennessee at Chattanooga, USAJohn Graef, Lingju Kong

We consider periodic boundary value problems for afourth order nonlinear di↵erence equations. We usevariational methods and critical point theory to getsu�cient conditions for the existence of at least onesolution, two solutions, and nonexistence of solutions.The advantage that primitive function of the nonlin-ear term is not involved in the conditions of our mainresults make it much easier to apply our theorems inspecific problems.

Existence of Solutions of SingularFractional Boundary Value Problem

Je↵rey LyonsNova Southeastern University, USAChristina A. Hollon, Je↵rey T. Neugebauer

We use iterative and fixed point methods to show theexistence of a positive solution of a fractional bound-ary value problem D↵

0+

x+ f(t, x,Dµ0+

) = 0, 0

Green’s Functions for 2n-OrderBoundary Value Problems

Lucıa Lopez SomozaUniversity of Santiago de Compostela, Spain

We express the Green’s functions of the 2n-orderlinear di↵erential equation coupled with di↵erentboundary value conditions as a linear combination ofGreen’s functions of Periodic problems. From thatexpressions we are able to deduce some results inspectral theory and to compare the solutions of theconsidered problems.

Heteroclinic Solutions for �-Laplacian Equations

Feliz MinhosEvora University, Portugal

In this paper we consider a second order discon-tinuous equation with a � -Laplacian, in the realline, with � an increasing homeomorphism such that�(0) = 0 and �(R) = R. We remark that the ex-istence of heteroclinic solutions is obtained withoutasymptotic or growth assumptions on the nonlinear-ities � and f. Moreover, as far as we know, our mainresult is even new when �(y) = y.

Existence and Comparison Resultsfor Eigenvalues of a Higher OrderFractional Boundary Value Problemwith a Fractional Boundary Condi-tion

Je↵rey NeugebauerEastern Kentucky University, USAAngela M. Koester

Let ↵ 2 (n � 1, n]. We show the existence ofand then compare smallest eigenvalues of the frac-tional boundary value problems D↵

0

+u+�1

p(t)u = 0,D↵

0

+u + �2

q(t)u = 0, t 2 (0, 1), satisfying boundary

conditions u(i)(0) = 0, i = 0, 1, . . . , n� 2, D�

0

+u(1) =0, � 2 [0, n � 1], where p and q are nonnegativecontinuous functions on [0, 1] which do not vanishidentically on any nondegenerate compact subinter-val of [0, 1]. Here D↵

0

+ and D�

0

+ denote the standardRiemann-Liouville fractional derivatives. The caseswhere � = 0 and � > 0 are treated separately andthen compared.


Positive Solutions for First-Order Di↵erential Equations withRiemann-Stieltjes Integral Bound-ary Conditions

Seshadev PadhiBirla Institute of Technology, India

This article concerns the existence of positive solu-tions of the first-order di↵erential equation

x‘(t) = r(t)x(t) + f(t, x(t)) t 2 [0, 1],

x(0) =

Z1

0

h(s, x(s)) d↵(s),

where the nonlinear boundary condition is aRiemann-Stieltjes integral. We use the Leray-Schauder and the Leggett-Williams fixed point theo-rems to obtain positive solutions.

Oscillation Criteria for Second Or-der Nonlinearneutral Di↵erentialEquations with Deviating Argu-ment

Saroj PanigrahiUniversity of Hyderabad, India

With a geometric idea, we obtain a set of new oscilla-tion criteria for the forced second order neutral delaydi↵erential equation with deviating argument of theform

(x(t) + p(t)x(�(t)))“ + q(t)f(x(⌧(t))) = g(t).

This criteria improves the results obtained by Q.Kong, M. Wang, Oscillation criteria for a forcedsecond order di↵erential equations with deviating ar-guments, Commu. App. Anal. 16 (2012), 459-470.

Recent Developments in Delay Dif-ferential Equations on Time Scales

Gnana Bhaskar TenaliFlorida Institute of Technology, USAT. Gnana Bhaskar

Time scale, arbitrary nonempty closed subset of R(with the topology and ordering inherited from R),is an e�cient and general framework to study dif-ferent types of problem to discover the common-

alities and highlight the essential di↵erences. Wediscuss some recent developments in the existencetheory of functional Dynamic Equations includinga few (counter) examples. In particular, we dis-cuss first order functional dynamic equations withdelay, namely, x�(t) = f(t, xt) on a time scale.Here xt 2 C([�⌧, 0],Rn) and is given by xt(s) =x(t + s), �⌧ s 0. We consider an appropriatetimescale on which such delay equations can be stud-ied meaningfully. We establish an existence result forthe solutions of such problems. We also present a fewexamples.

On a Fractional Boundary ValueProblem with a Positive Green’sFunction

Min WangEquifax, USAJohn R. Graef, Lingju Kong, Qingkai Kong

In this paper, the authors study a nonlinear fractionalboundary value problem. The associated Green’sfunction is derived as a series of functions. Criteriafor the existence and uniqueness of positive solutionsare then established based on it.

Existence of Positive Solutions forFirst-Order Resonant NonlocalProblem

Miroslawa ZimaUniversity of Rzeszow, Poland

We discuss the existence of positive solutions for afirst-order equation subject to a nonlocal conditionformulated in the terms of the Riemann-Stieltjes in-tegral. We are interested in the resonant case. Ourapproach relies on the application of the theoremon positive solutions for semi-linear equations due toO‘Regan and Zima (2006).


Special Session 64: Dynamics of Evolutionary Equations in the AppliedSciences

Tomas Caraballo, Universidad de Sevilla, SpainXiaoying Han, Auburn University, USA

Felipe Rivero, Universidade Federal Fluminense, Brazil

Evolutionary di↵erential equations are primary mathematical tools used to model various problems froma large variety of subjects such as biology, physics, engineering, material sciences, neurosciences, etc. Inadditional to fundamental questions such as the existence and well-posedness of evolution systems, moree↵orts in recent years have been devoted to the study of dynamics and qualitative/quantitative propertiesof the systems, e.g., asymptotic behaviors, special structures of solutions and stability/instability of specialstructures, travelling waves, temporal/spatial chaos, etc. Results in these areas have shown significantimpact on various interdisciplinary areas, such as atmosphere and ocean studies, macro-micro mechanicsof materials, cancer research, etc. The main goal of this special session is to present most recent resultson various dynamical aspects of evolutionary systems, with an emphasis on those arising from the appliedsciences.

On the Rate of Convergence of the2d Stochastic Alpha Model

Hakima BessaihUniversity of Wyoming, USAPaul Razafimandimby

We study the convergence of the solution of the twodimensional stochastic Leray-↵ model to the solutionof the 2-D stochastic Navier-Stokes equations. Weare mainly interested in the rate, as ↵ ! 0, of thefollowing error function

✏↵(t) = sups2[0,t]

ku↵(s)� u(s)kL2

where u↵ and u are the solution of stochastic Leray-↵ model and the stochastic Navier-Stokes equations,respectively. We show that when properly localizedthe error function ✏↵ converges in mean square as↵! 0 and the convergence is of order O(↵). We alsoprove that ✏↵ converges in probability to zero withorder at most O(↵).

Rumour Flow As a Dynamical Sys-tem on a Social Network

Bernard BrooksRochester Institute of Technology, USA

Mathematical models can provide some insight intohow rumours flow on a social network such as Face-book. Understanding rumour and information flowin situations of high anxiety and uncertainty is crit-ical. Disaster rumours such as occurred in the af-termath of Katrina in New Orleans hampered res-cue e↵orts and threatened public safety. Health ru-mours encourage African Americans in Chicago toavoid cancer screening. Rumours that American sol-diers distribute pornography to Iraqi children solidifythe mistrust and hatred that decimates any feelingsof safety or wellbeing both in Iraq and the world over.Our interdisciplinary team of mathematicians, socialpsychologists and computer scientists embarked ona multi-year e↵ort to understand how beliefs in ru-mours propagate across social networks. The diversenature of our research group resulted in mathemat-

ical models with empirically calibrated parameters.A summary of our models will be presented. Thesemodels include a generalized SIR style model of ru-mour as epidemic, rumour as a dynamical systemflowing over various network topologies and computerassisted panel studies (CAPS) which are used to cal-ibrate the models‘ parameters.

Construction of Quasi-Periodic Re-sponse Solutions for Forced Systemswith Strong Damping

Renato CallejaIIMAS-UNAM, MexicoLivia Corsi, Alessandra Celletti, Rafael de laLlave

I will present a method for constructing quasi-periodic response solutions (i.e. quasi-periodic so-lutions with the same frequency as the forcing) forover-damped systems. Our method applies to non-linear wave equations subject to very strong dampingand quasi-periodic external forcing and to the var-actor equation in electronic engineering. The strongdamping leads to very few small divisors which allowsto prove the existence by using a contraction mappingargument requiring very weak non-resonance condi-tions on the frequency. This is joint work with A.Celletti, L. Corsi, and R. de la Llave.

Impulsive Nonautonomous Dynam-ical Systems

Rodolfo CollegariUSP, BrazilE. M. Bonotto, M. Bortolan, T. Caraballo

In this work we define the notions of impulsive non-autonomous dynamical systems“ and impulsive co-cycle attractors“. We also establish conditions to en-sure the existence of an impulsive cocycle attractorfor a given impulsive non-autonomous dynamical sys-tem, which are analogous to the continuous case.


Stochastic Shell Models of Turbu-lence with Fractional Noise

Maria Garrido-AtienzaUniversity of Seville, SpainHakima Bessaih, Bjorn Schmalfuß

We consider some shell models of turbulence in a verygeneral form. These are phenomenological approxi-mations of the Navier-Stokes equations, with a vis-cous linear part that is dissipative and a nonlinearpart that is not globally Lipschitz. We assume thatthis model is perturbed by a multiplicative fractionalBrownian noise with Hurst parameter H 2 (1/2, 1).We will prove the existence and uniqueness of a path-wise mild solution, and the proof will be achieved intwo steps. In the first step we shall prove the exis-tence and uniqueness of variational solutions to theShell model but driven by smooth paths, for whichwe are able to get some important uniform estimatesin appropriate functional spaces. In a second step, byusing these estimates and a compactness argument,we are able to pass to the limit, showing that thislimit is a pathwise mild solution for the Shell modeldriven by a fractional Brownian motion.

Epsilon Optimal Controls forControl of Stochastic NonlinearSchrødinger Equations

Wilfried GreckschMartin-Luther-University Halle-Wittenberg, Ger-many

We introduce a stochastic Schrødinger equation inthe sense of variational solutions over rigged Hibertspaces (V,H, V ⇤) driven by a cylindrical Wiener pro-cess W for certain admissible controls U

dX(t) = �iAX(t)dt+ f1

(t,X(t))dt+ f2

(t)U(t)dt+

+ g(t,X(t))dW (t), X(0) = X0

2 V. (1)

The optimal control problem consists of the minimiz-ing of a integral functional

J(U,X) = E

Z T

0

(J1

(X(t)) + J2

(U(t))) dt (2)

with respect to U . We approximate (1) by the so-lutions of linearized problems (see [1]). Then wecalculate optimal controls by using of a maximumprinciple for the linearized problem (1) and the cor-responding goal functional (2). These controls are"-optimal for the original problem.

References

[1] Grecksch, W., Lisei, H., Linear Approximationof nonlinear Schrødinger Equations Driven byCylindrical Wiener Process. To appear.

Periodic Structure of Some Zero En-tropy Di↵eomorphisms: the Sphereand The Torus Case

Juan L.G. GuiraoTechnical University of Cartagena, SpainJaume Llibre

We consider discrete dynamical systems given by aself–di↵eomorphism f defined on a given compactmanifold M. In this setting usually the periodic or-bits play an important role. In dynamical systems of-ten the topological information can be used to studyqualitative and quantitative properties of the system.Perhaps the best known example in this direction arethe results contained in the paper entitle Period threeimplies chaos for continuous self–maps on the inter-val. For continuous self–maps on compact manifoldsone of the most useful tools for proving the existenceof fixed points, or more generally of periodic points, isthe Lefschetz Fixed Point Theorem and its improve-ments. The Lefschetz zeta function Zf (t) simplifiesthe study of the periodic points of f . This is a gen-erating function for the Lefschetz numbers of all it-erates of f . In talk we put our attention in the classof discrete smooth dynamical systems defined by theMorse–Smale di↵eomorphisms on the tori and thesphere.


Demographic Stochasticity andEvolution of Dispersion

Hyejin KimUniversity of Michigan-Dearborn, USAYen Ting Lin, Charles Doering

The selection of dispersion is a classical problem inecology and evolutionary biology. Deterministic dy-namical models of two competing species di↵eringonly in their passive dispersal rates suggest that thelower mobility species has a competitive advantagein inhomogeneous environments, and that dispersionis a neutral characteristic in homogeneous environ-ments. We consider models including local popula-tion fluctuations due to both individual movementand random birth and death events to investigatethe e↵ect of demographic stochasticity on the compe-tition between species with di↵erent dispersal rates.For homogeneous environments where deterministicmodels predict degenerate dynamics in the sense thatthere are many (marginally) stable equilibria withthe species‘ coexistence ratio depending only on ini-tial data, demographic stochasticity breaks the de-generacy. A novel large carrying capacity asymptoticanalysis, confirmed by direct numerical simulations,shows that a preference for faster dispersers emergeson a precisely defined time scale. While there is noevolutionarily stable rate for competitors to choose inthese spatially homogeneous models, the stochasticselection mechanism is the essential counterbalancein spatially inhomogeneous models including demo-graphic fluctuations which do display an evolution-arily stable dispersal rate.

Robustness of Attractors for a Fam-ily of Parabolic Equations withNonlocal Di↵usion

Pedro Marin-RubioUniversidad de Sevilla, SpainTomas Caraballo, Marta Herrera-Cobos

The existence of weak solutions to a family ofreaction-di↵usion problems with nonlocal viscosityterm is established. However, uniqueness of solutionis not guaranteed. Then, some continuity proper-ties and the long-time behavior of the solutions areanalyzed in the framework of the theory of attractors.

This is a joint work with T. Caraballo andMarta Herrera-Cobos, from Universidad de Sevilla(SPAIN). It has been partially supported by FEDERand Ministerio de Economıa y Competitividad(Spain) Grant MTM2011-22411 and by Junta de An-dalucia Grant P12-FQM-1492. M.H.-C. is a fellow ofPrograma de FPI del Ministerio de Economıa y Com-petitividad, reference BES-2012-053398.

On the Spectral Stability of Trav-eling Fronts for Reaction Di↵usionDegenerate Equations

Ramon PlazaUniversidad Nacional Autonoma de Mexico, MexicoJuan Francisco Leyva

Motivated by biological applications (e.g. spa-tial ecology, bacterial aggregation), several reaction-di↵usion models consider density-dependent di↵u-sion coe�cients which, in addition, are degeneratein one (or more) equilibrium points of the reaction.These degenerate di↵usions describe, for example,the avoidance of crowded areas by individuals of cer-tain biological populations. In this talk I present newresults and techniques in the study of spectral stabil-ity of traveling fronts for reaction-di↵usion equationswith degenerate di↵usion. I will explain the two mainideas to control, on one hand, the essential spectrumand, on the other hand, the point spectrum of the lin-earized operator around the wave. Both techniquesare designed to overcome the degeneracy of the dif-fusion at the end point. This is joint work with J.Francisco Leyva.

Colony and Evolutionary Dynamicsof a Two-Stage Model with BroodCannibalism and Division of Laborin Social Insects

Marisabel RodriguezArizona State University, USAYun Kang

Division of labor (DOL) is a major factor for thegreat success of social insects because it increases thee�ciency of a social group because di↵erent individ-uals perform di↵erent tasks repeatedly and presum-ably with increased performance. Cannibalism playsan important role in regulating colony growth anddevelopment by regulating the number of individu-als in a colony and increasing survival by providingaccess to essential nutrients and minimizing competi-tion among colony mates. To understand the synergye↵ects of division of labor and cannibalistic behavioron colony dynamic outcomes, we propose and studya compartmental two-stage model using evolution-ary game theory settings. Our analytical results ofthe evolutionary models suggest that: (1) Division oflabor can prevent colony death and natural selectioncan preserve strong Allee e↵ects by selecting the traitswith the largest investment on brood care and thelowest cannibalism rate. (2) Natural selection mayincrease the fitness of the colony, i.e. the successfulproduction of workforce which results in the increaseof total worker population size, colony survival andreproduction. Our results suggest both cannibalismand division of labors are adaptive strategies thatincrease the size of the worker population, and there-fore, persistence of the colony.


Topology of Foliations and De-composition of Stochastic Flows ofDi↵eomorphisms

Paulo Ru�noUniversity of Campinas, BrazilAlison Melo, Leandro Morgado

Let M be a compact manifold equipped with a pairof complementary foliations, say horizontal and ver-tical. In Catuogno, Silva and Ru�no (Stoch. Dyn.2013) it is shown that, up to a stopping time ⌧ , astochastic flow of local di↵eomorphisms 't in M canbe written as a Markovian process in the subgroupof di↵eomorphisms which preserve the horizontal fo-liation composed with a process in the subgroup ofdi↵eomorphisms which preserve the vertical foliation.Here, we discuss topological aspects of this decompo-sition. The main result guarantees the global decom-position of a flow if it preserves the orientation ofa transversely orientable foliation. In the last sec-tion, we present an Ito-Liouville formula for subde-terminants of linearised flows. We use this formulato obtain su�cient conditions for the existence of thedecomposition for all t � 0.

Initial and Boundary Value Prob-lems for the Deterministic andStochastic Zakharov-KuznetsovEquation in a Bounded Domain

Chuntian WangUniversity of California, Los Angeles, USARoger Temam, Jean-Claude Saut

In this talk I will focus on the well-posedness andregularity of the Zakharov-Kuznetsov (ZK) equationin the deterministic and stochastic cases, subjectedto a rectangular domain in space dimensions 2 and

3. Mainly we have established the existence, in 3D,and uniqueness, in 2D, of the weak solutions, and thelocal and global existence of strong solutions in 3D.Then we extend the results to the stochastic caseand obtain in 3D the existence of martingale solu-tions, and in 2D the pathwise uniqueness and exis-tence of pathwise solutions. The main focus is on themixed features of the partial hyperbolicity, nonlinear-ity, nonconventional boundary conditions, anisotrop-icity and stochasticity, which requires methods quitedi↵erent than those of the classical models in fluiddynamics, such as the Navier-Stokes equation, Prim-itive Equation and related equations.

Random Dynamics and Averagingfor Non-Autonomous StochasticWave Equations

Yuncheng YouUniversity of South Florida, USAHongyan Li

The existence of a random attractor for a non-autonomous stochastic wave equation with nonlineardamping and multiplicative noise on an unboundeddomain will be presented. The Hausdor↵ distancebetween the random attractors of the original equa-tion with rapidly oscillating external force and theaveraged equation is estimated.


Special Session 65: On Singular Problems Related to Distance Functionsand Very Weak Solutions

Rakotoson, University Poitiers LMA, France

The aim of this session is to present the recent developments concerning singular problems related to distancefunctions and the notion of very weak solutions. Some new notions like the Uniform Hopf inequality andUltra-weak solution should be presented.

Uniform Gradient Estimates forViscosity Solutions to the Normal-ized p-Poisson Problem

Amal AttouchiUniversity of Jyvaskyla, FinlandEero Ruosteenoja, Mikko Parviainen

This talk is concerned with regularity and uniformestimates of the gradient of viscosity solutions to thenormalized p-Laplacian equation:

��Np u = f in⌦ .

We present some recent results on the C1,↵ regularityof the solutions when f 2 Lq(⌦) \ C(⌦) and f > 0.The main idea to derive a gradient estimate relieson the study of an approximate problem, potentialtheory tools, some known regularity results of weaksolutions of the classical p-Laplacian equation (withdivergence structure) and the uniqueness of viscositysolutions.

The Uniform Hopf Inequality forDiscontinuous Coe�cients andOptimal Regularity in BMO forSingular Problems

Nada el BerdanLMA- Universite de Poitiers, FranceJ.M Rakotoson, J.I Diaz

We discuss the existence and non existence of the socalled Hopf uniform Inequality (variant of a maxi-mum principle) for the linear equation Lv = f withmeasurable coe�cients and under the homogeneousDirichlet Boundary condition. Then we apply suchinequality to prove the W 1,p

0

-regularity of a semi lin-ear singular problem Lu = F (u), F (0) is infinite (un-der a precise growth) and with the coe�cients of themain operator of L in the space of vanishing meanoscillation. Moreover , when those coe�cients areLipschitz , we show that the gradient of the solutionis at most in the space of bounded mean oscillation:bmor.This work is in collaboration with Professor J.I Diaz(Universidad Complutense de Madrid) and ProfessorJ.M Rakotoson , and the detailed paper The UniformHopf Inequality for discontinuous coe�cients and op-timal regularity in BMO for singular problems, willappear in Journal of Mathematical analysis and itsapplications).

Pointwise Estimates for G-GammaFunctions and Applications

Maria Rosaria FormicaUniversity of Naples Parthenope, ItalyA. Fiorenza, J.M. Rakotoson

We present some new results, based on a joint workwith A. Fiorenza and J. M. Rakotoson, on the reg-ularity of some linear PDEs, expressed by meansof certain nonstandard spaces, namely, the smallLebesgue spaces and their generalization, the G-Gamma spaces. We deal with the relative rearrange-ment technics introduced by Mossino and Temam,we use some borderline Sobolev embeddings relatedto the Grand Lebesgue spaces (which generalize theFusco-Lions-Sbordone results) and we get either newestimates for the gradient of weak solutions in thesmall Lebesgue spaces, either new estimates of veryweak solutions in the spaces G-Gamma.

Uniqueness of Very Weak Solu-tions of Linear Elliptic Equationswith Singular Absorption PotentialsWithout Boundary Conditions

David Gomez-CastroUniversidad Complutense de Madrid, SpainJ.I. Diaz

A question related to the famous 1928 study byGamow of the Schroedinger equation for the infinitewell potential is the consideration of solutions of thefollowing equation:

��u+ V (x)u = f (1)

where V (x) ⇠ d(x,@ ⌦)�↵ for some ↵ > 0. Fromhere on �(x) = d(x,@ ⌦). It is natural to assumeV u 2 L1(⌦, �) so that the solutions must be searchedfor in the class of local very weak solutions of the form

�Z

⌦

u�'+

Z

⌦

V u '=

Z

⌦

f' (2)

where ' 2 W 2,1c (⌦) in the case that V 2 L1

loc(⌦),V � 0. Existence and uniqueness in the case V 2L1(⌦, �) with Dirichlet boundary conditions requiresconsidering test functions ' 2 W 2,1(⌦) \W 1,inf

0

(⌦)(notice how the choice of test functions is adapted tothe boundary condition). For V = ��r this conditionmeans that

R|u|�1�r


Some Nonlinear Elliptic Systemswith Right-Hand Side IntegrableData with Respect to the Distanceto the Boundary

Dalian University of Techology, Peoples Rep of China

In this talk, we will discuss the very weak solutionsto some nonlinear elliptic systems with right-handside integrable data with respect to the distance tothe boundary. By using an approximate method anda priori estimates technology in the framework ofweighted spaces, we obtain the existence, uniquenessand regularity of very weak solutions.

Optimal Control of Very WeakSolutions to Elliptic Systems withSingular Data Related to NeumannBoundary Conditions

Jochen MerkerHTWK Leipzig University of Applied Sciences,Germany

In this talk, we discuss optimal L1-control of veryweak solutions u : ⌦ ! RM to elliptic systems withsingular data on a smooth bounded domain⌦ ⇢ RN .More precisely, we consider the linear optimizationproblem

hg, ui = max! subject to Au f and u � 0 , (*)

where g 2 L1(⌦,RM ), the operator A : Lp0(⌦) !W 2,p(⌦,RM )⇤, 1 < p < 1, is a very weak real-ization of a vector-valued elliptic di↵erential opera-tor � div(

PMj=1

aijruj) +PM

j=1

bijuj , and the right

hand side f 2 W 2,p(⌦,RM )⇤ is a singular integrallike hf, vi :=

R⌦

R@⌦

f(x, y)(v(x) � v(y)) dy dx with

f(x, y)|x � y|↵ 2 L1(⌦ ⇥ @⌦) for ↵ 2 (0, 1], but notneccessarily f 2 L1(⌦ ⇥ @⌦). As was shown in [1]for Poisson’s equation in the scalar case, such sin-gular integrals as right hand sides allow a rigorousdiscussion of singular Neumann boundary conditions⇣PM

j=1

aijruj

⌘· ⌫ = �1 on parts of @⌦, which

model an explosive flow into the domain. We for-mulate corresponding results for elliptic systems (see[2]) prove existence of optimizers for (*) and providesome numerical examples with an explosive inflowover the inner boundary of an annular domain [3].

References

[1] J. Merker, J.-M. Rakotoson: Very weak solu-tions of Poisson’s equation with singular dataunder Neumann boundary conditions. Calc. Var.PDE 52 (2015), 705–726.

[2] M. Giaquinta, L. Martinazzi: An Introductionto the Regularity Theory for Elliptic Systems,Harmonic Maps and Minimal Graphs. Publica-tions of the Scuola Normale Superiore, 2012.

[3] J. Merker: Linear programming with systemsof elliptic partial di↵erential inequalities as con-straints. Preprint.


Special Session 66: Mathematical Oncology

Alexander R. A. Anderson, Integrated Mathematical Oncology, Mo�tt Cancer Center,Tampa, FL, USA

Cancer is a complex, multiscale process, in which genetic mutations occurring at a subcellular level manifestthemselves as functional changes at the cellular and tissue scale. We believe that such a complex system canonly truly be understood via the integration of theory and experiments. Therefore mathematical oncology isfocused on integrating mathematical and computational modeling approaches with experimental and clinicaldata to better understand cancer growth and development and to translate this understanding into noveltherapies. Since there is no single right model of cancer, in fact by definition all cancer models are wrong, weutilize a diverse portfolio of mathematical and computational approaches that cover the gamut of biologicalscales. The aim of this session will be to showcase this diversity, with a clear focus on data driven modelsthat have direct translational potential.

Modelling the Vicious Cycle inBone Metastases

David BasantaH. Lee Mo�tt Cancer Center, USAArturo Araujo, Leah Cook, Pranav Warman,Conor Lynch

Prostate cancer is one of the most commonly diag-nosed cancers in men. Every year more than 30 thou-sand men in the US get diagnosed with cancer but ofthose, only 10cancer becomes lethal when it metasta-sizes to other organs and, in 90of the cases, patientsshow evidence of metastases to the bone. Under-standing prostate cancer metastasis to the bone isthus key if we want to find ways to improve treatmentand decrease mortality. Successful metastases comefrom prostate cancer cells that can interact with theresident stromal cells and take advantage of the hostphysical microenvironment. These interactions arekey but di�cult to model experimentally due to thecomplexity of the factors and actors involved. Math-ematical models can help by integrating experimentaland clinical data, biological insights and by balancingthe need to capture enough detail to model biologi-cal reality with the need for simplicity. In this talk Iwill describe two di↵erent mathematical models, oneagent-based and one game theoretical, that aim tosynergistically bring new understanding on the evo-lutionary dynamics of metastatic success as well astreatment resistance.

Interacting Scales in Modeling HPVand Oropharyngeal Cancer

Marisa EisenbergUniversity of Michigan, Ann Arbor, USAAndrew Brouwer, Rafael Meza

Human papillomavirus (HPV) is a sexually transmit-ted infection which is associated with several formsof cancer, including cervical and oropharyngeal can-cer. HPV-related oropharyngeal cancers have re-cently overtaken cervical as the most common HPV-related cancer in the US. Interactions between infec-tious diseases and cancer form an inherently multi-scale problem, with population-level disease trans-

mission driving within-host carcinogenesis, yieldingoverall population-level cancer trends. In this talk, Iwill discuss recent work examining the dynamics ofHPV and oropharyngeal cancer, at both the cellularand human population scales.

Abscopal Benefits of Localized Ra-diotherapy Depend on ActivatedT Cell Tra�cking and DistributionBetween Metastatic Lesions

Heiko EnderlingMo�tt Cancer Center & Research Institute, USAJan Poleszczuk

It remains unclear how localized radiotherapy forcancer metastases can occasionally elicit a systemicantitumor e↵ect, known as the abscopal e↵ect, buthistorically it has been speculated to reflect the gen-eration of a host immunotherapeutic response. Theability to purposefully and reliably induce abscopale↵ects in metastatic tumors could meet many un-met clinical needs. Here, we describe a mathemati-cal model that incorporates physiological informationabout T cell tra�cking to estimate the distributionof focal therapy-activated T cells between metastaticlesions. We integrated a dynamic model of tumor-immune interactions with systemic T cell tra�ckingpatterns to simulate the development of metastases.In virtual case studies, we found that the dissemina-tion of activated T cells among multiple metastaticsites is complex and not intuitively predictable. Fur-thermore, we show that not all metastatic sites par-ticipate in systemic immune surveillance equally, andtherefore the success in triggering the abscopal e↵ectdepends, at least in part, on which metastatic site isselected for localized therapy. Moreover, simulationsrevealed that seeding new metastatic sites may ac-celerate the growth of the primary tumor because Tcell responses are partially diverted to the developingmetastases, but the removal of the primary tumor canalso favor the rapid growth of pre-existing metastaticlesions. Collectively, our work provides the frame-work to prospectively identify anatomically-definedfocal therapy targets that are most likely to triggeran immune-mediated abscopal response, and there-fore may inform personalized treatment strategies inpatients with metastatic disease.


Targeting the Phenotype: Treat-ment Strategies for HeterogeneousCancer

Jill GallaherMo�tt Cancer Center, USAAlexander R. A. Anderson

Targeted cancer drugs attack pathway specific phe-notypes and can lead to very positive outcomes whena particular phenotype dominates the population ofa specific tumor. However, these drugs often fail be-cause not all cells express the targeted phenotype tothe same degree. This leads to a heterogeneous re-sponse to treatment, and ultimate recurrence of thecancer as sensitive cells die o↵ and resistant cells takeover. We explore how treatment strategies informedby a tumor’s phenotypic mix, can help slow the emer-gence of resistance and stave o↵ tumor recurrence.We use an o↵-lattice agent-based model that incor-porates inheritance of two phenotypes - proliferationrate and migration speed - and is modulated by aspace limiting selection force. We find how and whendistinct distributions of phenotypes require di↵erenttreatment strategies.

Exploitng Evolutonary Dynamics toImprove Cancer Therapy

Robert GatenbyMo�tt Cancer Center, USAPedro Enriquez-Navas, Ariosto Silva, JoelBrown, Jessica Reynolds, Jingsong Zhang

A number of successful systemic therapies are avail-able for treatment of disseminated cancers. However,tumor response to these treatments is almost invari-ably transient and therapy fails due to emergence ofresistant populations. The latter reflects the tempo-ral and spatial heterogeneity of the tumor microen-vironment as well as the evolutionary capacity ofcancer phenotypes to adapt to therapeutic perturba-tions. Interestingly, although cancers are highly dy-namic systems, cancer therapy is typically adminis-tered according to a fixed, linear protocol. Treatmentis changed only when the tumor progresses but suc-cessful tumor adaptation begins immediately uponadministration of the first dose. Applying evolution-ary models to cancer therapy demonstrate the poten-tial advantage of using more dynamic, strategic ap-proaches that focus not just on the initial cytotoxice↵ects of treatment but also on the evolved mech-anisms of cancer cell resistance and the associatedphenotypic costs. The goal of evolutionary therapyis to prevent or exploit emergence of adaptive tumorstrategies. Examples of this approach include adap-tive therapy and double bind therapy. The formercontinuously alters therapy to maintain a stable tu-mor volume using a persistent population of therapy-sensitive cells to suppress proliferation of resistantphenotypes. The latter uses the cytotoxic e↵ects ofan initial therapy to promote phenotypic adaptationsthat are then exploited using follow-on treatment.

In pre-clinical models, application of adaptive ther-apy permits indefinite tumor control with a singlecytotoxic drug. Clinical results from studies usingadaptive therapy and double bind therapy will bepresented.

Identifiability in CompartmentalModels of Cancer Chemotherapy

Harsh JainFlorida State University, USA

In this talk, we will present ordinary di↵erential equa-tion models of chemotherapy targeting solid tumors.The drugs considered include: platinum based com-pounds that induce cell death by inflicting DNA dam-age; and taxols that target cells undergoing mito-sis. Although such models are relatively simple informulation, it is still not clear whether model pa-rameters are identifiable from available experimen-tal data. Here, we show that while nonlinearities inmodel terms generally result in structural identifia-bility, the increased numbers of parameters lead toissues of practical identifiability (due to imperfect orincomplete data). We discuss methods to estimatepractically identifiable combinations of parameters insuch cases. An important result of our analysis ispredicting what sort of additional experimental datawould render the model completely identifiable.

Tailoring Delivery of TargetedTherapy to the Tumor: a Study ina Pancreatic Cancer and TLR2

Aleksandra KarolakMo�tt Cancer Center, USAVeronica Estrella, Tingan Chen, AmandaHuynh, David Morse, Katarzyna Rejniak

The pancreatic adenocarcinomas are particularly dif-ficult to detect and treat, and the 6% 5-year survivalrate for this cancer has not been improved in thelast twenty years. Recently, our experimental collab-orators reported the toll-like receptor 2 (TLR2) as abona fide cell-surface marker for targeting pancreaticcancer. In order to provide a quantitative assessmentof tumor penetration and uptake, we combined com-putational modeling with the real time in vivo imag-ing of the TLR2 nanoligand (L) conjugated to near-infrared fluorescent dye, Cyanine-5 (Cy5), in pancre-atic adenocarcinoma tumor xenografts in mice.The integration of intravital fluorescence microscopywith cell-level modeling allowed us to quantify thespatio-temporal dynamics of TLR2L-Cy5 complex.Our studies also led to evaluation of the nanoparticleintratumoral transport including extravasation, in-terstitial di↵usion and intracellular accumulation inrelation to explicitly defined tumor tissue architec-ture. The computational model o↵ers cost-e↵ectivemethod for determining dynamics of agent binding,internalization and intracellular uptake, and servesas a potential tool for future predictions of the ap-propriate nanoparticle schedules and dosage based onpatient-specific data.


The Role of M1/M2 Microglia inRegulation of Cell Infiltration inGlioblastoma

Yangjin KimKonkuk University, KoreaHyejin Jeon, Hans Othmer

Malignant gliomas are the most common type ofbrain cancer, which arise from glial cells, and intheir most aggressive form are called GBMs. GBMsare highly invasive and di�cult to treat becausecells migrate into surrounding healthy brain tissuerapidly, and thus these tumors are di�cult to com-pletely remove surgically. GIMs, which can com-prise up to one third of the total tumor mass, arepresent in both intact glioma tissue and necroticareas. They apparently originate from both resi-dent brain macrophages (microglia) and newly re-cruited monocyte-derived macrophages from the cir-culation. Activated GIMs exhibit several pheno-types: one called M1 for classically activated, tu-mor suppressive, and another called M2 for alterna-tively activated, tumor promoting, and immunosup-pressive. Within a tumor the balance between thesephenotypes is typically shifted to the M2 form. Nu-merous factors secreted by glioma cells can influenceGIM recruitment and phenotypic switching, includ-ing growth factors, chemokines, cytokines and matrixproteins. In this work, we focus on mutual interactionbetween a glioma and M1/M2 microglia mediated byCSF-1, TGFbeta, and EGF. Up-regulated TGFbetaleads to up-regulation of Smad within the tumor cellsand secretion of MMPs, leading to proteolysis forEMT process and cell infiltration. The mathemat-ical model consists of densities of glioma cells, M1type cells, M2 type cells, and concentrations of CSF-1, EGF, TGFbeta, Extracellular matrix, and MMPs.We developed the model to investigate the mutual in-teractions between tumor cells in the upper chamberand microglia in the lower chamber. In the experi-ments, Boyden invasion assay was used to show thatthis mutual interaction induces glioma infiltration invitro and in vivo. We show that our simulation re-sults are in good agreement with the experimentaldata and we generate several hypotheses that shouldbe tested in future experiments in vivo.

Phase I Trials in Melanoma: aFramework to Translate PreclinicalFindings to the Clinic

Eunjung KimH.Lee Mo�tt Cancer Center and Research Institute,USAVito Rebecca, Keiran Smalley, AlexanderAnderson

We present a mathematical model driven framework,phase i (virtual/imaginary) trials, that integratesthe heterogeneity of actual patient responses andpreclinical studies through a cohort of virtual pa-tients. The framework includes an experimentallycalibrated mathematical model, a cohort of heteroge-

neous virtual patients, an assessment of stratificationfactors, and treatment optimization. We show thedetailed process using both preclinical and clinicaldata of melanoma combination therapy (chemother-apy and an AKT inhibitor). The mathematicalmodel composed of ordinary di↵erential equationspredicts melanoma treatment response and resistanceto mono and combination therapies. Model param-eters were estimated by utilizing an optimization al-gorithm to identify parameters that minimized thedi↵erence between predicted cell populations and ex-perimentally measured cell numbers. The model wasthen validated with in vitro experimental data. Thevalidated model and a genetic algorithm were usedto generate virtual patients whose tumor volume re-sponses to the combination therapy matched statis-tically the actual heterogeneous patient responses inthe clinical trial. Analyses on simulated cohorts re-vealed key model parameters such as a tumor vol-ume doubling rate and a therapy-induced phenotypicswitch rate that stratified simulated virtual clinicaltrial outcomes. Finally, our approach predicts op-timal AKT inhibitor scheduling suggesting more ef-fective but lower cumulative doses of AKT inhibitortreatment strategies.

Drug-Resistance Strategies: Evolu-tion of Heterogeneity Over Spaceand Time

Mark Robertson-TessiMo�tt Cancer Center, USADan Nichol, Alexander Anderson

Drug resistance is implicated in the majority deathsdue to cancer. The failure of drugs to producecomplete regression leads to tumor recurrence. Themechanisms by which a tumor escapes a given ther-apy are many and varied, including genotypic andphenotypic heterogeneity, variegation in drug deliv-ery, and numerous other physiological barriers. Inthis talk, phenotypic heterogeneity will be surveyedfrom a theoretical perspective.A cellular automaton is used to probe the short-and long-term e↵ects of di↵erent mechanisms thatgenerate phenotypic heterogeneity in a populationof cells, including combinations of: genetic muta-tion; environmentally-driven plasticity; phenotypicbet-hedging; and probabilistic resistance. The cy-totoxic drug is applied to a simulation of cell culture.The success of a given strategy depends on the drugregimen, the re-plating protocol, and the size of thedish. Sequential versus combination therapies withtwo drugs alter the evolutionary dynamics and selectfor di↵erent drug-resistance strategies. Combinationsof mechanisms produce exceedingly robust cell popu-lations which are essentially impossible to completelydestroy with any drug regimen, suggesting that dis-abling the underlying mechanisms that generate theheterogeneity may be more successful than modifyingdrug dosing and treatment schedules.


Steering Evolution to Prevent theEmergence of Resistance

Jacob ScottDepartment of Integrated Mathematical Oncology,Mo�tt Cancer Center and Research Institute, USAAlexander Anderson, Daniel Nichol, QuanTran

To better understand, and prevent the emergence ofresistance, we construct a simple markov model ofevolution on fitness landscapes. We utilize an em-pirically derived dataset of E. coli resistance to beta-

lactam antibiotics to derive experimental designs forlung cancer resistance. We then attack the prob-lem of collateral sensitivity and resistance, and showthat the experiments done to date can be misleadingor even dangerous and suggest an alternate formula-tion. To address overly stringent assumptions neededfor the Markov model (strong selection weak muta-tion) we formulate a stochastic model which we use toconsider the issue of evolutionary convergence timesand drug ordering. We Finally, I will present earlyresults from evolution experiments done in ALK mu-tated non-small cell lung cancer and discuss theoret-ical extensions and ongoing experimental work


Special Session 67: Applications of Mathematical Modeling inDevelopmental and Cell Biology

Jia Zhao, University of South Carolina, USAXinfeng Liu, University of South Carolina, USAQi Wang, University of South Carolina, USA

This minisymposium aims to bring researchers in applied mathematics with diverse background to addressrecent advances in mathematical modeling and computational technologies for complex systems in devel-opmental and cell biology that include (but not limited to) cell signaling pathways, cell oscillation andpolarization, cancer stem cells, cancer tumor growth and cell mitosis etc. Such systems usually consist ofmultiple interacting components that exhibit complicated temporal and spatial dynamics with multiple timeand length scales, which are extremely di�cult to model and make faithful predictions. In this mini sympo-sium, the challenges of modeling these complex systems will be discussed and, moreover, the new analyticaland computational techniques to simulate these models will also be presented.

Modeling Dorsal Closure

Andreas AristotelousWest Chester University, USAStephanos Venakides

Dorsal closure is a part of the Drosophila embryo-genesis. It constitutes a model for cell sheet mor-phogenesis during development and wound healing.For this reason the study of the various mechanismsgoverning dorsal closure is of great importance. Herewe devise a flexible modeling platform employing in-dividual based techniques that allows us to incorpo-rate experimental data and gives us the freedom totest di↵erent kinds of modeling equations on the var-ious parts of the simulated Drosophila embryo. Ourmodel simulates the behavior of not only the entireamnioserosa, but its constituent cells, as well. Themodel considers the combination of elastic and con-tractile properties of the cells in the tissue on a ro-bust geometric platform, in order to capture variousexperimentally observed phenomena.

Robust Dynamics in Tissue Growthand Developmental Patterning

Weitao ChenUniversity of California, Irvine, USAArthur Lander, Qing Nie

Robustness is observed widely in biological systemsand the related study is essential in mathematicalmodeling. In particular, size control and pattern for-mation, both displaying strong robustness, can serveas good models to investigate the related mecha-nisms. Tissue and organ size is genetically specifiedwith remarkable precision, independent of growthrate, cell size, only weakly sensitive to initial con-ditions and relatively resistant to a variety of ex-ternal perturbations. The patterning of many de-veloping tissues is organized by morphogens and itsformation is often quite resistant to embryonic dif-ference, intrinsic or extrinsic noises. The robustnessof di↵erent systems can be enhanced by particularmechanisms. In this talk, I will use a multi-stage celllineage model to discuss general strategies that maycontribute in achieving large tissue size robustly. I

will also present two particular systems, the papillaeformation on a mouse tongue or the scaling behaviorduring the growth of a wing disc in drosophila, toreveal the mechanisms for obtaining specific patternswith robustness.

Moment Stability for NonlinearStochastic Growth Kinetics ofBreast Cancer Stem Cells withTime-Delays

Xinfeng LiuUniversity of South Carolina, USA

Solid tumors are heterogeneous in composition. Can-cer stem cells (CSCs) are a highly tumorigenic celltype found in developmentally diverse tumors thatare believed to be resistant to standard chemother-apeutic drugs and responsible for tumor recurrence.Thus understanding the tumor growth kinetics is crit-ical for development of novel strategies for cancertreatment. In this paper, the moment stability ofnonlinear stochastic systems of breast cancer stemcells with time-delays has been investigated. First,based on the technique of the variationof-constantsformula, we obtain the second order moment equa-tions for the nonlinear stochastic systems of breastcancer stem cells with time-delays. By the com-parison principle along with the established momentequations, we can get the comparative systems of thenonlinear stochastic systems of breast cancer stemcells with time-delays. Then moment stability the-orems have been established for the systems withthe stability properties for the comparative systems.Based on the linear matrix inequality (LMI) tech-nique, we next obtain a criteria for the exponentialstability in mean square of the nonlinear stochasticsystems for the dynamics of breast cancer stem cellswith time-delays. Finally, some numerical examplesare presented to illustrate the e�ciency of the results.


Kinetic Monte Carlo Simulationsof Multicellular Aggregate Self-Assembly in Biofabrication

Yi SunUniversity of South Carolina, USAXiaofeng Yang, Qi Wang

We present a three-dimensional lattice model tostudy self-assembly and fusion of multicellular aggre-gate systems by using kinetic Monte Carlo (KMC)simulations. This model is developed to describeand predict the time evolution of postprinting mor-phological structure formation during tissue or organmaturation in a novel biofabrication process (or tech-nology) known as bioprinting. In this new technol-ogy, live multicellular aggregates as bio-ink are usedto make tissue or organ constructs via the layer-by-layer deposition technique in biocompatible hydro-gels; the printed bio-constructs embedded in the hy-drogels are then placed in bioreactors to undergo theself-assembly process to form the desired functionaltissue or organ products. Here we implement ourmodel with an e↵cient KMC algorithm to simulatethe making of a set of tissues/organs in several de-signer’s geometries like a ring, a sheet and a tube,which can involve a large number of cells and variousother support materials like agarose constructs etc.We also study the process of cell sorting/migrationwithin the cellular aggregates formed by multipletypes of cells with di↵erent adhesivities.

Macroscopic Limits of Pathway-Based Kinetic-Transport Modelsin the Exponential Large GradientEnvironment

Min TangShanghai Jiao Tong University, Peoples Rep ofChinaWeiran Sun

It is possible to develop predictive agent-based mod-els thanks to the understanding of the intracellularsignaling pathway. It is of great biological interest tounderstand the molecular origins of chemotactic be-havior of E. coli by deriving population-level modelbased on the underlying signaling pathway dynamics.We derive macroscopic models for E.coli chemotaxisusing kinetic equations that incorporate intracellularchemosensory system. These macroscopic models areshown to be in good agreement with individual basedsimulations. They match the average speed for thewhole range of the exterior gradient. This in particu-lar gives an answer to the question about the averagespeed for large gradients and suggest that the bacte-ria have a maximum drift velocity at their favoritegradient.

Modeling the Dynamical Organiza-tion of the Genome in Live YeastCells

Paula VasquezUniversity of South Carolina, USAGreg Forest, Caitlin Hult, David Adalsteins-son, Josh Lawrimore, Kerry Bloom

The genome comprises the entire genetic informationthat makes up an organism. This information is en-coded in DNA and stored in the nucleus of every cellin that organism in a dynamical manner. Under-standing the three-dimensional, dynamic, structureof the genome is a crucial step in characterizing howcellular DNA adopts and transitions between di↵er-ent entropic functional states over the course of thecell cycle, facilitating vital functions such as geneexpression, DNA replication, recombination, and re-pair. In this talk we discuss our existing models ofchromosomes and protein function in live yeast cells.Our mathematical models show that enzymes do notcreate the topological and energetic landscapes inthe nucleus; rather they bias the entropy-dominatedstochastic dynamics into cycle-specific states. In thisparadigm, entropy and confinement dictate the lead-ing order structure and dynamics of the genome, andthe role of enzymes is to guide, stabilize, and sustaincycle-specific genome states.

Two Phase Flow Models of CancerGrowth and Polymer Solvent Inter-actions

Steven WiseThe University of Tennessee, USA

In this talk I will describe and analyze a model forpolymer-solvent interaction in an application in theproduction of organic photovoltaics. I will show thatthis model is closely related to another describingcancer growth in the presence of interstitial fluid flow.

A New Nonlocal Poisson-FermiModel for Ion Channel Studies

Dexuan XieUniversity of Wisconsin-Milwaukee, USA

Ion channels are the valves of cells and the main con-trollers of many biological functions. While advanceshave been made in their mathematical modeling, onesubstantial challenge we face to address is how toincorporate the e↵ects of ion sizes and polarizationcorrelations among water molecules into a dielectriccontinuum model in partial di↵erential equations. Inthis talk, I will review some progresses we made onthis important research issue. I then will introduce anew nonlocal Poisson-Fermi model (NPF), which is afourth order elliptic partial di↵erential equation sub-jected to a system of nonlinear algebraic equations.Here, each ion is treated as a hard sphere with di↵er-ent radii for di↵erent ionic species, and the solution


of NPF leads to the ionic concentration functions andthe electrostatic potential. Furthermore, I will showthat NPF includes the previous Poisson-Fermi mod-els as special cases, and its solution is the convolutionof a solution of the corresponding nonlocal Poissondielectric model with a Yukawa-like kernel function.Finally, some simulation results will be discussed toillustrate the potential application of our new NPFmodel in ion channel studies. This project was par-tially supported by the National Science Foundation,USA, through grant DMS-1226259.

Multiscale Modeling of Axonal Cy-toskeleton Dynamics in Disease

Chuan XueOhio State University, USABlerta Shytulla, Anthony Brown, JonathanToy, Wenrui Hao

The shape and function of an axon is dependent on itscytoskeleton, including microtubules, neurofilamentsand actin. Neurofilaments accumulate abnormallyin axons in many neurological disorders. An earlyevent of such accumulation is a striking radial seg-regation of microtubules and neurofilaments. Thissegregation phenomenon has been observed for over30 years now, but the underlying mechanism is stillpoorly understood. I will present a stochastic mul-tiscale model that explained these phenomena andgenerated testable predictions. I will also present ourprogress in the derivation and analysis of a contin-uum PDE model which yielded further insights intothe problem.

CQLM—A Novel Numerical Ap-proach to Solve the Gradient FlowProblem

Xiaofeng YangDepartment of math, U of South Carolina, USA

There are two commonly used numerical approachesto solve the gradient flow problem while preservingthe desired energy stability: the Convex Splittingapproach and the Stabilized approach. The ConvexSplitting approach is energy stable, however, it pro-duces a nonlinear scheme at most cases, thus the im-plementation is complicated and the computationalcost is high. Stabilized approach generates linearscheme that is extremely easy to implement, how-ever, the magnitude of the stabilizing term dependson the upper bound of the second order derivativeof the nonlinear potential. Therefore, such methodis particularly reliable for those models with maxi-mum principle. For many nonlinear models, both ofthe two methods are not optimal choices. We intro-duce a novel, so called Compulsory-Quasi-Lagrange-Multiplier approach, that can possess the advantagesof both the convex splitting approach and the stabi-lized approach, but avoid their imperfections men-

tioned above. More precisely, the schemes (i) areaccurate (up to second order in time); (ii) are sta-ble (unconditional energy dissipation law holds); and(iii) are e�cient and easy to implement (only need tosolve some linear equations at each time step.

A Mechanochemical Model for CellPolarity

Lei ZhangPeking Universtiy, Peoples Rep of China

Cell polarization toward the attractant is related toboth physical and chemical factors. Most existingmathematical models are based on reaction di↵u-sion systems and only focus on the chemical processduring cell polarization. However, experiments re-veal that membrane tension may act as a long-rangeinhibitor for cell polarization. Here we present amathematical model that incorporates the interplaysbetween Rac, filamentous actin (F-actin), and cellmembrane tension for the formation of cell polarity.We also test the predictions of this model with singecell measurements on the spontaneous cell polariza-tion of cancer stem cells (CSC) and non-cancer stemcells (NCSS) as the former have smaller cell mem-brane tension. Both our model and experimentalresults show that the cell polarization is more sensi-tive to stimuli under low membrane tension, and highmembrane tension improves the robustness and sta-bility of cell polarization so that polarization is per-sistent under random perturbations. Furthermore,our simulations for the first time reproduce the re-sults from the aspiration-release experiment and thepseudopod-neck-cell body morphology severing ex-periment, demonstrating that aspiration (elevationof tension) and release (reduction of tension) resultin decrease and recover of the activity of Rac-GTP,respectively, and relaxation of tension leads to theformation of new polarity of the cell body when thecell with morphology of pseudopod-neck-body is sev-ered. The joint work with Weikang Wang (PKU),Feng Liu (PKU).

A Multiphasic Complex FluidsModel for Cytokinesis of Eukary-otes

Jia ZhaoUniversity of South Carolina, USAQi Wang

In this presentation, we develop a full 3D multiphasehydrodynamic model to study the fundamental mi-totic mechanism in cytokinesis, the final stage of mi-tosis. The model describes the cortical layer, a cy-toplasmic layer next to the cell membrane rich in F-actins and myosins, as an active liquid crystal systemand integrate the extra cellular matrix material andthe nucleus into a multiphase complex fluid mixture.With the novel active matter model built in the sys-tem, our 3D simulations show very good qualitative


agreement with the experimental obtained images.The hydrodynamical model together with the GPUbased numerical solver provides an e↵ective tool forstudying cell mitosis theoretically and computation-ally


Special Session 68: Rate-Dependent and Rate-Independent EvolutionProblems in Continuum Mechanics: Analytical and Numerical Aspects

Giuliano Lazzaroni, SISSA, Trieste, ItalyMarita Thomas, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

In recent years, energetic criteria have been extensively used to study processes of both rate-dependent and-independent type. This session will illustrate di↵erent variational approaches to evolution problems arisingmainly from continuum mechanics. Such problems lead to systems of nonlinear partial di↵erential equations,often involving non-smooth constraints. Special focus will be given to di↵erent notions of quasistatic, viscous,gradient flows as well as inertial dynamic evolutions, and to their relationships. Analytical and numericalresults will be discussed and compared for these di↵erent notions. Several fields of applications will be takeninto account, such as damage, fracture, dislocations, plasticity, phase separation. Moreover, models arisingin other fields of application, which can be treated with comparable techniques, will also be addressed.

Thin-Film Limits in Elas-tic Solids: Issues Related toThermomechanical-Coupling andNon-Interpenetration

Barbora BenevsovaUniversity of Wurzburg, GermanyMartin Kruzık, Gabriel Patho

The talk will be concerned with a thin-film passageand will be divided into two parts: In the first partwe will consider a full thermo-mechanical model ofthe bulk with a non-simple elastic energy, i.e., withone that includes a higher-order-gradient surface-energy part. Such energy is typical if we considermicrostructure-forming materials on a scale on whichthe microstructure can be fully resolved. We willthen perform a passage to a model for thin films largein area in the following way: We will first perform thedimension reduction and in a second step scale thesurface energy part to zero.In the second part on the talk we have a closer lookon the dimension reduction in the elastic part of theenergy. Here, we will consider the so-called mem-brane regime in which the procedure has becomestandard by now; however, not if combined to non-interpenetration conditions. We will concentrate onexactly this case in the talk. Thus, we will needto find a definition of non-interpenetration in thethin film and show that it is fulfilled if the bulk de-formations in the thin film passage have been non-interpenetrative, too.

Gradient Damage Models forElastoplastic Materials

Vito CrismaleSISSA, Italy

The talk concerns gradient damage models for elasto-plastic materials. These are a generalization both ofthe coupling between damage and elasticity, and ofelastoplasticity models without damage. The cou-pling I consider has been studied in many engineer-ing and numerical works. Starting from the perfectplasticity in a linearized setting, I point out the im-portance of the damage in the elastic and plastic re-sponses, and how some fatigue phenomena can be

included in the formulation. Afterwards, I introducethe mathematical treatment of the model, based onthe two approaches of energetic solutions and of evo-lutions via vanishing viscosity. The work is in collab-oration with G. Lazzaroni.

Dynamics of Discrete Screw Dislo-cations Along Glide Directions

Lucia de LucaTechnical University of Munich, GermanyRoberto Alicandro, Adriana Garroni, Mar-cello Ponsiglione

We consider a zero temperature model for screw dis-locations in discrete lattices. Using a discrete-in-timevariational scheme, we study the motion of disloca-tions in the dilute regime, towards low energy config-urations. Letting the spacing and time parameters goto zero, we deduce an e↵ective fully overdamped dy-namics predicting motion along the glide directionsof the crystal.

On the Genesis of Directional DryFriction Through Bristle-Like Me-diating Elements

Paolo GidoniSISSA (Trieste), ItalyAntonio DeSimone

We propose an explanation of the genesis of direc-tional dry friction, as emergent property of the oscil-lations produced in a bristle-like mediating elementby the interaction with microscale fluctuations on thesurface. Mathematically, we extend a convergenceresult by Mielke for Prandtl–Tomlinson-like systems,showing the rate-independent nature of the limit ofa family of systems characterized by a vanishing vis-cosity and a wiggly perturbation in the energy, thatscales non-homothetically to zero. We then applythe result to some simple mechanical models, thatexemplify the interaction of a bristle with a surfacehaving small fluctuations. We find that the result-ing friction is the product of two factors: a geometricone, depending on the bristle angle and on the fluctu-


ation profile, and a energetic one, proportional to thenormal force exchanged between the bristle-like ele-ment and the surface. Finally, we explain the “withthe nap/against the nap“ asymmetry in terms of ourresult.

Phase Field Modeling of Fracture -Quasi-Static Vs. Dynamic Formula-tions

Charlotte KuhnUniversity of Kaiserslautern, GermanyAlexander Schlueter, Ralf Mueller

In phase field models of brittle fracture cracks arerepresented by means of a continuous scalar field,which resembles a damage variable. The evolutionequation of this fracture field is coupled to the me-chanical field equations in order to model the mutualinteraction of fracture propagation and the mechan-ical loading. The extension of the well-known quasi-static phase field fracture model to a dynamic set-ting is straightforward. Concerning numerical sim-ulations of fracture with the finite element method,the phase field modeling technique is beneficial sincethere are neither displacement jumps across the crackligaments nor stress singularities at crack tips. How-ever, simulations with the quasi-static formulationare faced with certain di�culties concerning numer-ical stability especially in situations when cracks donot propagate smoothly but extend by finite incre-ments. In this case special numerical techniques or aregularization of the model are required to stabilizethe simulations. On the other hand such events offinite crack extension do not occur in the dynamicphase field fracture model since the velocity of crackextension is naturally bounded in this setting. Theobserved crack speeds are in good agreement withexperimental data and analytic considerations fromclassical fracture mechanics.

Quasistatic and Dynamic Evolutionof Materials with Defects

Giuliano LazzaroniSISSA, Trieste, Italy

I will present some recent results on quasistaticand dynamic evolution of materials a↵ected by frac-ture, damage, and plasticity, studied with variationalmethods.

Gradient Flow of Fractional Inter-action Equations

Edoardo MaininiVienna University, ItalyStefano Lisini, Antonio Segatti

We prove existence of a global weak solution for afractional interaction equation. The equation is in-terpreted as the gradient flow of the square norm ofthe Sobolev space H�s, s 2 (0, 1). We provide dissi-pation estimates and decay rates of Lp norms as well.Finally, we show the convergence of the constructedsolutions to the solution of the standard porous me-dia equation, as s ! 0.

Quasistatic Evolution for a Cohe-sive Fracture Model with Fatigue: aProof of Global Stability by Meansof Optimal Transport of Youngmeasures

Gianluca OrlandoSISSA, ItalyVito Crismale, Giuliano Lazzaroni

We introduce a model for the evolution of a materialwhich may present a cohesive fracture on a prescribedcrack path. The main feature of this model is thatsome energy is dissipated both when the crack open-ing increases and when it decreases. This peculiarresponse leads to a fatigue behaviour, so that a com-plete fracture may occur even after small oscillationsof the jump. We prove the existence of a globallystable quasistatic evolution in terms of Young mea-sures: the proof of the global stability is carried outby employing Optimal Transport of Young measures.

Crystal Dislocations with Di↵erentOrientation and Collisions

Stefania PatriziUT Austin, USAEnrico Valdinoci

Dislocations are moving defects in crystals that canbe described at several scales by di↵erent models.We consider a 1D evolution equation arising in thePeierls-Nabarro model, which is a phase field modeldescribing dislocation dynamics at a microscopicscale. Di↵erently from the previous literature, wetreat the case in which dislocations do not occur allwith the same orientations (i.e. opposite orientationsare allowed as well). We show that, at a long timescale, and at a mesoscopic space scale, the disloca-tions have the tendency to concentrate as pure jumpsat points which evolve in time, driven by the externalstress and by a singular potential. Due to di↵erencesin the dislocations orientation, these points may col-lide in finite time. We provide an estimates on therelaxation times of the system after collision. Theresults that will be presented have been obtained insome papers in collaboration with E. Valdinoci.


Stability of a Three DimensionalVariational Model of SuperelasticShape-Memory Alloys

Kim PhamENSTA ParisTech, FranceRoberto Alessi

We present a variational framework for the three-dimensional macroscopic modeling of superelasticshape-memory alloys in an isothermal setting. Phasetransformation is accounted through a unique secondorder tensorial internal variable, acting as the trans-formation strain. Postulating the total strain energydensity as the sum of a free energy and a dissipatedenergy, the model depends on two material scalarfunctions of the norm of the transformation strainand a material scalar constant. The quasi-static evo-lution problem of a domain is formulated in terms oftwo physical principles based on the total energy ofthe system: a stability criterion which selects the lo-cal minima of the total energy and an energy balancecondition which ensures the consistency of the evo-lution of the total energy with respect to the exter-nal loadings. The local phase transformation laws interms of Kuhn-Tucker relations are deduced from thefirst-order stability condition and the energy balancecondition. Based on this variational framework, sta-bility of homogeneous states under for proportionaland non-proportional loadings is provided by meansof second-order stability conditions.

Slow Motion for the Nonlocal Allen-Cahn Equation in N Dimensions

Matteo RinaldiCarnegie Mellon University, USARyan Murray

Slow motion of solutions of the nonlocal Allen–Cahnequation in a bounded domain⌦ ⇢ Rn, for n > 1,is studied. The initial data is assumed to be close toa configuration whose interface separating the statesminimizes the surface area (or perimeter); both localand global perimeter minimizers are taken into ac-count. The evolution of interfaces on a time scale ✏�1

is deduced, where ✏ is the interaction length parame-ter. The key tool is a second-order �–convergenceanalysis of the energy functional, which providessharp energy estimates. New regularity results arederived for the isoperimetric function of a domain.Slow motion of solutions for the Cahn–Hilliard equa-tion starting close to global perimeter minimizers isproved as well.

Discretization of Evolutions of Crit-ical Points and Applications inFracture Simulation

Francesco SolombrinoTU Munchen, GermanyMassimo Fornasier, Filippo Cagnetti, MarcoArtina

We introduce a novel constructive approach to definetime evolution of critical points of an energy func-tional. Our procedure is prone to e�cient and con-sistent numerical implementations, and allows for anexistence proof under very general assumptions. Weconsider in particular rather nonsmooth and noncon-vex energy functionals, provided the domain of theenergy is finite dimensional. Nevertheless, in the in-finite dimensional case study of a cohesive fracturemodel, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evo-lution can be indeed recovered as limit of evolutionsof critical points of finite dimensional discretizationsof the energy, constructed according to our scheme.To illustrate the results, we provide several numericalexperiments both in one and two dimensions. Theseagree with the crack initiation criterion, which statesthat a fracture appears only when the stress over-comes a certain threshold, depending on the material.

Capabilities and Numerical Imple-mentation of a Variational Gradient-Damage Plasticity Model for Cohe-sive and Ductile fracture

Erwan TanneEcole Polytechnique, FranceR. Alessi, B. Bourdin, J.J. Marigo

It is now well established that gradient damage mod-els are very e�cient to account for the behavior ofbrittle and quasi-brittle materials. However suchmodels cannot describe the nucleation of cohesivecracks, i.e. the existence of surface of discontinuityof the displacement field with a non vanishing inter-facial stress. Moreover, these models do not accountfor residual strains, typically observed during duc-tile fracture. A natural way to include such e↵ectsis to introduce plastic strains into the model and tocouple their evolution with damage. In this talk, Iwill extend a coupled damage and plasticity modelowing these features [R. Alessi, J.J Marigo, S. Vi-doli, Springer 2014], by implementing general perfectplasticity criteria, and present some novel numericalsimulations. Specifically, the capability and versa-tility of this extended model to describe ductile andcohesive fracture behaviours is shown on a tractiontest in both one- and multi-dimensional settings bymeans of parametric analyses. Also crack path andcohesive e↵ects comparisons between numerical sim-ulations and experiments will be shown.


From Adhesive Contact toBrittle Delamination in Visco-Elastodynamics

Marita ThomasWeierstrass Institute Berlin, GermanyRiccarda Rossi

This contribution addresses two models describingthe rate-independent fracture of a material com-pound along a prescribed interface in a visco-elasticmaterial. This unidirectional process is modeled inthe framework of Generalized Standard Materialswith the aid of an internal delamination parame-ter. In the context of (fully) rate-independent sys-tems within the energetic formulation it has becomea well-established procedure to obtain solutions of thebrittle model via an adhesive-contact approximationbased on tools of evolutionary Gamma-convergence.This means that the non-smooth, local brittle con-straint, confining displacement jumps to the null setof the delamination parameter, is approximated bya smooth, non-local surface energy term. Here, wediscuss the extention of this approach for systemsthat couple the rate-independent evolution of the de-lamination parameter with a viscous and dynamicevolution of the displacements in the bulk.

Evolutionary Convergence of Dis-crete Dislocation Dynamics in 2D

Patrick van MeursKanazawa University, JapanA. Garroni, M. A. Peletier, L. Scardia

Plasticity of metals is facilitated by the collective mo-tion of many dislocations, which we model as pointparticles in two-dimensions. The starting point isa gradient flow description for the dislocation posi-

tions, which is driven by an interaction energy. Ourmain result is the evolutionary convergence of thesegradient flows as the number of dislocations tends toinfinity. Our proof strategy relies on variational toolssuch as Gamma-convergence and the framework forevolutionary convergence developed by Sandier andSerfaty in 2004. The main challenge in the proof isto control the logarithmic nature of the interactionpotential, which is unbounded and non-local.

Numerical Approximation of aMesoscale Approach for the Dislo-cation Density Evolution in SingleCrystal Plasticity

Christian WienersKIT, Germany

Classical macroscopic approaches in continuum me-chanics for single crystal plasticity fail to describethe physical mechanisms induced by dislocation mo-tion. On the other hand, small-scale approachesdirectly simulating single dislocations require ex-tremely high computational costs. Here, we considera mesoscale model for the evolution of dislocationdensities based on small-strain single-crystal plastic-ity combined with Orowan’s relation for the plasticshear strain in every slip system of the crystal. Thesystem is closed by an evolution system for averageddislocation densities and dislocation curvature densi-ties (derived from the higher order continuum dislo-cation theory developed by Hochrainer et al.) and aconstitutive law for the dislocation velocity. We in-troduce a fully coupled numerical method combininga conforming finite element approximation of elasto-plasticity with an implicit Runge-Kutta discontin-uous Galerkin discretization of the dislocation sys-tem. The mesoscale method is evaluated for several3D benchmark configurations and compared with re-sults obtained from simulations on smaller scales.


Special Session 69: Dispersive E↵ects in Nonlinear PDEs

Paolo Antonelli, Gran Sasso Science Institute - INFN, Italy

The aim of this session is to discuss the analysis of dispersive behavior emerging in the study of somenonlinear PDEs. Those equations arise in the description of various phenomena in physics, from nonlinearoptics to low temperature physics, to acoustic waves in compressible fluid dynamics. Particular attentionwill be given to gather young mathematicians working in this active research area.

Novel Schrodinger-Strichartz Esti-mates for Liouville Equations, andWell-Posedness for Rough and Non-linear Transport

Agissilaos AthanassoulisUniversity of Leicester, England

Strichartz estimates for Schrodinger equations havebeen used to construct a powerful well-posedness the-ory for a very large class of nonlinear problems andproblems with very general space-time dependent po-tentials. Several attempts to replicate this approachfor di↵erent PDEs have appeared in recent years,with various degrees of success.Kinetic transport equations are a natural candidatefor a Strichartz-type-estimates theory, since in thesimplest case they describe the same dynamics asthe Schrodinger equation, simply unfolded in phase-space. Indeed, there exist “kinetic Strichartz esti-mates”, e.g.

kfkLq

t

Lp

x

Lr

k

kf0

kLa

x,k

,

2q= n

✓1r� 1

p

◆,

1a=

12

✓1r+

1p

◆,

but they do not seem to drive many of the nonlinearor rough problems that are currently under investi-gation.We present a new class of dispersive estimatesfor kinetic equations, in direct analogy with theSchrodinger-Strichartz estimates on R2n, e.g.

kfkLq

t

Lr

x,k

kf0

kL2 ,1q= n

✓12� 1

r

◆,

along with applications in the well-posedness of roughand nonlinear kinetic problems. Using these esti-mates, a large class class of nonlinear Liouville equa-tions is essentially transformed to derivative-NLSequations.

Invariant Measure for Cubic NLSon the Real Line

Federico CacciafestaUniv. Milano Bicocca, ItalyAnne-Sophie de Suzzoni

The existence of invariant measures has proven toyield significant improvements in the deterministicanalysis of nonlinear dispersive equations. In thistalk I will try to explain the general ideas of the the-ory, focusing on some recent results we obtained incollaboration with A. S. de Suzzoni concerning thecubic nonlinear NLS on the real line.

Modulational Instability inDispersion-Kicked Optical Fibers

Guillaume DujardinInria, FranceS. Rota-Nodari, M. Conforti, A. Kudlinski,A. Mussot, S. Trillo, S. De Bievre

We present a theoretical, numerical and experimen-tal study of modulational instability in dispersion-kicked optical fibers. In our setting, the evolutionalong the fiber of a weakly perturbed continuous waveis described by a cubic nonlinear Schrodinger equa-tion with periodically-modulated dispersion. In thecontext of normal averaged dispersion, using Floquettheory, we provide simple analytical estimates on thepositions of the gain bands. In particular, we showthat their positions do not depend on the form ofthe modulation of the dispersion. Moreover, whenthe dispersion takes the form of a Dirac comb, weprovide analytical estimates on the bandwiths of thegain bands. We conclude by illustrating our theoret-ical analysis with numerical experiments as well asactual physical experiments that are in good agree-ment of the analysis.

Growth of Sobolev Norms for theNon-Linear Schrodinger Equationon the Two-Dimensional Torus

Emanuele HausUniversity of Naples “Federico II”, ItalyMarcel Guardia, Michela Procesi

We study the non-linear Schrodinger equation (withanalytic nonlinearity of any order) on the two-dimensional torus and exhibit orbits whose Sobolevnorms grow with time. The main point is to makeuse of an accurate combinatorial analysis in orderto reduce to a su�ciently simple toy model, which


generalizes the one discussed in the paper by J. Col-liander, M. Keel, G. Sta�lani, H. Takaoka and T.Tao for the case of the cubic NLS. We also give esti-mates of the time needed to obtain such growth, byrefining and adapting to this more general case thetechniques used for the cubic case in the work by M.Guardia and V. Kaloshin.

On a Singularly Perturbed Gross-Pitaevskii Equation

Stefan le CozUniversity of Toulouse 3, FranceIsabella Ianni, Julien Royer

We consider the 1D Gross-Pitaevskii equation per-turbed by a Dirac potential. Using a fine analysis ofthe properties of the linear propagator, we study thewell-posedness of the Cauchy Problem in the energyspace of functions with modulus 1 at infinity. Thenwe study existence and stability of the black solitonswith a combination of variational and perturbativearguments. This is a joint work with Isabella Ianniand Julien Royer.

Freezing of Energy of a Soliton inan External Potential

Alberto MasperoUniversity of Nantes, FranceDario Bambusi

We study the dynamics of a soliton in the generalizedNLS with a small external potential ✏V of Schwartzclass. We prove that there exists an e↵ective mechan-ical system describing the dynamics of the soliton andthat, for any positive integer r, the energy of such amechanical system is almost conserved up to times oforder ✏�r. In the rotational invariant case we deducethat the true orbit of the soliton remains close to themechanical one up to times of order ✏�r.

Quasi-Periodic Standing Wave So-lutions for Gravity Capillary WaterWaves

Riccardo MontaltoUniversity of Zurich, SwitzerlandMassimiliano Berti

I will present a recent result concerning the existenceand the stability of small-amplitude quasi-periodicsolutions for the water waves equations with surfacetension. The core of the proof is the reduction of thelinearized equation (at any approximate solutions) toconstant coe�cients. Such a reduction procedure isachieved by using Pseudo di↵erential operators the-ory and a KAM reducibility scheme.

Dispersive Analysis of Kink Solu-tions

Claudio MunozUniversity of Chile, ChileMichal Kowalczyk and Yvan Martel

The purpose of this talk is to review some recentresults about the asymptotic dynamics of kink so-lutions for the well-known physical model called �4

in 1 1 dimensions. We will show that orbital stabil-ity leads to a particular case of asymptotic stabilityunder suitable assumptions on the initial data.

Stein-Tomas and Strichartz Esti-mates for Orthonormal Functions

Julien SabinUniv. Paris-Sud Orsay, FranceRupert L. Frank

We generalize the Stein-Tomas and Strichartz in-equalities to systems of orthonormal functions, withan optimal dependence on the number of such func-tions. These can be used to study dispersive proper-ties in infinite quantum systems.

Schroedinger Regularization of aVan Der Waals Gas

Marta StraniUniversite Paris Diderot, IMJ-PRG, FranceBenjamin Texier

We consider a dispersive regularization of the com-pressible Euler equations in Lagrangian coordinates.We assume a Van der Waals pressure law, whichpresents both hyperbolic and elliptic zones. The dis-persive regularization is of Schroedinger type. Thisimplies in particular that the regularized system sat-isfies the same conservation law as the original, phys-ical system. For this system, we prove local-in-timeexistence of solutions generated by initial data of am-plitude one, over time intervals that depend on theregularization parameter.

Bound on the Slope of Steady WaterWaves with Vorticity

Miles WheelerNYU Courant Institute, USAWalter Strauss

We consider the maximum angle between the freesurface of a steady two-dimensional water wave andthe horizontal. In the absence of vorticity, McLeodproved that this angle exceeds 30 degrees for verylarge waves, while Amick proved that it can neverexceed 31.15 degrees. With adverse vorticity, on theother hand, there is numerical evidence that steadywaves can become much steeper and even overturn.We prove an upper bound of 45 degrees for a largeclass of waves with favorable vorticity, in particularfor constant vorticity of the appropriate sign.


Special Session 70: Vortex Dynamics and Geometry: Analysis,Computations and Applications

Takashi Sakajo, Kyoto University, JapanBartosz Protas, McMaster University, Canada

This session will survey recent developments in vortex dynamics focusing on problems in which geometry,understood in a broad sense, plays an important role. Examples of such problems include flows definedon various manifolds as well as free-boundary problems arising in the dynamics of finite-area vortices.Geometric aspects are also important for applications such as superfluid turbulence and vortex-dominatedflows in domains with complex boundaries. The topics of the session will be selected to emphasize theinteraction between mathematical analysis and computations.

Existence and Regularity of V-States for the Euler and SQG PatchEquations

Javier Gomez-SerranoPrinceton University, USAAngel Castro, Diego Cordoba

The evolution of the motion of a patch is completelydetermined by the evolution of the boundary, allow-ing the problem to be treated as a non-local onedimensional equation for the contour. In this talkwe will discuss the existence and regularity of uni-formly rotating solutions - also known as V-states -for the vortex patch and generalized surface quasi-geostrophic (gSQG) patch equation.

New Solutions for Hollow Vorticesin an Infinite Channel.

Christopher GreenQueensland University of Technology, Australia

We will present new analytical solutions for a co-travelling hollow vortex pair and a single row of hol-low vortices in an infinite channel. These new solu-tions generalise several known classical solutions forhollow vortices. The mathematical problems to besolved are particular types of free boundary problemover a multiply connected domain. We have foundconcise formulae for the conformal mapping deter-mining the shape of the boundaries of the hollowvortices in both channel geometries by employing freestreamline theory in combination with the functiontheory of the Schottky-Klein prime function. Variousproperties of the solutions will also be presented.

Axisymmetrization of a Non-Uniform Elliptic Vortex and InverseEnergy Cascade in 2D Turbulence

Yoshifumi KimuraNagoya University, Japan

A non-uniform elliptic vortex tends to shed fila-ments of vorticity while rotating, and reshapes itinto a circular vortex. This process is often calledthe axisymmetrization process. This process is in-vestigated by using two di↵erent methods, pseudo-spectral DNS (Eulerian) and a system of point vor-

tices (Lagrangian). In particular, the physical cas-cade mechanisms in 2D turbulence, i.e. inverse en-ergy cascade and forward enstrophy cascade, arestudied concerning with the axi-symmetrization pro-cess. With the DNS, it will be shown that the squaredvorticity gradient called palinstrophy grows along thefilament ejection, and the development of palinstro-phy is analyzed. With a system of point vortices, itcan be demonstrated that an averaged wave numberdecreases during the process, which indicates that thedistribution of energy tends towards a lager scale.

A General Theory of Viscous Selec-tion in Two-Dimensional VorticalFlows

Paolo Luzzatto-FegizUC Santa Barbara, USA

Vortical equilibria prove useful in modeling prob-lems including aircraft wakes, biolocomotion, andgeophysical flows. A common approximation is toneglect viscosity; while this enables steady vortices(whose shape must be found), it does not predict thevorticity distribution (which must be assumed). Un-fortunately, model results can be very sensitive to thechoice of distribution.Interestingly, simulations have shown that viscousvortices (such as vortex pairs) can relax to nearlyself-similar states. A similarity ansatz decouples thevorticity equation into a steady advection equation,together with a di↵usion equation. Aside from thespecial cases of axisymmetric or parallel flow (whereadvective terms vanish), no other solution is known,to the best of our knowledge.To address these issues, we discretize the flow by su-perposing uniform-vorticity regions, yielding a weakformulation. This reveals more equations than un-knowns, such that an exact solution is not possible.We circumvent this by retaining the advection equa-tion, while enforcing an averaged version of the dif-fusion equation, such that the distribution is selectedby viscosity. This revised problem is solved with lowcomputational e↵ort, finding flows in good agreementwith full DNS results, thereby yielding a general ap-proach to find viscously selected equilibria.


Linear Stability of Hill’s Vortex toAxisymmetric Perturbations

Bartosz ProtasMcMaster University, CanadaAlan Elcrat

We consider the linear stability of Hill’s vortex withrespect to axisymmetric perturbations. Given thatHill’s vortex is a solution of a free-boundary prob-lem, this stability analysis is performed by applyingmethods of shape di↵erentiation to the contour dy-namics formulation of the problem in a 3D axisym-metric geometry. This approach allows us to system-atically account for the e↵ect of boundary deforma-tions on the linearized evolution of the vortex underthe constraint of constant circulation. The resultingsingular integro-di↵erential operator defined on thevortex boundary is discretized with a highly accu-rate spectral approach. This operator has two un-stable and two stable eigenvalues complemented bya continuous spectrum of neutrally-stable eigenval-ues. By considering a family of suitably regularized(smoothed) eigenvalue problems solved with a rangeof numerical resolutions we demonstrate that the cor-responding eigenfunctions are in fact singular objectsin the form of infinitely sharp peaks localized at thefront and rear stagnation points. These findings thusrefine the results of the classical analysis by Mo↵att& Moore (1978). It is also shown that the magnitudeof the eigenvalues associated with the unstable andstable eigenmodes is proportional to the translationvelocity of the vortex.

Vortex Knots Cascade by HOM-FLYPT Polynomial

Renzo RiccaU. Milano-Bicocca, ItalyXin Liu

Since Mo↵att’s original work of 1969, it is well-knownthat the kinetic helicity of a vortex filament admitstopological interpretation in terms of Calugareanu-White self linking number, and geometric decompo-sition in terms of writhe and twist. By applying knottheoretical techniques Liu & Ricca (2012; 2015) de-rived well-known knot polynomials, most notably theHOMFLYPT polynomial, from the helicity of fluidsystems, hence showing that these are new invariantsof ideal fluid mechanics. In the case of HOMFLYPTthe two polynomial variables are shown to be relatedto the writhe and twist of the vortex knot.In presence of dissipation topology is bound tochange by the continuous interaction and reconnec-tion of fluid strands. A prototype reconnection mech-anism is based on the anti-parallel recombination ofthe interacting strands. During such an event writhehelicity remains conserved, thus showing that anychange in helicity is due to a change in intrinsic twistof the interacting strands. We show that the naturalvortex cascade process seen in real experiments canbe detected by a monotonically decreasing sequenceof HOMFLYPT values.

Words and Trees: Symbolic Clas-sifications of Streamline Topologiesfor 2D Incompressible Vortex Flows

Takashi SakajoKyoto University, JapanTomoo Yokoyama

I will provide a mathematical theory of topologi-cal pattern characterizations for 2D incompressibleflows that has recently been developed in our researchproject. It enables us to identify topological flowpatterns, i.e. streamline patterns, with simple sym-bolic expressions called maximal words and regularexpressions with graph (tree) representations. Thesymbolic descriptions bring us new qualitative infor-mation on the correspondence between flow patternsand their functions/evolutions, that are applicable tomany flow phenomena appearing in material and lifesciences.

Drift Due to a Viscous Vortex Ring

Jean-Luc Thi↵eaultUniversity of Wisconsin – Madison, USAThomas Morrell, Saverio Spagnolie

Biomixing is the study of fluid mixing caused byswimming organisms. The swimming of large organ-isms can lead to mixing by the turbulent flows in theirwakes, but the wakes created by small swimming or-ganisms are not turbulent. Instead, the main mech-anism of mixing by smaller organisms is the net par-ticle displacement (drift) induced by the swimmer.Several experiments have been performed to exam-ine this drift for small jellyfish; these produce vortexrings which trap and transport a fair amount of fluid.However, since inviscid theory implies infinite par-ticle displacements, the e↵ects of viscosity must beincluded to understand the damping of real vortexmotion. We use a model viscous vortex to computeparticle displacements and other relevant quantities,such as the integrated moments of the displacement.Fluid entrainment at the tail end of a growing vortex‘envelope‘ is found to play an important role in thetotal fluid transport and drift.

Self-Similar Vortex Filament Mo-tion Under the Non-Local Biot-Savart Model

Robert van GorderUniversity of Oxford, England

The self-induced motion of a thin vortex filament isgoverned by the Biot-Savart model which, due to in-herent non-locality and nonlinearity, is often approx-imated by the local induction approximation (LIA).While very regular filaments, such as those exhibit-ing helical or planar geometries, have been studiedanalytically under both formulations, more compli-cated filament structures are often only studied underLIA. One type of vortex filament structure to haveattracted interest in recent years is that which obeys


a self-similar scaling. Among various applications,these filaments have been used to model the motion ofquantized vortex filaments in superfluid Helium afterreconnection events. While similarity solutions havebeen described analytically and numerically using theLIA, they have not been studied (or even shown toexist) under the non-local Biot-Savart model. Wewill show not only that self-similar vortex filamentsolutions exist for the non-local Biot-Savart model,but that such solutions are qualitatively similar totheir LIA counterparts. This suggests that the vari-ous LIA similarity solutions found previously shouldbe valid physically (at least when they are of su�-cient bounded variation), since they agree well withthe dynamics from the Biot-Savart model. Exten-sions to more complicated models involving mutualfriction and an imposed normal fluid, which are use-ful for describing vortex filament dynamics in super-fluid Helium above the zero-temperature limit, maybe discussed.

The Interplay of Curvature andVortices in Flow on Curved Sur-faces

Axel VoigtTechnische Universitat Dresden, GermanySebastian Reuther

Incompressible fluids on curved surfaces are consid-ered with respect to the interplay between topol-ogy, geometry and fluid properties using a surfacevorticity-stream function formulation, which is solvedusing parametric finite elements. Motivated by de-signed examples for superfluids, we numerically con-sider the influence of a geometric potential on vorticesfor fluids with finite viscosity and show examples inwhich a change in the geometry is used to manipulatethe flow field.


Special Session 71: Elliptic Equations and Systems, and ConcentrationPhenomena

Cyril Tintarev, Uppsala University, SwedenKanishka Perera, Florida Institute of Technology, USA

Marco Squassina, University of Verona, Italy

Existence Results for NonlinearElliptic Problems

Pasquale CanditoMediterranean University of Reggio Calabria, ItalyG. Bonanno, R. Livrea, D. Motreanu

The aim of this talk is to present a recent coinci-dence point theorem for sequentially weakly contin-uous maps defined on Banach spaces. As a conse-quence, some existence results for nonlinear bound-ary value problems are showed.

Radial Solutions for Elliptic Prob-lems with Symmetry

Florin CatrinaSt. John’s University, USA

We discuss the existence of radial solutions for cer-tain second order semilinear elliptic PDEs. An en-ergy balance identity is employed to prove nonexis-tence of such solutions. These nonexistence cases aredue to the loss of compactness in embeddings of theappropriate functional spaces. The loss of compact-ness is manifested in the concentration of minimizingsequences at singularities of the potential.

Two Non-Zero Solutions to EllipticDirichlet Problems

Giuseppina DaguiUniversity of Messina, ItalyGabriele Bonanno

This talk deals with the existence of two non-zerocritical points for an appropriate class of di↵eren-tiable functionals. Our main tools are a recent lo-cal minimum theorem and the classical Ambrosetti-Rabinowitz theorem. Our main result is an appro-priate combination of such results in order to obtaintwo non-zero critical points. In fact, once obtainedthe first non-zero critical point by the local minimumtheorem, a direct application of the mountain passtheorem allows to get the second critical point thatin general can be zero. Instead, we verify that alocal minimum actually is a global minimum for asuitable restriction of the functional and, hence, weprove that all the paths starting from it have a highlevel greater than zero, and this guarantees the sec-ond critical point must be non-zero. As an applica-tion, we get two non-zero weak solutions to nonlinearelliptic Dirichlet problems.

Three Nontrivial Solutions for Non-linear Fractional Laplacian Equa-tions

Antonio IannizzottoUniversity of Cagliari, ItalyF. Gamze Duzgun

We study a Dirichlet-type boundary value prob-lem for a pseudodi↵erential equation driven by thefractional Laplacian, proving the existence of threenonzero solutions. When the reaction term is sub-linear at infinity, we apply the second deformationtheorem and a recent characterization of the secondeigenvalue of the fractional Laplacian. When the re-action term is superlinear at infinity, we apply themountain pass theorem and Morse theory.

Second Order Derivatives of Solu-tions of Uniformly Elliptic IsaacsEquations

Jay KovatsFlorida Institute of Technology, USA

In this talk, we discuss continuity and integrabilityproperties of second order derivatives of viscosity so-lutions of uniformly elliptic Isaacs equations of theform F (D2u) := max

y2Yminz2Z

tr[A(y, z)D2u] = 0 in a

domain D ⇢ Rd, where Y, Z are finite sets. Here,8 y 2 Y, z 2 Z, A(y, z) is a symmetric d ⇥ d matrix,satisfying �Id A(y, z) ⇤Id, for some constants 0

On a Dirichlet Problem with thep-Laplacian

Roberto LivreaUniversity of Reggio Calabria, ItalyS. Carl, P. Candito

The aim of the talk is to show some existenceand multiplicity results for a class of parameter-dependent Dirichlet problems involving the p-Laplacian. By using di↵erent approaches based onvariational and topological methods, possible di↵er-ent intervals of parameters for which the problem un-der examination admits solutions are detected.


On a Robin Problem with IndefiniteWeight and Asymmetric Reaction

Salvatore Angelo MaranoUniversity of Catania, ItalyNikolaos S. Papageorgiou

The existence of multiple smooth solutions to a semi-linear Robin problem with indefinite unbounded po-tential and asymmetric nonlinearity is established.Both crossing and resonance are allowed. Proofs ex-ploit variational methods, truncation techniques, andMorse theory.

The Brezis-Nirenberg Problem forthe Fractional p-Laplacian

Kanishka PereraFlorida Institute of Technology, USASunra Mosconi, Marco Squassina, Yang Yang

We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional p-Laplacian op-erator, extending some results in the literature for thefractional Laplacian. The quasilinear case presentstwo serious new di�culties. First an explicit formulafor a minimizer in the fractional Sobolev inequalityis not available when p 6= 2. We get around this dif-ficulty by working with certain asymptotic estimatesfor minimizers. The second di�culty is the lack ofa direct sum decomposition suitable for applying theclassical linking theorem. We use an abstract linkingtheorem based on the cohomological index to over-come this di�culty.

A Survey of Cocompactness Resultsand Uses

Ian SchindlerUniversity of Toulouse 1 Capitole, France

Cocompact embeddings and profile decompositionsare two fundamental tools in concentration analysison a functional analytic level . Cocompact embed-dings (relative to a non-compact group) are similarbut weaker than compact embeddings. In addition to

the classical cases of compact embeddings of Sobolevspaces relative to the non-compact groups of transla-tions and dilations, cocompact embeddings have beenestablished in a wide range of spaces. We surveysome of these embeddings and give their applicationsto PDE.

Uniqueness of Positive Solutions ofBrezis-Nirenberg Problems on theHyperbolic Space

Naoki ShiojiYokohama National University, JapanKohtaro Watanabe

Uniqueness of positive solutions for the equations likethe scalar field equation have been studied by manyresearchers. Recently, the author and Prof. KohtaroWatanabe introduced a generalized Pohozaev iden-tity and gave uniqueness results which are applicableto various equations including the scalar field equa-tion; see JDE 252 (2012) and CVPDE 55 (2016). Inthis talk, we study the uniqueness of positive solu-tions of

�Hn'+ �'+ 'p = 0

on the n-dimensional hyperbolic space Hn, wheren 2 N with n � 2,� Hn is the Laplace-Beltrami op-erator on Hn, � (n� 1)2/4, and p is subcritical orcritical. In particular, in the case n = 2, we improveMancini and Sandeep’s uniqueness result in ASNSP7 (2008).

On Degenerate P(x)-Laplace Equa-tions Involving Critical Growthwith Two Parameters

Inbo SimUniversity of Ulsan, KoreaKy Ho

After establishing a concentration-compactness prin-ciple for weighted variable exponent spaces, we showthe existence of a nonnegative solution and infinitelymany solutions for degenerate p(x)-Laplace equationsinvolving critical growth with two parameters usingvarious techniques in Calculus of Variations.


Special Session 72: Optimal Control and its Applications

Alexander J. Zaslavski, The Technion - Israel Institute of Technology, IsraelMonica Motta, University of Padua, Italy

Franco Rampazzo, University of Padua, Italy

A special session on “Optimal Control and its applications” will bring together a selected group of expertsin this area. The growing importance of optimal control has been realized in recent years. This is duenot only to theoretical developments in this area, but also because of numerous applications to engineering,economics and life sciences. Approximately 25 participants from Australia, Brazil, France, Germany, Israel,Italy, Russian and USA will participate in the session. The topics which will be discussed include infinitehorizon control problems, turnpike phenomenon, averaging in optimal control, optimality conditions incontrol problems, qualitative and quantitative aspects of optimal control, control and stabilization of PDEs.

Optimal Feedback Control Law forUncertain Dynamic Systems onBanach Spaces

Nasiruddin AhmedUniversity of Ottawa, Canada

In this talk we consider a class of partially ob-served dynamic systems on infinite dimensional Ba-nach spaces subject to dynamic and measurement un-certainty. The problem is to find an output feedbackcontrol law, an operator valued function, that min-imizes the maximum risk. This involves optimiza-tion on the space of operator valued functions. Wepresent a recent result on existence of optimal feed-back law. Inspired by the existence result, we con-sider the question of characterization of the optimalfeedback operator. We develop the necessary con-ditions of optimality which an optimal feedback lawmust satisfy. A conceptual algorithm for computa-tion of the optimal operator valued function is alsopresented.

Second Order Analysis of Control-A�ne Problems with Scalar StateConstraint

Maria Soledad AronnaSchool of Applied Mathematics, FGV/Rio, BrazilJ. Frederic Bonnans, Bean-San Goh

In this article we establish new second order neces-sary and su�cient optimality conditions for a classof control-a�ne problems with a scalar control and ascalar state constraint. These optimality conditionsextend to the constrained state framework the Gohtransform, which is the classical tool for obtainingan extension of the Legendre condition. We provideexamples to illustrate the theory.

Useful Conditions for Non TrivialPontryagin’s Adjoint Variable inInfinite Dimension

Mohammed BachirUniversity Paris 1 Pantheon-Sorbonne, FranceJ. Blot

We give some usable conditions on a sequence ofnorm one in a dual Banach space under which thesequence does not converges to the origin in the w⇤-topology. These requirements help to ensure thatthe Lagrange multipliers are nontrivial, when we areinterested for example on the infinite dimensionalinfinite-horizon Pontryagin principles for discrete-time problem.

Pontryagin Principles for SystemsGoverned by Delay Functional Dif-ferential Equations

Joel BlotUniversity Paris 1 Pantheon-Sorbonne, FranceJ. Blot, M.I. Kone

We provide new results on Pontryagin principleswhen the di↵erential equation which governs the sys-tem is a delay functional di↵erential equation. Thenew results are based on the study of precise proper-ties of a linearization of the di↵erential equation toobtain results under lightest possible assumptions.

Minimizers for Nonconvex Varia-tional Problems in the Plane ViaConvex/Concave Rearrangements

Dean CarlsonMathematical Reviews, American MathematicalSociety, USA

Recently, A. Greco utilized convex rearrangementsto present some new and interesting existence resultsfor noncoercive functionals in the calculus of varia-tions. Moreover, the integrands were not necessarilyconvex. In particular, using convex rearrangementspermitted him to establish the existence of convexminimizers essentially considering the uniform con-vergence of the minimizing sequence of trajectoriesand the pointwise convergence of their derivatives.The desired lower semicontinuity property is now a


consequence of Fatou’s lemma. In this paper we pointout that such an approach was considered in the late1930’s in a series of papers by E. J. McShane forproblems satisfying the usual coercivity condition.Our goal is to survey some of McShane’s results andcompare them with Greco’s work. In addition, wewill update some hypotheses that McShane made bymaking use of a result due to T. S. Angell on theavoidance of the Lavrentiev phenomenon.

A New Su�cient Optimality Con-dition for Infinite Horizon OptimalControl Problems

Valeriano de OliveiraUNESP - Univ. Estadual Paulista, Brazil

The work is devoted to introduce a new su�cient op-timality condition for infinite horizon optimal controlproblems. It is showed that normal extremal pro-cesses are optimal under this new condition, termedas MP-pseudoinvexity. Moreover, problems in whichevery normal extremal process is optimal necessarilyobey the definition of MP-pseudoinvexity.

An Optimal Control Problem forHybrid Dynamical Systems withPolynomial Impulses

Elena GoncharovaInstitute for System Dynamics and Control Theoryof the Siberian Branch of the Russian Academy ofSciences, RussiaMaxim Staritsyn

We consider an optimal control problem for a mea-sure driven hybrid dynamical system. The dynamicsis a BV-relaxation of a system of a polynomial struc-ture. The relaxed system is described by a measuredi↵erential equation subject to nonstandard mixedconstraints imposed on a state and a control measure.The main results concern a special space-time trans-formation technique aimed at reducing the optimalimpulsive control problem to an equivalent conven-tional one, and nesessary conditions for optimality.

Estimating the Number of Zeros ofthe Switching Functions in OptimalControl Problems for CompartmentModels

Ellina GrigorievaTexas Woman’s University, USA

Estimation of the number of switchings of the op-timal controls under assumption of the absence ofsingular arcs is a major challenge. Such estimates al-low us to reduce the initial complex optimal controlproblem to a simpler problem of finite-constrainedoptimization. In turn, the number of the switchingsof optimal controls is associated with the estimationof the number of zeros of the corresponding switchingfunctions, which completely determine the behavior

of these controls. Currently there are two approachesactively used for estimating the number of zeros ofthe switching functions. The first approach is to con-vert a linear non-autonomous system of di↵erentialequations for the switching function and related aux-iliary functions to the linear non-autonomous di↵er-ential equation with bounded coe�cients with sub-sequent application of the Vallee-Poussin’s theoremto such equation. The second approach is to reducethe matrix of the mentioned non-autonomous linearsystem to an upper-triangular form on a given timeinterval and then apply the generalized Rolle’s Theo-rem to the transformed linear system. In this report,we propose our new results associated with estimat-ing the number of zeros of the switching functions inoptimal control problems for various compartmentmodels. These results are compared with the resultsobtained earlier by the authors for SIR and SEIRmodels.

The Relationship Between RevealedPreference and The Slutsky Matrix

Yuhki HosoyaKanto-Gakuin University, Japan

This study provides a calculation method for utilityfunction from a smooth demand function whose Slut-sky matrix is negative semi-definite and symmetric.Moreover, this study presents an axiom of demandfunctions, and show that under the strong axiom,this axiom is equivalent to the existence of the cor-responding continuous preference relation. If the de-mand function obeys this axiom, then such a pref-erence relation is unique, and our calculating util-ity function is continuous and represents its prefer-ence relation. These results are obtained even if thedemand function is not income-Lipschitzian. Fur-ther, this study shows that the mapping from de-mand function into continuous preference relation iscontinuous, which assures the applicability of our re-sults for econometrics. Moreover, this study showsthat if this demand function satisfies the rank con-dition, then our utility function is smooth. Lastly,this study shows that under an additional axiom, theabove results hold even if the demand function has acorner solution.

Equilibrium Locus of the Flow onCircular Networks of Cells

Yirmeyahu KaminskiHolon Institute of Technology, Israel

We perform a geometric study of the equilibrium lo-cus of the flow that models the di↵usion process overa circular network of cells. We prove that when con-sidering the set of all possible values of the parame-ters, the equilibrium locus is a smooth manifold withboundary, while for a given value of the parameters,it is an embedded smooth and connected curve. Fordi↵erent values of the parameters, the curves are allisomorphic.


Moreover, we show how to build a homotopy be-tween di↵erent curves obtained for di↵erent valuesof the parameter set. This procedure allows the ef-ficient computation of the equilibrium point for eachvalue of some first integral of the system. This pointwould have been otherwise di�cult to be computedfor higher dimensions. We illustrate this constructionby some numerical experiments.Eventually, we show that when considering the pa-rameters as inputs, one can easily bring the systemasymptotically to any equilibrium point in the reach-able set, which we also easily characterise.

Case Studies for Optimal Control ofCoupled ODE-PDE Systems

Sven-Joachim KimmerleUniversitaet der Bundeswehr Muenchen, GermanyMatthias Gerdts, Roland Herzog

We consider several case studies for optimal controlproblems subject to coupled systems of ordinary andpartial di↵erential equations. As a key example wefocus on a crane transporting a load.

The crane beam is considered as elastic and the loadis modelled by a pendulum that is moved by a trol-ley running along the crane beam. The goal is toconsider time-optimal control subject to initial andterminal conditions. Other examples are a truckwith a fluid container where the fluid is subject tothe shallow water equations and an elastic bridgeunder moving loads.

We derive mathematical models, formulate the opti-mal control problems and discuss numerical and an-alytic approaches for optimal control. We considerfirst-discretize-then-optimize methods as well as first-optimize-then-discretize methods.

Approximations with BoundedFunctions in Scalar Multidimen-sional Variational Problems

Carlo MaricondaUNIPD, Italy

We consider an integral functional I(u) =Z

⌦

L(x, u(x),ru(x)) dx (the “energy“) defined in the

space of Sobolev functions on an open and boundedset, and that agree with a prescribed Lipschitz func-tion � on the boundary of the domain ⌦, a boundedopen subset of Rn. We deal with the problem of ap-proximating a given Sobolev function both in energyand in the strong topology with bounded functionsthat agree with � on the boundary of ⌦. As a conse-quence we prove the non occurrence of the Lavrentievphenomenon for a class of non convex problems.

Small-Time Local Attainability fora Class of Control Systems withState Constraints

Antonio MarigondaUniversity of Verona, ItalyThuy Thi Le

We consider the problem of small time local attain-ability (STLA) for nonlinear finite-dimensional time-continuous control systems in presence of state con-straints. More precisely, given a nonlinear controlsystem subjected to state constraints and a closedset S, we provide su�cient conditions to steer to Severy point of a suitable neighborhood of S alongadmissible trajectories of the system, respecting theconstraints, and giving also an upper estimate of theminimum time needed for each point to reach thetarget. The results presented generalize previous re-sult obtained by Krastanov and Quincampoix withdi↵erent techniques. Special emphasis is given to thecontrol-a�ne systems, in which more explicit condi-tions can be given exploiting the natural Lie alge-braic structure associated with the system. Methodsof nonsmooth analysis are used.

Minimum Time and UnboundedVariation

Monica MottaPadua University, ItalyCaterina Sartori

Given a control system with dynamics a�ne in the(unbounded) derivative of the control u and a closedtarget set, we study the minimum time problem with-out constraints on the total variation of u. Togetherwith inputs u in AC, we introduce piecewise, lo-cally AC inputs (with possibly unbounded variation)and consider the corresponding Charateodory solu-tions and extended Charateodory solutions, respec-tively. We embed such controls and solutions into aset of space-time solutions with unbounded variationand prove that they all belong to the larger class of(simple) limit solutions, recently introduced by M.S.Aronna and F. Rampazzo. Then we compare theminimum times over limit solutions and space-timesolutions, with the infima over either AC or piece-wise, locally AC inputs, respectively. We introducesu�cient conditions to have either that all these val-ues coincide or that the minimum times over limit so-lutions and space–time solutions are equal just to theinfimum over piecewise, locally AC controls. We con-clude with a regularization result, where the (unique)minimum time function is approximated by penalizedminimum time functions with minimizing inputs ofbounded variation.


Maharam-Types and Lyapunov’sTheorem for Vector Measures onLocally Convex Spaces WithoutControl Measures

Nobusumi SagaraHosei University, JapanM. Ali Khan

We formulate the saturation property for vector mea-sures in locally convex Hausdor↵ spaces as a nonsep-arability condition on the derived Boolean sigma-algebras by drawing on the topological structure ofvector measure algebras. We exploit a Pettis-likenotion of vector integration in locally convex Haus-dor↵ spaces, the Bourbaki–Kluvanek–Lewis integral,to derive an exact version of the Lyapunov convexitytheorem in locally convex Hausdor↵ spaces withoutthe Bartle–Dunford–Schwartz property. We applyour Lyapunov convexity theorem to the bang-bangprinciple in Lyapunov control systems in locally con-vex Hausdor↵ spaces to provide a further character-ization of the saturation property.

On Second Order Necessary Condi-tions in Optimal Control

Daniela TononCEREMADE Universite Paris Dauphine, FranceHelene Frankowska

This talk is devoted to second order necessary opti-mality conditions for the Mayer optimal control prob-lem when the control set is a closed subset of Rn andendpoint constraints are present. Admissible controlsare supposed to be just measurable. We show howto exploit the properties of particular second ordervariations, obtained using the adjacent tangent coneand the second order adjacent tangent subset to thecontrol constraint, in order to obtain several di↵er-ent formulations of second order necessary conditionsin integral form. These second order necessary con-ditions are then applied to obtain Goh condition, apointwise second order necessary optimality condi-tion which turns out to be useful when the optimalcontrol for the Mayer problem is singular, i.e. whenthe classical Legendre-Clebsch condition is no moreof use.

Linear Control Systems with Peri-odic Convex Integrands

Alexander ZaslavskiThe Technion - Israel Institute of Technology, Israel

We study the structure of approximate optimal tra-jectories of linear control systems with periodic con-vex integrands and show that these systems possessa turnpike property. To have this property means,roughly speaking, that the approximate optimal tra-jectories are determined mainly by the integrand, andare essentially independent of the choice of time inter-val and data, except in regions close to the endpointsof the time interval. We also show the stability ofthe turnpike phenomenon under small perturbationsof integrands and study the structure of approximateoptimal trajectories in regions close to the endpointsof the time intervals.

Injectivity Properties of Pole Place-ment Maps of Linear Control Sys-tems

Igor ZelenkoTexas A&M University, USAYanhe Huang, Frank Sottile

Our original question is for which linear control sys-tems generically among all configurations of polesthat can be realized by a static output feedback thereis another static feedback that realizes the same con-figuration? For this goal we introduce the notion ofgeneralized Wronski maps (GWM) on Grassmaniansand study its degree. We distinguish a subclass ofthese maps, called self-adjoint GWM, for which thedegree of the map is at least 2. This class general-izes Wronski maps on the Grasmannians of the spaceof solutions of self-adjoint di↵erential operators andpole placement maps of symmetric linear control sys-tems. The main question: are there non-selfadjointGWM with degree greater than 1? We give a nega-tive answer for small-dimensional Grassmannians ofhalf-dimensional subspaces and in this way answerour original question in the case when the input andoutput spaces have the same small dimension.


Special Session 73: Mathematical Modeling and Computations

Hi Jun Choe, Yonsei University, KoreaSeick Kim, Yonsei University, Korea

Considering nonlinear phenomena like fluid flow, we make mathematical model by partial di↵erential equa-tions. As Hilbert announced in his 6th problem of Paris ICM, foundation of modeling and related computa-tions are main themes. Hamiltonian dynamics, dynamical system and continuum mechanics are discussed tounderstand their relations. Our scope includes particle transport, nonlinear flow, turbulence, mathematicalquestions and their computations.

Asymptotic Self-Similarity of Sin-gular Solutions for QuasilinearLane-Emden Equations

Soohyun BaeHanbat National University, Korea

We consider the asymptotic self-similarity of singularsolutions for quasilinear Lane-Emden equations. Themain result is described for p� Laplace Lane-Emdenequation.

E↵ects of White Noise in Multi-stable Dynamics

Carey CaginalpBrown University, USAXinfu Chen, Jianghao Hao, Yajing Zhang

The concept of stable equilibrium plays a key role inthe theory of ordinary di↵erential equations. A giveninitial condition uniquely determines how the systemevolves to a particular stable equilibrium point. Animportant question that one can ask involves intro-ducing white noise into the problem, and how evena perturbation by a very small amount of noise caninfluence which particular equilibrium point to whichthe system will evolve. Multiscale dynamics are well-known to describe practical examples such as pat-terns in physics, chemistry, and biology. This talkwill address a prototypical multistep dynamics withmultiple equilibrium points where the probabilities ofending up change depending on initial conditions.

Initial and Boundary Values forLq

↵

(Lp) Solution of the Navier-StokesEquations in the Half-Space

Tongkeun ChangYonsei University, KoreaBum Ja Jin

In this paper, we study the initial and boundary valueproblem of the Navier-Stokes equations in the half-space.

Incompressible Flow Computationsby VIP(virtual Interpolation PointCollocation)

Hi Jun ChoeCMAC, Yonsei University, KoreaGahyung Jo, Seongkwan Park

In this talk, we introduce meshfree method for fluidcomputations. Developing virtual interpolation tech-nique, we formulate Navier-Stokes and Stokes equa-tions in virtual nodes and prove the inf-sup condi-tion for virtual interpolation scheme. The stationaryStokes flow and Navier-Stokes flow can be computedeasily in complicated geometries. If time is allowedwe continue discussing projection method and stabil-ity for time dependent flows.

Holder Continuity of a BoundedWeak Solution for Degenerate andSingular Equations.

Sukjung HwangYonsei University, KoreaGary Lieberman

We generalise quasilinear parabolic p-Laplace typeequations to obtain the prototype equation

ut � div

✓g(|Du|)|Du| Du

◆= 0

where g is a nonnegative, increasing, and continu-ous function trapped in between two power functions|Du|g0�1 and |Du|g1�1 with either 2 g

0

g1

< 1or 1 < g

0

g1

2. We provide a proof for theHoolder continuity of such solutions emphasising ongeometrical properties, so called, the expansion ofpositivity. Also we discuss the same regularity forthe porous medium type equations.

Staggered Discontinous GalerkinMethods for Flow Problems

Hyea Hyun KimKyung Hee University, KoreaEric T. Chung, Chak Shing Lee, Siu WunCheung, Yue Qian

Staggered discontinuous Galerkin methods are devel-oped for flow problems. By introducing additionalunknowns, a first order system for the Stokes problemis obtained and then discreteized by using staggered


discontinuous finite element functions. The staggereddiscontinuous finite element functions give a flux con-dition on the element boundaries without the needof artificial flux or tuning of penalty parameters incontrast to the standard discontinous finite elementfunctions. Optimal order of errors can be obtainedfor a given polynomial degree k for all the unknowns.In addition, the method is applied to Navier-Stokesequations. Numerical results are presented to showthe capability of the proposed method for various testexamples.

Convergence Proof for AugmentedHodge Projection of Fluid-SolidInteraction

Chohong MinEwha Womans University, KoreaGangjoon Yoon, Seick Kim

In this work, we study a Fluid-Solid interaction basedon the Helmholtz-Hodge decomposition. Fluid andsolid dynamics interact intimately during the move-ment and the interaction rather makes then a newphysical object combined than a simple union of two

separated objects. For this reason, we take the mono-lithic treatment on fluid-solid interaction that thetwo-way couplings are imposed at the same time. Weintroduce a novel decomposition of the state variableinto two orthogonal components, which is a varia-tion of the Hodge decomposition. We refer to thedecomposition as an augmented Hodge decomposi-tion. The decomposition enables us to decouple thecomputations of the velocity and the pressure in theincompressible Navier-Stokes equation, giving an el-liptic equation for the pressure non-local Robin typeboundary condition, Then, the decomposition is ful-filled by solving the equation. We show the existence,uniqueness and the regularity of the solution to theequation. The monolithic treatment leads to the sta-bility that the kinetic energy does not increase in theprojection step by taking one of two components ofthe composition. Using an Heaviside function, we ex-press the boundary condition independent of the in-terfaces of the fluid and the solid. Also, we proposes anumerical method and shows that the unique decom-position and orthogonality also hold in the discretesetting. Also, we show that the numerical methodproduces the numerical solution at least with firstorder accuracy. We carry out numerical experimentsthat validate our analysis and arguments.


Special Session 74: Infinite Dimensional Stochastic Systems andApplications

Wilfried Grecksch, Martin Luther University Halle-Wittenberg, Germany

This special session should give a general overview as well about new tendencies in the field of infinite di-mensional stochastic systems as applications to sciences, economic sciences and problems of optimal control.The invited speakers will discuss especially issues related to stochastic partial di↵erential equations, stochas-tic di↵erential equations with memory, stochastic integral equations, stochastic integrodi↵erential equationsand stochastic di↵erence equations in infinite dimensional spaces.

A Stochastic Optimal Control Prob-lem in UMD-Banach Space Lq byUsing Maximum Principl

Mahdi AzimiMartin Luther University Halle-Wittenberg, Ger-many

We consider an optimal control problem in Banachspace E, where E = Lq(S,S, µ), µ is �-finte mea-sure. The state process will be defined as Ito Volterrastochastic integral equation and the stochastic inte-gral is defined with respect to a cylindrical Wienerprocess. The concept of Lp-stochastically integrabil-ity in Banach spaces will be used. By using appropri-ate assumptions, corresponding backward stochasticintegral equations will be derived and a duality prin-ciple between forward and backward stochastic inte-gral equations will be calculated. Then we use max-imum principle method to solve the optimal controlproblem.

Well-Posedness and Regularizationby Noise for Nonlinear PDE

Benjamin GessMax Planck Institute for Mathematics in the Sci-ences, Germany

In this talk we will revisit regularizing e↵ects of noisefor nonlinear SPDE. In this regard we are interestedin phenomena where the inclusion of stochastic per-turbations leads to increased regularity of solutionsas compared to the unperturbed, deterministic cases.Closely related, we study e↵ects of production ofuniqueness of solutions by noise, i.e. instances ofnonlinear SPDE having a unique solution, while non-uniqueness holds for the deterministic counterparts.The talk will concentrate on these e↵ects in the caseof nonlinear scalar conservation laws.

A Control Theory Approach to theSchrodinger Equation

Jeanette KoeppeMLU Halle-Wittenberg, GermanyWolfgang Paul

In 1966, E. Nelson [1] established a new interpre-tation of quantum mechanics, whereby the parti-cles follow some conservative di↵usion process, i.e.forward-backward stochastic di↵erential equations(FBSDEs), which are equivalent to the Schrodingerequation. Until now, this equivalence has been ap-plied in such a way that a known solution to theSchrodinger equation is used to integrate the stochas-tic di↵erential equations numerically and analyze thestatistical properties of the sample paths.

However, in analogy to classical mechanics, thestochastic equations of motion can be derived froman optimal control problem, whereby the variation ofstochastic action functionals leads to the Schrodingerequation as the Hamilton-Jacobi-Bellman equation ofthe problem. We show that the stochastic equationsof motion, i.e. the stochastic Hamilton equations,can be determined from optimal control problems byusing the maximum principle, which leads to equa-tions for the adjoint processes. The equations forthe di↵usion process as well as the equations of thecorresponding conjugated momentum constitute theHamilton equations of motion in the stochastic case.Solving these coupled FBSDEs numerically, we de-termined the probability distribution of the process,i.e. the square of the absolute value of the wavefunction.

References

[1] E. Derivations to the Schrodinger Equation fromNewtonian Mechanics, Phys. Rev., 1966, 150,1079-1085

Using Stochastic Di↵erential Equa-tions to Model the Cell Cycle

Rachel LeanderMiddle Tennessee State University, USAEdward Allen, Darren Tyson, Shawn Garbett,Vito Quaranta, Zack Jones

Given that division is one of the most important tasksa cell performs, it is surprising that, even within aclonal population, there is considerable variability inthe time it takes a cell to divide. Furthermore, indi-


vidual responses to a disturbance, such as drug treat-ment, are dynamic and heterogeneous. A better char-acterization of the distribution of division times canhelp researchers compare populations of proliferatingcells, describe growth at the population level, and in-vestigate cell cycle control mechanisms. In this talk,I will discuss the development, parameterization, andimplications of a simple stochastic di↵erential equa-tion model of the cell cycle.

Parameter Estimation for a MeanReversion Model Driven by Frac-tional Processes

Jens LueddeckensMLU Halle-Wittenberg / Institute of Mathematics,Germany

The aim of this talk is to estimate the mean rever-sion level µ of an inhomogeneous linear fractionalstochastic di↵erential equation with fractional Brow-nian motion BH(t) and two fractional compensatedPoisson processes NH

�1(t) and NH

�2(t). BH(t), NH

�1(t)

and NH�2(t) are stochastically independent. The es-

timation of µ, based on observed data and on thecondition of a positive constant mean reversion rateK, is done using a generalization of the least squaremethod. Finally the unbiasedness and consistency ofthe estimator for µ is proved.

Abstract Fractional Stochastic Evo-lution Equations with Applicationsto Nonlinear Beam Dynamics andStochastic Wave Equations

Mark MckibbenWest Chester University, USA

A class of abstract stochastic evolution equationsdriven by a fractional Brownian motion process withPoisson jumps will be the focus of this talk. Resultsconcerning the existence and uniqueness of solutionscorresponding to di↵erent types of nonlinear forcingterms and under various growth conditions are estab-lished. Results concerning stochastic stability andvarious convergence schemes will also be discussed.The motivation of the general theory established isrooted in the study of nonlinear beam dynamics andstochastic wave equations. The applicability of thegeneral theory to such applications will be illustrated.

Random Attractors for StochasticDi↵erential Inclusions in BanachSpaces

Alexandra NeamtuFriedrich-Schiller University Jena, Germany

We consider on separable Banach spacesX stochasticdi↵erential inclusions constituted by

⇢dx(t) 2 Ax(t)dt+ F (t, x(t))dt+ �(t, x(t))dWH(t)x(0) = x

0

, t 2 [0, T ].

Here A is the generator of a compact analytic C0

-semigroup, the nonlinear term F : [0, T ] ⇥ X !2X\{;} is a set-valued map, � : [0, T ]⇥X ! L(H;X)and WH is an H-cylindrical Wiener process. By us-ing fixed-point theorems for set-valued mappings, weestablish the existence of at least one solution forthe given evolution inclusion. Furthermore, undersuitable assumptions we derive a multivalued randomdynamical system and show that it has a random at-tractor.

Asymptotic Growth and Fluctu-ation for a Class of NonlinearStochastic Volterra Equations

Denis PattersonDublin City University, IrelandJohn Appleby

Motivated by economic applications, we develop pre-cise bounds on the growth rates and fluctuation sizesof unbounded solutions of forced nonlinear Volterraequations. The nonlinearity is assumed sublinear forlarge values of the state and we prove general resultsapplicable to a range of state–independent pertur-bations, random or deterministic. If an appropriatefunctional of the forcing term has a limit L at infinity,solutions of the forced equation behave asymptoti-cally like the unforced equation when L = 0, like theforcing term when L = +1, and inherit properties ofboth the forcing term and unperturbed equation forL 2 (0,1). In the special case of Brownian drivennoise, for L su�ciently large, solutions are recurrenton the real line and obey a type of time–changed it-erated logarithm law.This talk is based on joint work with Prof. John Ap-pleby (DCU) and is supported by the Irish ResearchCouncil under the project GOIPG/2013/402.

On a Rescaling Transformation forStochastic Partial Di↵erential Equa-tions

Michael RocknerBielefeld University, GermanyViorel Barbu

In the talk we shall give a survey on the rescalingtransformation to stochastic partial di↵erential equa-tions. We shall also present some recent new appli-cations.


4th Order Stochastic PDEs withQ-Regular Space-Time Noise

Henri SchurzSIU, USABoris Belinskiy

Some stochastic PDEs with Q-regular space-timenoise are discussed (with applications to Mechani-cal Engineering). Existence and uniqueness of ap-proximate strong solutions, some stability and energyestimates are presented under Dirichlet-type bound-ary conditions and with L2-integrable initial dataon bounded rectangular domains D. The resultsare connected to a joint work with Boris Belinskiy(UTC).

Invariance of Closed Convex Conesfor Stochastic Partial Di↵erentialEquations

Stefan TappeUniversity of Hannover, Germany

The goal of this talk is to clarify when a closed convexcone is invariant for a stochastic partial di↵erentialequation (SPDE) driven by a Wiener process and aPoisson random measure, and to provide conditionson the parameters of the SPDE, which are necessaryand su�cient. Our result is accompanied by severalexamples arising in natural sciences and economics.

Optimal Control of a StochasticHeat Equation Driven by Q-WienerProcesses

Christoph TrautweinMax Planck Institute Magdeburg, GermanyPeter Benner

We consider a linear quadratic control problem ofthe stochastic heat equation with Neumann bound-ary condition, where controls and noise terms are de-fined inside the domain as well as on the boundary.The noise terms are given by independent Q-Wienerprocesses. The control problem is given by a so calledtracking problem such that we can utilize the spe-cific cost functional to derive necessary and su�cientconditions stochastic optimal controls have to sat-isfy. Using these optimality conditions, we deduceexplicit formulas to obtain that stochastic optimalcontrols are given by feedback controls. Therefore,we conclude that the optimal controls are adapted toa certain filtration and we ensure that the state is anadapted process as well.

Cylindrical Continuous Martingalesand Stochastic Integration in Infi-nite Dimensions

Mark VeraarTU Delft, NetherlandsIvan Yaroslavtsev

In this talk we define a new type of quadratic vari-ation for cylindrical continuous local martingales onan infinite dimensional spaces. It is shown that alarge class of cylindrical continuous local martingaleshas such a quadratic variation. For this new class ofcylindrical continuous local martingales we develop astochastic integration theory for operator valued pro-cesses under the condition that the range space is aUMD Banach space. We obtain two-sided estimatesfor the stochastic integral in terms of the gamma-norm. In the scalar or Hilbert case this reduces tothe Burkholder-Davis-Gundy inequalities. An appli-cation to a class of stochastic evolution equations isgiven at the end of the paper.

On the Stochastic Heat Equationwith Sticky Reflected BoundaryCondition

Robert VosshallUniversity of Kaiserslautern, GermanyMartin Grothaus, Torben Fattler

In this talk we study the stochastic heat equationwith sticky reflected boundary condition. Dirichletform techniques are used to construct and charac-terize its solution. The obtained process already forsome time is conjectured to be the scaling limit of thedynamical wetting model, also known as Ginzburg-Landau dynamics with pinning and reflection com-peting on the boundary. For the second part of thistalk it is planned to discuss the progress on this prob-lem.

Minimization of Distance Mea-sures to E�ciently Capture theMacroscale Behavior of StochasticSystems.

Przemyslaw ZielinskiKU Leuven, BelgiumKristian Debrabant, Tony Lelievre, GiovanniSamaey

I will present a method to accelerate the simulationof observables of processes determined by stochas-tic ODEs, motivated by the recent development ofgeneric multiscale frameworks. The technique ex-ploits the separation between the fast time scale onwhich we compute trajectories and the slow evolutionof observables. The algorithm combines short burstsof paths simulation with extrapolation of a numberof moments forward in time.


I will concentrate on the crucial step – how to ob-tain, after extrapolation, a new ensemble of parti-cles/replicas compatible with given moments. Todeal with this inference problem, we introduce thematching operator based on the minimization of suit-able distance between probability distributions. I am

going to discuss the generic properties of this oper-ator that allow to establish the theorem on the or-der of convergence of our method. As a particularexample, I will present the matching based on thelogarithmic entropy that also provides a convenientnumerical approach.


Special Session 75: Recent Trends on PDEs Driven by GaussianProcesses with Applications

Hakima Bessaih, University of Wyoming, USAMaria J. Garrido-Atienza, University of Seville, Spain

We are interested in the theory and applications of stochastic systems driven by Gaussian noise. Of particularimportance are systems driven by fractional Brownian motion (fBm), and other long-memory and/or self-similar noise models. Classical probabilistic tools cannot be used to analyze fBm, because it fails to havethe martingale and Markov property. In recent years, there have been major breakthroughs in integrationtheory for fBm, particularly in rough path, fractional calculus, and random dynamical systems. This specialsession will be devoted to have a discussion about these various techniques and to what extent they can beused jointly. Employing these theories to solve dependent-noise-driven SDEs and SPDEs will be the firstmain goal of this session, as well as the study of stability properties of the solutions by using the theory ofrandom dynamical systems. The second goal will be to bring together the theoretical and applied communityof researchers in order to bridge the gap between theory and applications. On the applied side, a particularattention will be devoted to porous media and data assimilation.

Reynolds Number Dependence inHomogeneous Turbulence

Bjorn BirnirUC Santa Barbara, USAGregory Bewley, John Kaminsky, ShahabKarmini

We prove the existence of (local) solutions toa stochastic Navier-Stokes equation, in Sobolevspaces corresponding to the Kolmogorov-Obukhov-She-Leveque scaling. This permit a computation ofthe structure function of turbulence including theirdependance on the Reynolds-Taylor number. Thetheory is applied to the data obtained in a vari-able density wind-tunnel in the Max-Planck Institutefor Dynamics and Self-Organization in Gottingen,Germany, that is a modern version of the classicalPrandtl wind-tunnel experiment. A good agreementis found for a range of Reynolds-Taylor numbers, andsome surprising result on the smoothness of the so-lutions in turbulent flow.

Intermittency Properties for a Classof SPDEs Driven by FractionalNoise.

Daniel ConusLehigh University, USA

A space-time random-field is called physically inter-mittent if it develops high-valued peaks concentratedon small spatial islands as time gets large. In thistalk, we will illustrate this notion by discussing sev-eral examples of intermittent random fields givenby solutions to a class of parabolic and hyperbolicSPDEs. In particular, we will present some resultsrelated to equations driven by fractional noise.

Stochastic Navier-Stokes Equationsin Rd with Not Regular Multiplica-tive Noise

Benedetta FerrarioUniversity of Pavia, ItalyZ. Brzezniak

We consider the Navier-Stokes equation in Rd (d =2, 3) with multiplicative Gaussian white noise of verylow space regularity. We present results on solutionsand invariant measures/stationary solutions. This isa joint work with Z. Brzezniak.

Accuracy of Filters for QuadraticDissipative Systems

Kody LawORNL, USA

The solution to the problem of nonlinear filtering maybe given either as an estimate of the signal (and ide-ally some measure of concentration), or as a full pos-terior distribution. Similarly, one may evaluate thefidelity of the filter either by its ability to track thesignal or its proximity to the posterior filtering dis-tribution. Hence, the field enjoys a lively symbio-sis between probability and control theory, and thereare many applications which benefit from algorith-mic advances. This talk will survey some recent the-oretical results involving accurate signal tracking forquadratic dissipative deterministic PDE (as well assome related ODE). In the limit of continuous-timeobservations, the equation for the filter is a stochasticPDE (SPDE).


Dynamics of Non-Densely DefinedStochastic Evolution Equations

Alexandra NeamtuFriedrich-Schiller University Jena, Germany

We consider on separable Banach spaces a class ofstochastic evolution equations with a non-densely de-fined linear part. In this case, the C

0

-semigroup the-ory can no longer be applied. Such situations oc-cur when certain additional restrictions are incorpo-rated in the domain of a linear operator. The noiseterm is constituted by a Banach space-valued Brow-nian motion. A suitable transformation allows us toreduce the stochastic equation into a random one,from which we can derive a random dynamical sys-tem and investigate the existence of random attrac-tors. Our theory is based on the integrated semi-group approach, considered in the deterministic caseby P. Magal and S. Ruan (2009). As applications, wediscuss parabolic equations with nonlinear boundaryconditions.

On the Sochastic Transport Equa-tion

Christian OliveraIMECC UNICAMP, BrazilE. Fedrizzi, W. Neves, C. Tudor

We study the existence and uniqueness of solutionsto the stochastic transport equations with irregularcoe�cients driven by standard Brownian motion andFractional Brownian motion. Also we study the regu-larity in law of the solutions using Malliavin Calculus.

Data Assimilation Using Stochasti-cally Noisy Data

Eric OlsonUniversity of Nevada, USAHakima Bessaih, Edriss Titi

We analyze the performance of a data-assimilation al-gorithm based on a linear feedback control when usedwith observational data that contains measurementerrors. Our model problem consists of dynamics gov-erned by the two-dimension incompressible Navier-Stokes equations, observational measurements givenby finite volume elements or nodal points of the ve-locity field and measurement errors which are repre-sented by stochastic noise.

Sample Path Properties for SDEsDriven by Fractional Brownian Mo-tions

Cheng OuyangUniversity of Illinois at Chicago, USAFabrice Baudoin, Shuwen Lou, Eulalia Nu-alart, Samy Tindel

In this talk, we present some of our recent resultson sample path properties for SDEs driven by frac-tional Brownian motions, including hitting probabil-ity, Hausdor↵ dimension of sample paths and levelsets, and existence and regularity of local times.

Fractional White-Noise Limit andParaxial Approximation for Wavesin Random Media

Olivier PinaudColorado State University, USAChristophe Gomez

This talk is devoted to the asymptotic analysis ofhigh frequency wave propagation in random mediawith long-range dependence. We are interested intwo asymptotic regimes, that we investigate simulta-neously: the paraxial approximation, where the waveis collimated and propagates along a privileged direc-tion of propagation, and the white-noise limit, whererandom fluctuations in the background are well ap-proximated in a statistical sense by a fractional whitenoise. The fractional nature of the fluctuations isreminiscent of the long-range correlations in the un-derlying random medium. A typical physical settingis laser beam propagation in turbulent atmosphere.Starting from the high frequency wave equation withfast non-Gaussian random oscillations in the velocityfield, we derive the fractional Ito-Schrodinger equa-tion, that is a Schrodinger equation with potentialequal to a fractional white noise. The proof in-volves a fine analysis of the backscattering and ofthe coupling between the propagating and evanes-cent modes. Because of the long-range dependence,classical di↵usion-approximation theorems for equa-tions with random coe�cients do not apply, and wetherefore use moment techniques to study the con-vergence.


Irreducbility and Ergodicity ofSome Stochastic HydrodynamicalSystems Driven by Tempered StableProcesses

Paul RazafimandimbyMontanuniversitaet Leoben, Austria

In this talk, we present several results related to thelong-time behavior of a class of stochastic semilinearevolution equations in a separable Hilbert space H:

du(t)+[Au(t)+B(u(t), u(t))]dt = dL(t), u(0) = x 2 H.

The driving noise L is basically a general temperedstable process satisfying several technical assump-tions, A is a positive self-adjoint operator and Bis a bilinear map. By using a density transforma-tion theorem type for Levy measure, we first provea support theorem and an irreducibility property ofthe Ornstein-Uhlenbeck processes associated to thenonlinear stochastic problem. Second, by exploit-ing the previous results we establish the irreducibil-ity of nonlinear problem provided that for a certain� 2 [0, 1/4] B is continuous on D(A�)⇥D(A�) withvalues in D(A�1/2). Proving a Bismut-Elworthy-Litype lemma for SDEs driven by pure jump noise andusing a method from the 1995 paper of Flandoli andMaslowski, the uniqueness of invariant measure isalso proved under the assumption, which is a muchstronger than the previous one, that B is continuouson H⇥H. While the latter condition is only satisfiedby the nonlinearities of GOY and Sabra Shell mod-els, the assumption under which irreducibility prop-erty holds is verified by several hydrodynamical sys-tems such as the 2D Navier-Stokes, Magnetohydro-dynamic equations, the 3D Leray-↵ model, the GOYand Sabra shell models.The talk is based on joint works with H. Bessaih, E.Hausenblas and P. Fernando.

Dynamics of SPDE Driven by aFractional Brownian Motion

Bjorn SchmalfußFriedrich Schiller University of Jena, Germany

We consider the stochastic evolution equation on aHilbert-space V

du+Audt = G(u)d!, u(0) = u0

2 V

where ! is a fractional Brwonian motion with Hurst-parameter H 2 (1/2, 1), A generates an exponen-tially stable analytic semigroup and G is su�cientlysmooth. We discuss several aspects of the dynamicsof such an equation, like the existence of a randomattactor.

Exit Times for Solutions of Stochas-tic Navier-Stokes Equations

Padmanabhan SundarLouisiana State University, USA

Exponential estimates for exit times are obtained forsolutions of the stochastic Navier-Stokes system in2D and for related equations in hydrodynamics. Thestudy involves the asymptotic behavior of solutions intime as well as in a parameter. Equations perturbedby fractional Brownian noise will also be discussed.

Filtering Problems in StochasticTomography

Jason SwansonUniversity of Central Florida, USATyler Gomez, Alexandru Tamasan

We consider the X-ray transform of a function-valuedrandom variable, and study the conditional distribu-tion of the inversion in the presence of noise. Dueto its medical applications, the mathematical theoryof the X-ray transform is rich with numerous resultsspanning analytical and numerical aspects of inver-sion. However, these results assume a deterministicapproach. Here, we introduce a stochastic approachto the tomography problem. This is joint work withTyler Gomez and Alexandru Tamasan.

Quadratic Variations for theFractional-Colored Stochastic HeatEquation

Frederi ViensPurdue University, USASoledad Torres, Ciprian Tudor

Using multiple stochastic integrals and Malliavin cal-culus, we analyze the quadratic variations of a class ofGaussian processes that contains the linear stochas-tic heat equation on Rd driven by a non-white noisewhich is fractional Gaussian with respect to the timevariable (Hurst parameter H) and has colored spatialcovariance of ↵-Riesz-kernel type. The processes inthis class are self-similar in time with a parameter Kdistinct from H, and have path regularity propertieswhich are very close to those of fractional Brownianmotion (fBm) with Hurst parameter K (in the heatequation case, K = H�(d�↵)/4 ). However the pro-cesses exhibit marked inhomogeneities which causenaive heuristic renormalization arguments based onK to fail, and require delicate computations to estab-lish the asymptotic behavior of the quadratic varia-tion. A phase transition between normal and non-normal asymptotics appears, which does not corre-spond to the familiar threshold K = 3/4 known inthe case of fBm. We apply our results to constructan estimator for H and to study its asymptotic be-havior.


Special Session 76: Advances in the Numerical Solution of NonlinearEvolution Equations

Mechthild Thalhammer, Universitat Innsbruck, AustriaWinfried Auzinger, Technische Universitat Wien, Austria

The intention of this special session is to bring together mathematicians and theoretical physicists, inter-connected through their field of application, relevant analytical tools, or the numerical methods used. Thescope of topics includes, but is not limited to, evolutionary Schrodinger type equations, highly oscillatoryequations, and adaptive integration methods for partial di↵erential equations.

A Hybrid WKB-Based Method forthe Stationary Schroedinger Equa-tion in the Semi-Classical Limit

Anton ArnoldVienna Univ. of Technology / Analysis & ScientificComp., AustriaClaudia Negulescu

We are concerned with the e�cient numerical inte-gration of ODEs of the form ✏2u” + a(x)u = 0 for0 < ✏0 ) with evanescent regions (i.e. for a(x) < 0). In the oscillatory case we use a marching methodthat is based on an analytic WKB-preprocessing ofthe equation. And in the evanescent case we use aFEM with WKB-ansatz functions. We present a fullconvergence analysis of the coupled method, show-ing that the error is uniform in epsilon and secondorder w.r.t. h. We illustrate the results with numer-ical examples for scattering problems for a quantum-tunnelling structure.

Local Error Estimation and Adap-tive Splitting in Time

Winfried AuzingerTechnische Universitaet Wien, AustriaHarald Hofstaetter, David Ketcheson OthmarKoch, Mechthild Thalhammer

We consider the integration of evolution equationsby adaptive time-splitting methods, where the right-hand side is split into two or more components.Adaptive integrators are based on di↵erent tech-niques for estimating local error, including embeddedor adjoint pairs of schemes or defect-based estima-tors. Theoretical as well as implementation aspectsare discussed. Several numerical examples are pre-sented, in particular equations of Schroedinger typeand parabolic equations.

Geometric Integration of a DampedDriven Nonlinear SchrodingerEquation

Ashish BhattUniversity of Central Florida, USABrian E. Moore

A damped driven nonlinear Schrodinger equation(NLSE) is known to possess rich dynamics includingsolitary and shock wave solutions, periodic solutions,bifurcation and chaos. Numerical methods aim to re-duce quantitative errors whereas geometric integra-tors aim to reduce or eliminate qualitative errors (e.g.error in preservation of a conserved quantity) as wellin order to improve accuracy. Geometric integratorshave attracted a lot of attention in the past severaldecades. They have been shown to be advantageousas compared to non-structure preserving methods inresolving the dynamics of a problem. The purposeof this talk is to present application of a geometricintegrator to a damped driven NLSE and discuss nu-merical results.

Exponential Asymptotic Splittingfor the Linear Schroedinger Equa-tion

Karolina KropielnickaUniversity of Gdansk, PolandPhilipp Bader, Arieh Iserles, Pranav Singh

The discretization of a linear Schroedinger equationis di�cult due to the presence of a small parame-ter which induces high oscillations. A standard ap-proach consists of a spectral semidiscretization, fol-lowed by an exponential splitting. This, however,is sub-optimal, because the exceedingly high preci-sion in space discretization is married by low orderof the time solver. In this talk we sketch an alter-native approach. Our analysis commences not withsemi-discretisation, but with the investigation of thefree Lie algebra generated by di↵erentiation and bymultiplication with the interaction potential: it turnsout that this algebra possesses a structure which ren-ders it amenable to a very e↵ective form of exponen-tial asymptotic splitting: exponential splitting whereconsecutive terms are scaled by increasing powersof the small parameter. The semi-discretisation indeferred to the very end of computations. We willfocus on the method for the time dependant linear


Schroedinger equation with potential non-dependingon time, however we will also discuss the di�cultiesthat appear with time dependant potential and willbriefly propose the remedy to that stage of an a↵air.

Stochastic Homogenization of aVisco-Elastic Model for Strain-Stress Hysteresis

Frederic LegollEcole des Ponts, FranceThomas Hudson, Tony Lelievre

Motivated by the modelization of hysteresis in filledrubber, we consider a time-dependent viscoelasticmodel in which the constitutive law for the solidvaries randomly on a small lengthscale. At the finescale, the model includes an elastic energy, viscousfriction and solid friction, all with random highly os-cillatory coe�cients. It exhibits hysteretic behaviourwhich persists under slow loading. We identify thehomogenized limit of this model, and demonstratethat, at the coarse scale, the model again exhibitshysteretic behaviour which persists under slow load-ing.

Structure-Preserving Algo-rithms for Perturbed NonlinearSchrodinger Equations

Brian MooreUniversity of Central Florida, USALaura Norena, Constance Schober

Some nonlinear Schrodinger equations with addedlinear damping and/or convection terms have a spe-cial structure, which is known in the ODE literatureas conformal symplectic. This special structure leadsto several conserved quantities that can be preservedexactly by various discretizations. Presentation ofmethods of this type will be followed by a brief ex-planation of the benefits of using such discretizations,including both theoretical and numerical results.

Comparison of Convergence of FullyDiscrete Splitting Schemes for theNonlinear Schrodinger EquationUsing Fourier Transform and Itera-tive Linear Solvers

Benson MuiteUniversity of Tartu, EstoniaE. Vainikko

Convergence of splitting schemes for the nonlinearSchrodinger equation are examined. Rates of conver-gence and accuracy are measured for several serialand parallel algorithms based on Fourier pseudospec-tral discretizations and on finite di↵erence discretiza-tions.

Non-Local NLS of Derivative Typefor Modeling Highly Nonlocal Opti-cal Nonlinearities

Hans Peter StimmingUniv. of Vienna, WPI, AustriaIgor Mazets, Ephraim Shamoon, GershonKurizki, Pjotrs Grisins

A new NLS type equation is employed for model-ing long-range interactions in nonlinear optics, in acollaboration with experimental physicists. It is ofquasilinear type, and models fluctuations around a‘continuous-wave polariton‘ which are chosen accord-ing to Bogoliubov excitation theory. Mathematicaltheory / analysis for this equation is work in progress,results exist however for other quasilinear NLS at thesame order of derivatives. A similar model was usedfor numerical simulation of time-dependent decoher-ence in split BECs.

References

[1] Extremely nonlocal optical nonlinearities inatoms trapped near a waveguide, E. Shahmoon,P. Grisins, H. P. Stimming, I. Mazets, G. Kur-izki, arxiv.org:1412.8331v1

[2] Relativistic Nonlinear Schroedinger equation, A.DeBouard, N. Hayashi, J. C. Saut, Comm.Math. Phys. 189 (1997) 73-105

[3] Dephasing in coherently-split quasicondensates,H.P. Stimming, N.J. Mauser, J. Schmiedmayer,I.E. Mazets, Phys.Rev.A 63 (2011) 023618


Special Session 77: Delay Di↵erential Equations with State-DependentDelays and their Applications

Qingwen Hu, The University of Texas at Dallas, USABernhard Lani-Wayda, University of Giessen, Germany

Eugen Stumpf, University of Hamburg, Germany

Delay di↵erential equations with state-dependent delays have become an active research field of mathematics.More and more models in biology, machining dynamics, physiology, tra�c control and laser dynamics havebeen reported. The special session aims at bringing together researchers in this field and interested scholarsin mathematics and/or applied sciences for communications on the current state of this subject.

On Oscillation, Stability, Bounded-ness and Persistence of NonlinearEquations with Several Delays

Elena BravermanUniversity of Calgary, CanadaLeonid Berezansky

In the first part of the talk, we explore nonlinearequations and systems with a delayed positive feed-back, where the delay, is, generally, distributed. Suchequations are globally asymptotically stable (and in-trinsically non-oscillatory), under some natural con-ditions on the unique positive equilibrium, when thedelay is bounded. If there are several equilibriumpoints, multistability is observed. In the case of theunique positive equilibrium and monotonicity, similarresults are obtained for a system of two equations.In the second part, nonlinear equations with morethan one delay

x(t) =mX

k=1

fk(t, x(h1

(t)), . . . , x(hl(t)))� g(t, x(t)),

where the functions fk increase in some variables anddecrease in the others, we obtain conditions whena positive solution exists on [0,1), as well as ex-plore boundedness and persistence of solutions. Ex-amples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solu-tions can be neither persistent nor bounded, unlikethe well studied case when these two delays coincide.

A PDE of Variational Electrody-namics

Jayme de LucaUFSCar, Brazil

The electromagnetic two-body problem is an infinite-dimensional problem with four state-dependent de-lays of neutral type, which system can have periodicorbits with discontinuous velocities. Con- tinuousmaps of light-cone type from R4 to points along suchperiodic orbits naturally inherit the derivative dis-continuities and our distributional construct to avoidthese discontinuities involves a second order PDE.We will discuss the distributional construction andthe importance of such PDE to variational electro-dynamics.

Stability of Linear Discrete Systemswith Delays

Josef DiblikCEITEC, Brno University of Technology, Czech Rep

In the talk we give su�cient conditions for the ex-ponential stability of linear di↵erence systems withdelays

x (k + 1) = Ax (k) +Ps

i=1

Bi(k)x (k �mi(k)) , k = 0, 1, . . .

where A is an n⇥n constant matrix, Bi(k) are n⇥nmatrices, mi(k) 2 N, mi(k) m for an m 2 N, s 2 Nand x = (x

1

, . . . , xn)T : {�m,�m+ 1, . . . } ! Rn.

The results are compared with some previously pub-lished results. The exponential stability is studied bythe second Lyapunov method.

On Neutral Di↵erential Equationswith State-Dependent Delays

Ferenc HartungUniversity of Pannonia, Hungary

In this talk we consider a class of nonlinear neutraldi↵erential equations with state-dependent delays inboth the neutral and the retarded terms. We studywell-posedness and continuous dependence issues anddi↵erentiability of the parameter map with respect tothe initial function and other possibly infinite dimen-sional parameters.

Quasi-Periodic Solutions for State-Dependent Delay Di↵erential Equa-tions

Xiaolong HeGeorgia Institute of Technology, USARafael de la Llave

The existence of quasi-periodic solutions for state-dependent delay di↵erential equations is investigatedby using the parameterization method, which is dif-ferent from the usual way-working on the solutionmanifold. Under the assumption of finite-times dif-ferentiability of functions and exponential dichotomy,the existence and smoothness of quasi-periodic so-lutions are investigated by using contraction argu-ments. Meanwhile, we show that there are Lindst-edt series under some non-degeneracy conditions forthe analytic case. In particular, a KAM theory is


developed to seek analytic quasi-periodic solutions,which gets involving the theory of foliation preserv-ing torus mapping. Moreover, we prove that theset of parameters which guarantee the existence ofanalytic quasi-periodic solutions is of positive mea-sure. All of these results are given in an a-posteriorform. Namely, given an approximate solution satis-fying some non-degeneracy conditions, there is a truesolution nearby.

On Di↵usion Processes withThreshold-Type State-DependentDelay

Qingwen HuThe University of Texas at Dallas, USA

We develop a Hopf bifurcation theory of di↵usionprocesses with threshold-type delay and investigatethe e↵ect of the state-dependent di↵usion time onintracellular regulatory dynamics. A general modelwhich is an extension of the classic di↵erential mod-els with constant or zero time delays is developed tostudy the stability of steady state and the occurrenceand stability of oscillations in regulatory dynamics.Using the multiple time scale method, we computethe normal form of the general model and show thatthe state-dependent di↵usion time may stabilize ordestabilize the bifurcated periodic solutions of thecorresponding models which do not incorporate thestate-dependent di↵usion time.

Lyapunov-Razumikhin Techniquesfor State-Dependent DDEs

Tony HumphriesMcGill University, CanadaFelicia Magpantay

We present theorems for the Lyapunov and asymp-totic stability of the steady state solutions to generalstate-dependent delay di↵erential equations (DDEs)using Lyapunov-Razumikhin methods. The Lya-punov stability result applies to nonautonomousDDEs with multiple discrete state-dependent delaysof the form⇢

u(t) = f�t, u(t), u(t� ⌧

1

(t, u(t))), . . . , u(t� ⌧N (t, u(t)))�, t > t

0

,u(t) = '(t), t 6 t

0

,

and is proved by a contradiction argument which isadapted from a previous result of Barnea for retardedfunctional di↵erential equations (RFDEs).

Our asymptotic stability result applies to au-tonomous DDEs with multiple state-dependent dis-crete delays. Its proof is entirely new, and is based ona contradiction argument together with the Arzela-Ascoli theorem. This alleviates the need for an aux-iliary function to ensure the asymptotic contraction,which is a feature of the other Lyapunov-Razumikhinasymptotic stability results of which we are aware.We apply our results to the state-dependent modelequation

u(t) = µu(t) + �u(t� a� cu(t)), t � 0,u(t) = '(t), t 6 0,

which includes Hayes equation as a special case (whenc = 0), to directly establish asymptotic stability inparts of the stability domain along with lower boundsfor the basin of attraction. We also generalise ourtechniques to derive a condition for global asymp-totic stability of the zero solution to the model prob-lem, and also to find bounds on the periodic solutionswhen the steady-state solution is unstable.

Dynamical Behavior in Some Low-Dimensional Delay Systems

Anatoli IvanovPennsylvania State University, USA

The system of delay di↵erential equations of the form

x 0i (t) = ��i xi(t) +

nX

j=1

wijfi(xj(t� ⌧ij)), (⇤)

for i = 1, 2, . . . , n appears as a mathematical modelof numerous real world phenomena. It exhibits richdynamical behavior such as global asymptotic sta-bility of equilibria, existence and stability of peri-odic solutions, and existence of complex/chaotic so-lutions among others. We discuss some known re-sults, derive several new results, and propose severalconjectures for the cases of low-dimensional system(⇤), n = 3, n = 2, and n = 1.

Oscillation Speed and Slowly Os-cillating Solutions for a Class ofEquations with State-DependentDelay

Benjamin KennedyGettysburg College, USA

We consider a class of di↵erential delay equationswith state-dependent delay and negative feedbackthat includes the equation

x‘(t+ g(x(t))) = f(x(t))

introduced by Walther in 2008. We define a non-increasing oscillation speed for this class of equa-tions. Motivated primarily by known results inthe constant-delay case, we discuss the behavior ofslowly-oscillating solutions in the case that f is mono-tonic. This work is in a similar spirit to work byKrisztin and Arino from the early 2000s on the equa-tion x‘(t) = �µx(t) + f(x(t� d(x(t)))).


Lyapunov Method for Stability ofFDEs with State-Dependent Delay

Xiaodi LiShandong Normal University, Peoples Rep of ChinaJianhong Wu

This talk will address the stability problem for func-tional di↵erential systems with state-dependent de-lay, which is not assumed the be a priori bounded.Using some Lyapunov functions coupled with di↵er-ential inequality techniques tailed at state-depedentdelay, we shall present some su�cient conditions forexponential stability and Lipschitz stability. Then weshall introduce some specific examples to illustratethe e↵ectiveness of the approach and our general re-sults.

Intra-Specific Competition and In-sect Larval Development: a Modelwith Time-Dependent Delay

Yijun LouThe Hong Kong Polytechnic University, Hong KongStephen A. Gourley, Rongsong Liu

This talk presents a stage-structured model for aninsect population in which a larva matures on reach-ing a certain size, and in which there is intra-specificcompetition among larvae which hinders their devel-opment thereby prolonging the larval phase. Themodel, a system of delay di↵erential equations forthe total numbers of adults and larvae, assumes twoforms. One of these is a system with a variable state-dependent time delay determined by a threshold con-dition, the other has constant and distributed delays,a size-like independent variable and no threshold con-dition. We prove theorems on boundedness and onthe linear stability of equilibria.

State Dependent Delays in Infec-tious Disease Modeling

Gergely RostUniversity of Szeged, Hungary

We discuss two recent examples from mathematicalepidemiology where state dependent delays arise nat-urally. The first is an SIRS type model with waningand boosting of immunity, the second is malaria dy-namics with long incubation periods.

An Inverse Problem for Delay Dif-ferential Equations: ParameterEstimation and Sensitivity Analysis

Fathalla RihanUnited Arab Emirates University, United ArabEmirates

This contribution presents a theoretical framework tosolve inverse problems for Delay Di↵erential Equa-tions (DDEs). Given a parametrized DDE and ex-perimental data, we estimate the parameters appearin the model, using Least Squares approach. We alsoinvestigate the sensitivity and robustness of the mod-els to small perturbations in the parameters, usingVariational approach. The results may provide guid-ance for the modelers to determine the most infor-mative data for a specific parameter, and select thebest fit model. The nonlinearity may make the prob-lem ill-posed. Discontinuity and noisy data are alsochallenges. The consistency of neutral delay di↵er-ential equations with bacterial cell growth is shown,as a numerical example, by fitting the model to realobservations.

Existence of Unbounded Solutionsof Delayed Linear Di↵erential Equa-tions.

Zdenek SvobodaCEITEC, Brno University of Technology, Czech Rep

Recently, much attention was paid to a new formal-ization of the well-known method of steps in the the-ory of linear di↵erential equations with constant co-e�cients and single delay. In the first part of the talkwe will study solutions of linear di↵erential systemsof the first and second orders with constant matri-ces and constant delays. Solutions of such systemscan be represented by special delayed matrix func-tions defined on intervals (k � 1)⌧ t0 is a delay,as matrix polynomials continuous at nodes t = k⌧ .In the second part of the talk we will discuss prob-lems of asymptotic analysis of solutions, representedby delayed matrix functions. We show that thatthe sequence of values of solutions at nodes t = k⌧ ,k = 0, 1, . . . , can be approximately represented bya geometric-type sequence. Its “quotient” factor isdefined by the principal branch of the Lambert func-tion. Obtained asymptotic properties of the delayedexponential of a matrix are modified (utilizing an Eu-ler identity for delayed matrix functions) for the casesof delayed matrix sine and delayed matrix cosine.


Stability and Stabilization in Func-tional Di↵erential Equations withState-Dependent Delays

Janos TuriUTD, USA

In this talk we show that a constant steady-state of anautonomous state-dependent delay di↵erential equa-tion is stable (stabilizable) if the trivial solution ofa corresponding linear autonomous equation is sta-ble (stabilizable). The result is applied to a state-dependent distributed-delay model of turning opera-tions.

Hydra E↵ect for Delay Di↵erentialEquations

Gabriella VasUniversity of Szeged, HungaryTibor Krisztin, Monika Polner

We consider a delay di↵erential equation modelingproduction and destruction: y(t) = �µy(t)+ f(y(t�1)) with µ > 0 and nonincreasing f : R ! (0,1).We prove the presence of the paradoxical hydra ef-

fect, namely, we show that increasing the value of thedestruction parameter µ can result in an increase ofthe mean value of certain solutions. The nonlinearityf in the equation is a step function or a smooth func-tion close to a step function. This particular form off allows us to construct periodic solutions, and toevaluate the mean values of the periodic solutions.Our result explains how the global form of the pro-duction term f induces the appearance of the hydrae↵ect.

Existence and Uniqueness of Mildand Strict Solutions for AbstractDi↵erential Equations with StateDependent Delay

Jianhong WuYork, CanadaMichelle Pierri, Eduardo Hernandez

In this talk, we present some results related the exis-tence and uniqueness of mild and strict solutions forabstract di↵erential equations with state dependentdelay, with particular focus on alpha-Holder strictsolutions.


Special Session 78: Advances in Analysis of Mathematical Problemsarising from Materials and Biological Science

Toyohiko Aiki, Japan Women’s University, JapanAdrian Muntean, Karlstad University, Sweden

Tomomi Yokota, Tokyo University of Science, Japan

This special session focuses on technologically important and mathematically interesting problems arisingfrom material and biological sciences. Besides well-posedness and asymptotic studies of nonlinear PDEsand related particle systems, the session also includes presentations on averaging processes in porous media,non-periodic homogenization, statistical mechanics of particle systems, and multiscale dynamics

Recent Results Related to theTwo-Scale Model for Concrete Car-bonation

Toyohiko AikiJapan Women’s University, JapanYusuke Murase, Naoki Sato

From civil engineering point of view there exists sev-eral kinds of serious issues on concrete, for example,carbonation, cracks, corrosion and etc. In our re-cent works we have proposed and studied the math-ematical model for concrete carbonation. The modelconsists of two conservation laws for moisture andcarbon dioxide. Here, we discuss only the moisturetransport equation. On modeling for the moisturetransport process it is a crucial step how to repre-sent the relationship between the relative humidityand the degree of saturation. For this di�culty wealready gave the two-scale model containing a one-dimensional free boundary problem and shown thewell-posedness of the initial boundary value problemfor the model. The main aim of this talk is to estab-lish results on large time behavior of the solution ofthe free boundary problem.

Numerical Simulation of WavesPropagation in Materials with Mi-crostructure

Mihhail BerezovskiEmbry-Riddle Aeronautical University, USA

Results of numerical experiments are presented in or-der to compare direct numerical calculations of wavepropagation in a laminate with prescribed proper-ties and corresponding results obtained for an ef-fective medium with the microstructure modelling.These numerical experiments allowed us to analysethe advantages and weaknesses of the microstructuremodel.

Global Existence and Boundednessin a Mathematical Model of UrbanCrime Model

Kentarou FujieTokyo University of Science, Japan

In this talk we consider a mathematical model de-scribing burglary in residential areas. Based on twosociological e↵ects the broken window e↵ect and therepeat near-repeat e↵ect, this model was proposed inShort et al.(2008). From a mathematical view point,the system is similar as the chemotaxis system andinvolves a logarithmic sensitivity function and spe-cific interaction and relaxation terms. Under somecondition, global existence and boundedness will beshown.

Equation and Dynamic BoundaryCondition of Cahn-Hilliard Typewith Related Topics

Takeshi FukaoKyoto University of Education, Japan

In this talk, the perspective for some equation anddynamic boundary condition of Cahn-Hilliard typeis treated:

@u@t

� �µ = 0 in Q := (0, T )⇥ ⌦,

µ = ��u+W 0(u)� f in Q,

u�

= u|� , µ�

= µ|� ,@u

�

@t+ @⌫µ� �

�

µ�

= 0

on ⌃ := (0, T )⇥ �,

µ�

= @⌫u� ��

u�

+W 0�

(u�

)� f�

on⌃ ,

u(0) = u0

in⌦ , u�

(0) = u0�

on� .

Firstly, based on the result of the well-posedness byColli-Fukao (2015), the asymptotic limit of Cahn-Hilliard systems to the degenerate parabolic equa-tions is discussed under the dynamic boundary con-dition. Secondly, the boundary control problem isdiscussed. Both of them, the structure of the totalmass conservation, namely the volume in the bulkplus the boundary is the key of the problem.


The Solvability of the Perfect Plas-ticity Model with Time DependentConstraints

Risei KanoKochi University, JapanTakeshi Fukao

In this talk, We discuss problems with the plas-tic deformation of perfect materials. This problemhas been discussed by many scholars. In particular,G.Duvaut and J.L.Lions showed the solutions of theevolution problem that has the constraints on thresh-old of stress, in 1976. We think about the solvabilityof the extended problem having the function f(t,x)of the threshold. Here follows the way of Duvaut-Lions, we talk about the existence of solutions of theparabolic approximation problem.

Continuous Models of Graph-LikeMatter

Ivan KryvenUniversity of Amsterdam, Netherlands

The duality of concepts that emerges from two seem-ingly contradicting descriptions of matter: discreteversus continuous, is commonly accepted by view-ing each of the concepts at its own “spatial scale”.Nonetheless, substances consisting of randomly in-terconnected network of small units (e.g. polymermaterials) are hard to analyse from exclusively dis-crete or continuous point of view. This is mainlydue to the fact that the graph topology that standsbehind the network plays definitive role in shapingthe properties of continuum. I shall focus on modelsthat measure important characteristics of such net-works with distributions and predict their evolutionemploying continuous balance equations. The talkwill be fortified with examples of analytical, numeri-cal, and stochastic nature.

On a Multiscale Model for MoistureTransport Appearing Concrete Car-bonation Process

Kota KumazakiTomakomai National College of Technology, JapanToyohiko Aiki, Naoki Sato, Yusuke Murase

In this talk, we consider a mathematical model formoisture transport in concrete carbonation process.In this process, it is known that the relationship be-tween the relative humidity and the degree of satu-ration is a hysteresis, and this is caused by a wettingand drying phenomenon which occurs in each pore in-side of concrete (Microscopic level). Recently, a freeboundary problem describing the wetting and dryingprocess is proposed, and is shown the existence of asolution of this problem. In this talk, we propose a

mathematical model for moisture transport consist-ing of a di↵usion equation of the relative humidityand the above free boundary problem, and discussthe existence of a time local solution of this problem.

Chemotaxis with Logistic Source

Johannes LankeitPaderborn University, Germany

Chemotaxis is the directed movement of e.g. cellsor bacteria in response to concentration di↵erencesof a chemical in their neighbourhood and plays animportant role in a multitude of biological contexts,ranging not only from aggregation and pattern for-mation in colonies of bacteria or slime mold to cancerinvasion of a tumour into surrounding healthy tissue.In this talk we will deal with a cross-di↵usive sys-tem of PDEs that combines the e↵ects of chemotaxiswith those of (logistic) population growth. We willconsider qualitative properties of solutions, in partic-ular their asymptotic behaviour and transient growthphenomena.

Mathematical Modeling of GrowthPatterns and Consecutive BoneDestruction in Multiple Myeloma

Anna Marciniak-CzochraUniversity of Heidelberg, GermanyFlip Klawe, Andro Mikelic

The talk is devoted to mathematical modeling ofgrowth and transport dynamics in a heterogenousevolving structure. The model is motivated by spa-tiotemporal dynamics of multiple myeloma, a malig-nant disease of plasma cells causing a variety of clin-ical signs and symptoms including loss of bone massand mineral density and osteolytic lesions. The pro-cess is related not only to growth and spread of can-cer cells but also to changes of bone microstructure.We present derivation and well-posedness of a mul-tiscale model coupling surface and volume reactions.It leads to a moving boundary problem with densitydependent velocity. Coupling volume processes withreaction-di↵usion dynamics on moving surfaces pro-vides challenges for mathematical modeling, analysisand simulation.

Asymptotic Stability in a Two-Species Chemotaxis System withLogistic Source

Masaaki MizukamiTokyo University of Science, Japan

This talk is concerned with asymptotic bahavior ofsolutions to a two-species chemotaxis system with lo-gistic source. This system describes a situation inwhich multi populations react on single chemoattrac-tant. Moreover we assume that both populations re-


produce themselves, and mutually compete with theother, according to the classical Lotka-Volterra ki-netics. Zhang and Li proved existence of boundedglobal-in-time solutions to the system. The main re-sult asserts asymptotic stability in the system.

Periodic Homogenisation of Filter-ing Combustion in Fast Drift andFast Reaction Regimes

Adrian MunteanKarlstad University, Sweden

We discuss a basic homogenisation scenario thatrefers to periodic approximations of porous mediahosting combustion. The special feature of our set-ting is the presence of both fast drifts and fast reac-tions. Using suitable two-scale convergence-like con-cepts, we obtain e↵ective model equations and trans-port coe�cients. Additionally, we briefly discuss thenumerical simulation of the limit combustion prob-lem. This is a joint work with Ekeoma Ijioma (Univ.of Meiji, Tokyo, Japan).

Mathematical Model for BrewingJapanese Sake with Stirring E↵ectand Its Analysis

Yusuke MuraseMeijo University, Japan

Our main theme is a solvability of mathematicalmodel for brewing process of Japanese Sake with stir-ring e↵ect. In this model, staring e↵ect is representedby advection terms. Our full model is formulated by15 advection-di↵usion equations and one constraintcondition, but we take up a simple model (corre-sponding to 1st brewing stage) made by 6 equationsand a constraint condition, in this talk. The solu-tion of this model is written in variational inequalityform. Moreover, the solution depends upon the solu-tion self in consequence of the constraint condition.It means that the model has a character which isso-called quasi-variational inequality.

On the Existence Theorem of Peri-odic Solution to One-DimensionalFree Boundary Problem for Ad-sorption Phenomena in a Two-ScaleModel for Concrete CarbonationProcess

Naoki SatoNagaoka National College of Technology, JapanToyohiko Aiki

Our main interest is to study the moisture transportof two-scale model for concrete carbonation processin a three dimensional domain from a non-liner an-alytic point of view. In this model we consider abounded macro domain occupied with concrete. Wesuppose that for any point of this domain one poreis corresponded and regards the pore as a micro do-

main interval decomposed to water domain and theair region. The humidity in the macro domain con-trols the boundary between the water domain andthe air region in the micro domain interval. We canexpress the change as a one-dimensional free bound-ary problem by law of conservation of mass and Fick’slaws. In the free boundary condition, the value of freeboundary depends on the value of the time derivativeof the free boundary. In this talk, we are going totalk about the existence theorem of periodic solutionto this free boundary problem. We show that fromSchauder’s fixed point theorem, there is at least onefixed point. It is a periodic solution to our problem.Moreover, we are going to introduce physical back-ground of our model in detail.

Lateral Self-Contacts of SlenderViscoelastic Rods

T SeidmanUMBC, USA

We consider the dynamics of a slender viscoelasticCosserat rod admitting strong enough geometric vari-ation to create lateral self-contact while prohibitingself-penetratration. This is developed from collabo-rations with S.Antman and K.Ho↵man.

Energy Dissipation in anAnisotropic Kobayashi–Warren–Carter Type Model of Grain Bound-ary Motion

Ken ShirakawaChiba University, JapanSalvador Moll, Hiroshi Watanabe

In this talk, a type of Kobayashi–Warren–Cartermodel of grain boundary motion is considered withthe anisotropic e↵ect in a polycrystal. The math-ematical model is formulated as a gradient systemof a governing free-energy, consisting of a parabolicvariational inequality, and a type of quasi-variationalinequality. Here, it is particularly worth noting thatthe quasi-variational situation enables to reproducenot only the shapes of stable structures of grains, butalso the dynamic changes of the structures workingwith the rotations of crystalline orientations.Recently, the mathematical validity of ouranisotropic model is verified as the existence resultof time-global solutions. On this basis, we set theobjective of this talk to discuss about the contin-uing topics, concerned with energy-dissipation andlarge-time behavior for the anisotropic model.


Di↵usion Phenomena for the WaveEquation with Space-DependentDamping in an Exterior Domain

Motohiro SobajimaTokyo University of Science, JapanYuta Wakasugi

In this talk we consider the initial-boundary valueproblem to the N -dimensional wave equation withspace-dependent damping in an exterior domain.The coe�cient of the damping term is radially sym-metric, strictly positive and behaves like |x|�↵ with↵ 2 [0, 1) in a neighborhood of spatial infinity. Thepurpose of this talk is to give weighted energy esti-mates and precise asymptotic behavior of solutionsby using corresponding semigroup generated by anelliptic operator with an unbounded di↵usion coe�-cient. This is a joint work with Dr. Yuta Wakasugi(Nagoya University).

Nonlinear Evolution Equations withSome Singular Potential Ralated tothe Hardy Inequality

Toshiyuki SuzukiKanagawa University, Japan

We consider the nonlinear evolution equations withinverse-square potentials a|x|�2:

@u@t

+ ei✓(Pau+ f(u)) = 0 in R⇥ RN ,

where i =p�1, N � 3, ✓ 2 [0,⇡ /2], Pa = �� +

a|x|�2 and a = �(N � 2)2/4. We see that �� andthe strongly singular potential a|x|�2 are the samescale symmetry and hence scaling argument can notbe applied to Pa. a = �(N � 2)2/4 is the thresh-old of the selfadjointness of Pa in the sense of form-sum in L2(RN ). Here we note that D((1 + Pa)

1/2)does not coincide with H1(RN ). In this talk we showthe global existence in the energy space and relatedtopics in a view of ✓ = ⇡/2 (nonlinear Schrodingerequations).

Existence of Solutions to Problemsfor Charged Particles in Plasmas

Yutaka TsuzukiTokyo University of Science, JapanAlexander Leonidovich Skubachevskii

We consider local existence of solutions to the Vlasov-Poisson system in a half space, which is a model ofcharged ions and elements in plasmas. Local exis-tence of solutions to this system has been establishedby A. L. Skubachevskii (2012), where the existencetime T is small even if some conditions for given dataare strict. In view of physics, the existence time Tshould not be small. This talk will give a result whichenables us to admit large T under some conditions.

Weak Solutions to a NonlinearParabolic-Psedoparabolic SystemModeling Gypsum Growth in Con-crete

Arthur VromansEindhoven University of Technology, NetherlandsAdrian Muntean, Fons van de Ven, Jan Zeman

In this talk, I will use mixture theory from classi-cal continuum mechanics to derive the structure of acoupled parabolic-pseudoparabolic PDEs system de-scribing the growth of a gypsum layer out of a cementmixture exposed to acid attack. The system is spe-cial in the sense that it captures, for each componentof the mixture, the simultaneous evolution of di↵u-sion, bulk and surface chemical reactions, mechanicaldisplacements, as well as acid flow velocity. Using avariant of Rothe’s method, we show the existence ofweak solutions to our system.This represents joint work with Adrian Muntean(Karlstad, Sweden), Fons van de Ven (Eindhoven,The Netherlands) and Jan Zeman (Prague, CzechRepublic).

Parabolic-Hyperbolic ConservationLaws with Variable and NonlocalType Coe�cients

Hiroshi WatanabeOita University, Japan

Parabolic-hyperbolic conservation laws are regardedas a linear combination of the time-dependent con-servation laws (quasilinear hyperbolic equations) andthe porous medium type equations (nonlinear de-generate parabolic equations). Thus, these equa-tions have both properties of hyperbolic equationsand those of parabolic equations and describe var-ious nonlinear convective di↵usion phenomena suchas filtration problems, Stefan problems and so on. Inthis talk we consider parabolic-hyperbolic conserva-tion laws with variable and nonlocal type coe�cients.In particular, we show the existence and uniquenessof generalized solutions to the equations.

Stability Criteria for NumericalSimulations of Allen–Cahn Equa-tion with Double-Obstacle Con-straint Via Yosida Approximation

Noriaki YamazakiKanagawa University, JapanTomoyuki Suzuki, Keisuke Takasao

We consider a two-dimensional Allen–Cahn equationwith double-obstacle constraint, numerically. Theconstraint is provided by the subdi↵erential of the in-dicator function on the closed interval, which is themultivalued function. Therefore, it is very hard toperform a numerical simulation of our problem. In


this talk we approximate the constraint by the Yosidaapproximation. Then, we give the stability criteriafor the standard forward Euler method to providethe stable numerical experiments of the approximat-ing equation.

An Embedding Estimate for theRepulsive Hamiltonian

Kentarou YoshiiTokyo University of Science, JapanMotohiro Sobajima

We consider the repulsive Hamiltonian H0

:= �� |x|2, which describes the quantum particle a↵ectedby a strong repulsive force from the origin. In thistalk, we prove an embedding estimate for the func-tions in the domain D(H

0

). For simplicity we focusour attentions to the one-dimensional case.

Nonlinear M-Accretive OperatorTheoretic Approach to Parabolic-Elliptic Keller-Segel Systems withNonlinear Di↵usion

Noriaki YoshinoTokyo University of Science, Japan

In this talk, we deal with parabolic-elliptic chemo-taxis systems with nonlinear di↵usion by usingthe theory for nonlinear m-accretive operator.Marinoschi (2013) established m-accretive theoreticapproach to the solvability of chemotaxis systemswith Lipschitz di↵usion. However the method didnot work when the di↵usion coe�cient is not Lips-chitz continuous, such as D(r) = rm (m > 1). Wedevelop the method and show existence of solutionsto the systems in the case of non Lipschitz di↵usion.


Special Session 79: Nonlinear Di↵erential Dynamic Systems in FluidDynamics

Zheng Ran, Shanghai University, Shanghai Institute of Applied Mathematics and Mechanics,Peoples Rep of China

Jiezhi Wu, Peking University, Peoples Rep of China

There are many challenges in fluid mechanics, including vortex dynamics, unsteady flow separation, turbu-lence and aeroacoustics, even a great progress have been achieved in the last thirty years due to the fastdevelopment of numerical method and parallel computing. The studies demonstrated that many phenomenain fluid mechanics can be described by di↵erential dynamic system and o↵er us better understanding. Themain objective of this special session is to bring together the experts who apply di↵erential dynamic systemto fluid mechanics to present their latest results and discuss future directions.

The Reynolds Number Dependanceof the Townsend-Perry Constants inTurbulent Boundary Layers

Bjorn BirnirUC Santa Barbara, USAXi Chen, Can Li, John Kaminsky, ShahabKarmini

The existence of (local) solutions to a stochasticNavier-Stokes equation, in Sobolev spaces corre-sponding to the Kolmogorov-Obukhov-She-Levequescaling, allows us to develop a theory to explain thesub-Gaussian behavior of the Townsend-Perry con-stants recently measured for high-order fluctuationmoments in turbulent boundary layers. It yields thegeneralized (Prandtl-von Karman) logarithmic lawfor the high-order moments. The Reynolds-Taylornumber depencence of the Townsend-Perry constantsis computed and compared with experimental resultsand simulations at di↵erent Reynolds-Taylor num-bers. The predicted constants are in good agreementwith experimental and simulation data.

On the Steady-State Nearly-Resonant Water Waves with Time-Independent Spectrum

Shijun LiaoShanghai Jiaotong University, Peoples Rep of ChinaDali Xu, M. Stiassnie

The steady-state nearly-resonant water waves aregained by means of the homotopy analysis method(HAM), a analytic approximation method for highlynonlinear equations. Our strategy is to mathemati-cally transfer the steady-state nearly resonant waveproblem into the steady-state exactly resonant ones.By means of choosing a generalized auxiliary lin-ear operator that is a little di↵erent from the linearpart of the original wave equations, the small divisor,which is unavoidable for nearly resonant waves in theframe of perturbation theory, is avoided, or movedfar away from low wave frequency to rather highwave frequency with physically negligible wave en-ergy. It is found that the steady-state nearly resonantwaves have nothing fundamentally di↵erent from thesteady-state exactly resonant ones, from physical andnumerical viewpoints.

Nature of Vortex Bifurcation andCascade in Isotropic Turbulence

Zheng RanShanghai University, Peoples Rep of China

The central problem of fully developed turbulenceis understanding the energy cascading process andmultiscale interaction. Update, there is no deductivetheory which leads to a full physical understandingor mathematical formulation. The definition, devel-opment, challenge and the corresponding status ofturbulence cascade were briefly reviewed. The limita-tion of present methods were emphasized. Based onthe Karman-Howarth equation in 3D incompressiblefluid, a new isotropic turbulence scale evolution equa-tion and its related theory progress, the existence ofnonlinear dynamic system measured by turbulenceTaylor microscale was proven. The present results in-dicate that the energy cascading process has remark-able similarities with the deterministic constructionrules of the logistic map. The cascade appears as aninfinite sequence of period-doubling vortex bifurca-tions.

Lie-Group Symmetry Analysis ofWall-Bounded Turbulent Flow

Zhen-Su ShePeking University, Peoples Rep of ChinaXi Chen, Fazle Hussain

Canonical wall-bounded turbulent flows includechannel, pipe and turbulent boundary layer (TBL),the latter may extend to realistic situations involv-ing e↵ects of pressure gradient, compressibility, buoy-ancy, etc. Searching for universal principle governingthe mean flow (velocity and fluctuation intensity) dis-tribution has been one of the central goals for turbu-lence studies over one hundred years since the pio-neer work of Prandtl in 1904. We report a recent at-tempt to formulate a Lie-group analysis of Reynoldsaveraged Navier-Stokes (RANS) equation in the pres-ence of an infinite wall, where an invariant dilationgroup in the direction normal to the wall is identifiedand a generalized dilation invariant solution is pos-tulated to describe a multi-layer structure which isgeneric to all wall-bounded flows. Evidence for thevalidity of this description will be reported, from de-tailed comparison with experimental and numerical


data, involving all canonical flows mentioned above.Furthermore, it will be shown that the physical andmathematical understanding of fluid turbulence de-rived from this symmetry-based approach helps toimprove engineering models for aerodynamic simula-tion of turbulent flows.

Convection Onset of Two-Dimensional Rayleigh-Benard Sys-tem with Non-Oberbeck-BoussinesqE↵ects

De-Jun SunUniversity of Science and Technology of China,Peoples Rep of ChinaShuang Liu, Shu-Ning Xia, Zhen-Hua Wan

Convection onset of two-dimensional Rayleigh-Benard system with perfectly conducting horizontalwalls and adiabatic sidewalls is studied by meansof direct numerical simulation of low-Mach-numberequations near a codimension-2 point where single-vortex and double-vortex modes become unstablesimultaneously. The Prandtl number is fixed at0.71. Particular attention is paid to the e↵ects ofbreaking of up-down symmetry due to non-Oberbeck-Boussinesq e↵ects on mode interaction of single-vortex and double vortex modes. Under Oberbeck-Boussinesq approximation bifurcations to single-vortex and double-vortex patterns are supercriticaland mode interaction is weak with various solu-tion branches uncorrelated. When non-Oberbeck-Boussinesq e↵ects are taken into account, bifurcationgiving rise to one double-vortex pattern becomes sub-critical. For strong non-Oberbeck-Boussinesq e↵ectsthe single-vortex solution branch is disrupted due tomode interaction. Various flow patterns are identifiedand their hydrodynamic characteristics are analyzed.Multiple-solution and hysteresis phenomena are ob-served. Bifurcation diagrams corresponding to vary-ing degrees of symmetry breaking are presented. Fur-thermore, convection in the weakly nonlinear regimescan be qualitatively described by amplitude equa-tions constructed by symmetry considerations.

Theoretical and Numerical Studieson Vorticity Dynamics of Two-Dimensional Flows

Xilin XieFudan University, Peoples Rep of ChinaQian Shi, Yu Chen

Two dimensional incompressible and compressibleflows on fixed and moving smooth surfaces have beenstudied in the point of view of vorticity dynam-ics. The unified vorticity and dilatation algorithmfor general two dimensional flows have be attainedthrough the decompositions of vorticity and dilata-tion. Based on the Levi-Civita connection operatorfor surface tensor fields, the deformation kinematicsof continuous mediums with surface configurationscan be presented in an intrinsic way. On the otherhand, the conservation laws can be obtained basedon the surface gradient operator and the generalizedintrinsic Stokes formula. Some primary conclusionsin classical vorticity dynamics have been extendedto the case of two dimensional flows, including thegoverning equations of vorticity and dilatation, La-grange theorem on vorticity and representation ofrate-of-strain tensor on solid boundaries. The newlydeveloped theories are characterized by the appear-ance of surface curvatures in some primary relationsand governing equations. Some numerical results oftwo-dimensional flows based on our theories will bepresented in order to reveal the intrinsic features oftwo-dimensional flows.

Multi-Vortices in Separation Sur-face

Shuhai ZhangChina Aerodynamics Research and DevelopmentCenter, Peoples Rep of China

In this talk, I will introduce our recent study on un-steady flow separation. Based on nonlinear dynamicanalysis and numerical simulation, we studied theflow structure of unsteady flow separation. The tra-jectories of fluid particles are computed based on theunsteady flow velocity to study Lagrangian structure.It is found that there are many vortices on the sepa-ration line on the body surface, which will developedto vortex tubes in the separation surfaces. Secondly,One or more limit cycle will appear on body surfaceat certain condition, which becomes a new type offlow separation. It will develop into a tornado-likestructure.


Special Session 80: Inverse Limits in Dynamical Systems

Krystyna Kuperberg, Auburn University, USAJudy Kennedy, Lamar University, USA

The principal topic of the session is the role of inverse limits and generalized inverse limits in discretedynamical system. We will consider topological group actions, laminations, attractors, rotation number,inverse limit of a branched coverings, McCord solenoids, and algebraic properties of generalized inverselimits. Talks on related research in continuous dynamics and complex dynamics may be included as well.

Homeomorphic Restrictions of Uni-modal Maps

Lori AlvinBradley University, USA

In this talk we provide a symbolic characterizationof a class of unimodal maps whose restriction to theomega-limit set of the turning point is a minimalhomeomorphism on a Cantor set. This characteri-zation is given in terms of the shift space generatedby the kneading sequence of the unimodal map. Iftime provides, we will discuss how these results canbe used to better understand the topological struc-ture of the associated inverse limit space.

Golden Bee Tilings from the Pointof View of Inverse Limits

Michael BarnsleyAustralian National University, AustraliaAndrew Vince

Recent results of the authors concerning dynamicsand inverse limits associated with certain tilings ofthe plane by will be presented.

New Homogeneous Continua andSemi-Conjugacies of Interval Maps

Jan BoronskiIT4 Innovations, Ostrava, Czech RepM. Smith

In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdor↵ homogeneous, hereditary equiva-lent and hereditary indecomposable continuum. Tak-ing advantage of a decomposition theorem of W.Lewis, he obtained it as a long inverse limit of met-ric pseudo-arcs with monotone bonding maps. Ex-tending his approach, and the results of Lewis onlifting homeomorphisms, we construct a nonmetricpseudo-circle, and new examples of homogeneous 1-dimensional continua: a circle and solenoids of non-metric pseudo-arcs. Among many corollaries we alsoobtain an analogue of another theorem of Lewis from1984: any interval map is semi-conjugate to a home-omorphism of the nonmetric pseudo-arc.

Fundamental Groups and General-ized Covering Spaces

James KeeslingUniversity of Florida, USAJerzy Dydak, Joanna Furno

Classical covering spaces have played an invaluablerole in the study of spaces that are locally connectedand locally simply connected. There is good reasonto generalize these spaces and considerable work hasbeen done in this direction. We contribute to this ef-fort in two di↵erent directions. We give examples ofwhat we believe are the ideal examples of generalizedcovering spaces. That is, these have almost all of theproperties one works with in covering spaces exceptthe locally simply connected property. The examplesthat we give have a topological group structure on⇡1

(X) rather than being discrete groups. The worstexamples for generalized covering spaces are thosespaces X that are arcwise connected and locally ar-cwise connected with non-trivial fundamental group,but which have no covering spaces. We develop toolsto analyze such spaces and show that such spaces canalso have a rich structure.

Horseshoes and Lambda-Dendroidsin Generalized Inverse Limits

Judy KennedyLamar University, USAVan Nall

Set-valued functions from an interval into the closedsubsets of an interval arise in various areas of sci-ence and mathematical modeling. When studyingthe dynamical properties of a set-valued function, theproblem is that if one iterates in the standard way,the orbit of a point is not well defined. We studyinstead the dynamical system of the shift map de-fined on the inverse of the set-valued function. Theinverse limit is the space of all possible inverse or-bits, and the shift map is a continuous function fromthis compact subset of the Hilbert cube onto itself.The inverse limit can be very complicated topologi-cally, and here again, as in so many other settings,complicated topology and complicated dynamics gohand-in-hand. We explore the relationship betweenthe topological structure of the inverse limit spaceand the dynamics of the shift map.


Vietoris Theorems for GeneralizedInverse Limits

Krystyna KuperbergAuburn University, USA

In 1927, L. Vietoris introduced homology groups forgeneral metric spaces and proved a his famous iso-morphism theorem. Since then, many formulationsof the theorem appeared in the literature. We willdiscuss Vietoris theorems for generalized inverse lim-its.

Inverse Limits of Forward InvariantLaminations of the Unit Disk

John MayerUA-Birmingham, USA

Laminations of the unit disk were introduced byWilliam Thurston as a topological/combinatorialmodel for understanding the (connected) Julia setsof polynomials. To each locally connected Julia setof a degree d polynomial there corresponds a lamina-tion with a dynamical “angle-d-tupling“ map �d on itsemiconjugate to the polynomial on its Julia set. Inmost cases, a finite amount of information (a forwardinvariant finite lamination) allows for construction by“pullback“ of the entire lamination corresponding tothe Julia set. The pullback lamination is a directlimit of a sequence of finite laminations in the unitdisk, and the limit lamination carries the semicon-jugate �d dynamics. In contrast, we investigate theinverse limit of the finite laminations in the pullbacksequence with bonding map the �d map, restricted tothe finite laminations, corresponding to the degree dpolynomial on its Julia set. Each point of the inverselimit corresponds to an inverse orbit of the dynamics.We seek to determine how much information aboutorbits of the polynomial dynamics on the Julia setcan be thus obtained. We consider polynomial Juliasets of degree d = 2 and d = 3.

Inverse Limits and Shadowing

Jonathan MeddaughUniversity of Birmingham, England

Inverse limits are excellent models for the structure ofthe orbit space for certain dynamical systems. In thistalk we will discuss some connections between inverselimit spaces with various topologies and shadowingproperties of dynamical systems.

Euclidean Fractals

Magdalena NowakJan Kochanowski University in Kielce, PolandTaras Banakh

An Iterated Function System (IFS) on the metricspace X is a finite family F of contractive self-mapson X. Consider the space H(X) of nonempty, com-pact subsets of X. We can define a dynamical system(H(X),F) such that F(K) =

Sf2F f(K) for each

K 2 H(X).By the attractor of Iterated Function System (IFS-attractor) we understand a nonempty compact setA ⇢ X such that

A = F(A) =[

f2F

f(A)

and for every compact set K 2 H(X) the sequence�Fn(K)

�1n=1

converges to A with respect to theHausdor↵ metric on H(X).We deal with the question of which compact metriz-able spaces are homeomorphic to attractors of Iter-ated Function Systems in the Euclidean space. Wecalled such spaces Euclidean fractals. We extend theresult obtained by Duvall and Husch, and proved thatgiven any compact metrizable finite-dimensional dou-bling spaceX that contains an open zero-dimensionaluncountable subset, we can topologically transformthe space X such that the image is the attractor ofsome IFS in Rn. The proof bases on the representa-tion of the spaceX as the inverse limit of some fractalstructure and on the morphisms of the normed col-ored trees.

On Almost Specification and En-tropy

Piotr OprochaAGH University, Poland

Almost specification property is a generalization ofBowens specification property, which is present in awider class of systems. For example, all beta-shiftshave this property while it is know that some of themdo not satisfy specification property. In this talk wewill present some relations between almost specifica-tion property and topological entropy, and its conse-quences to other notions from topological dynamics.

Invariant Measures for Set-ValuedMaps

Brian RainesBaylor, USAJonathan Meddaugh, Tim Tennant

We consider the problem of constructing invariantmeasures for multi-valued dynamical systems, in par-ticular we are interested in non-atomic invariant mea-sures with full support. We examine multi-valued dy-namical systems, (X,F ), with the specification prop-erty. The associated inverse limit space and for-


ward orbit space are useful for studying multi-valueddynamical systems because they have an associatedshift map that is a continuous function rather thanan upper-semi-continuous relation. These associatedspaces also have the specification property, and, thus,on these spaces the set of invariant measures con-tains a denseG�-set of non-atomic invariant measureswith full support. Unfortunately these measures donot always induce a measure on the original systemthat is non-atomic. One of the main obstructions isthe presence of multiperiodic points for the originalmulti-valued dynamical system.

Quantization for Probability Distri-butions

Mrinal RoychowdhuryUniversity of Texas Rio Grande Valley, USA

Quantization for probability distributions refers tothe idea of estimating a given probability by a dis-crete probability with finite support. Recently, sev-eral works have been done in this direction. I willtalk about it.

About the Cohomology of Attrac-tors

Francisco Ruiz del PortalUniversidad Complutense de Madrid, SpainJaime J. Sanchez Gabites

Suppose K is an asymptotically stable attractor fora dynamical system on Rm. In the continuous case(a flow) it is well known that the inclusion of K inits basin of attraction A(K) induces isomorphismsin Cech cohomology. We discuss whether the sameholds true in the discrete case (a homeomorphism).We show that (i) it is true if coe�cients are takenin Q or Zp (p prime) and (ii) it is true for integralcohomology if and only if the Cech cohomology of Kor A(K)are finitely generated.

Three Classic Theorems

David RydenBaylor University, USA

In this talk we will consider three properties from theclassical theory of inverse limits that do not hold forinverse limits with upper semicontinuous set-valuedbonding maps, and we provide necessary and su�-cient conditions for them to hold in the more gen-eral setting. The properties in question are (1) theclosed set theorem, (2) the full projection property(and weak full projection property), and (3) a ver-sion of the subsequence theorem.

Some Constraints on the Topologyof Isolated Invariant Sets and HowThey Sit in Phase Space

Jaime J. Sanchez-GabitesUniversidad Autonoma de Madrid, Spain

Let K ✓ R3 be an isolated invariant set (in the senseof Conley) for a flow. Suppose K has finitely gen-erated Cech cohomology with Z

2

–coe�cients. Thenit is possible to construct isolating neighbourhoodsN of K that are compact 3–manifolds and such thatthe inclusion K ✓ N is a shape equivalence. Usingthis and other related results we shall obtain someconstraints on the topology of K and how it sits inR3. As an application, we generalize a result by E.S. Thomas (One-dimensional minimal sets. Topol-ogy, 12:233-242, 1973) concerning isolated general-ized solenoids.

Shape, Bifurcations and Splittingsof Some Isolated Invariant Sets

Jose SanjurjoUniversidad Complutense de Madrid, SpainHector Barge

We study the dynamics and topology of the isolatedinvariant sets created by bifurcations consisting inimplosions of the region of attraction of asymptoti-cally stable equilibria of flows. The topological prop-erties are formulated in terms of Borsuk’s shape the-ory. We also study attractor-repeller splittings ofnon-saddle sets.

Decompositions of Non-Metric Con-tinua

Michel SmithAuburn University, USA

Hernandez-Gutierrez (2014) showed that there isa generalized Whitney map from the non-metricpseudo-arc M of Smith (1985) onto the long arc. Us-ing this map a continuous decomposition of M maybe constructed. We show that any continuous decom-position of M into non-degenerate continua is a met-ric pseudo-arc. A technique is presented using inverselimits for producing a class of non-metric continua sothat if X is one of them and G is a continuous de-composition of X into non-degenerate subcontinua,then the decomposition space X/G is metric. Uppersemi-continuous decompositions are also considered.


Chaotic Dynamics in the TwistedHorseshoe Map Via a TopologicalApproach

Elisa SovranoUniversity of Udine, Italy

In recent years, several di↵erent approaches havebeen proposed in order to extend, in a topologicaldirection, the classical geometry of the Smale horse-shoe. These generalizations have been motivated bythe investigation of some new problems in chaotic dy-namics. The concept of topological horseshoe, devel-oped in the works of Burns and Weiss, Kennedy andYorke, and other authors, lead to a powerful theoryuseful to detect chaos in dynamical systems. In thisframework, we show, in an analytical way, the pres-ence of complex behaviors for a discrete dynamicalsystem arising form the study of the twisted horse-shoe map considered by Guckenheimer, Oster andIpaktchi in 1977.

When the Topology of the MapDetermines the Dynamics

James YorkeUniversity of Maryland, USAS. Das, Y. Saiki

Let F : T d ! T d where T d is a d-dimensional torus.The mapping always can be written as linear termMx, where M is a d⇥ d matrix with integer entries,plus a bounded termG that is periodic with period 1in each variable, so that F (x) = Mx + G(x) mod 1in each variable. We report on how M can determinemuch of the dynamics even when G is very large.We can think of M as determining the topology ofthe map since it is the homology matrix. It can alsodetermine some of the dynamics.


Special Session 81: Advances in Computer Assisted Proofs forDynamical Systems and Di↵erential Equations

Gianni Arioli, Politecnico di Milano, ItalyJason Mireles-James, Florida Atlantic University, USA

The last 20 years have seen a significant development of techniques for computer assisted proofs in dynam-ical systems theory. Such techniques are used to prove the existence of solutions of ordinary and partialdi↵erential equations, and to study invariant sets in a mathematically rigorous way using the computer.These techniques use two main fundamental approaches: one is based on topological tools, the other onfunctional analysis. The first approach, when applied to di↵erential equations, relies on the developmentof e�cient rigorous integrators and studies how sets in phase space map across one another. The secondapproach relies on transforming the equations into fixed points problems in some functional space. Theseapproaches are somehow complementary, and the aim of this session is to discuss recent advances in orderto learn how to exploit the best features of both.

Continuation of Connecting Orbitsfor Maps

Ronald AdamsEmbry Riddle Aeronautical University, USAJason Mireless James

I will discuss a validated continuation method forconnecting orbits between fixed points for finite di-mensional maps. The argument exploits high orderparameterizations of one parameter families of localstable/unstable manifolds. The argument can alsobe modified to compute saddle node bifurcations forconnection orbits (i.e. prove the existence of col-lisions of the stable/unstable manifolds giving riseto connections). I will illustrate the utility of thismethod by applying it to a one parameter family ofHenon maps which contains the classic parameters.

On a Nonlinear Nonlocal Hyper-bolic System Modeling SuspensionBridges

Gianni ArioliPolitecnico di Milano, ItalyFilippo Gazzola

This talk does not concern (yet) computer assistedproofs; instead, it presents some open problems thatwe hope to be able to address with computer as-sistance. We suggest a new nonlinear model for asuspension bridge and we perform numerical experi-ments with the parameters corresponding to the col-lapsed Tacoma Narrows Bridge. We show that thethresholds of instability are in line with those ob-served the day of the collapse. Our analysis enablesus to give a new explanation for the torsional insta-bility, only based on the nonlinear behavior of thestructure.

Recent Results Verifying Stabilityof Traveling Waves with ComputerAssisted Proof

Blake BarkerBrown University, USABjorn Sandstede, Kevin Zumbrun

We describe recent results in which we rigorously ver-ify spectral stability of travelling waves, a su�cientcondition to imply non linear stability in the systemsexamined. We outline our general approach and thesolutions we used to overcome such challenges as therapping e↵ect and computational cost of the study.In particular, we approach these studies with an eyetoward a general strategy for rigorously verifying sta-bility of travelling waves.

E�cient Methods for Long TermVerified Integration in HyperbolicSystems

Martin BerzMichigan State University, USAKyoko Makino

Hyperbolic systems, including those of the non-uniform kind, represent one of the most importantclass of problems encountered in the study of chaoticdynamical systems. For rigorous self-verified study,the practically appearing elongation in one of the di-rections typically represents a formidable challenge,since solution sets even under the best of circum-stances will necessarily elongate in a similar way. Wepresent a Taylor model based method to overcomethis problem for the computation of orbits in au-tonomous hyperbolic systems. As an example, theapproach allows the rigorous self-verified computa-tion of stable and unstable manifolds of systems for inprinciple often unlimited times, and in practice for farlonger times than conventional verified orbit integra-tion is possible. We show practical results and theiruse for some typical systems including the Henon andLorenz families, and compare the resulting possibleintegration lengths with those of other state of theart methods.


Computer Proof of HeteroclinicConnection in 1D Diblock Copoly-mers PDE Model

Jacek CyrankaRutgers University, USA

We present a computer proof of a heteroclinic connec-tion in the one dimensional diblock copolymers PDEmodel (Cahn-Hillard type) with periodic boundaryconditions

ut = ��("2�u+ f(u))� �(u� µ) in ⌦.

For some (�," )’s the equation exhibit rich attractorstructure. We establish existence of a heteroclinicconnection between the homogeneous state u ⌘ 0representing a perfect mixture of copolymers and alocal energy minimizers. This is a nongeneric phe-nomenon in Cahn-Hillard PDE models. It has an in-triguing physical interpretation – given a perfect mix-ture of copolymers, shake it a bit, and the stable stateresulting from the time evolution of this model can-not be anymore predicted. Due to the complicatedstructure of the global attractor the system exhibitseveral stable states, and which one is the result oftime evolution of an initial condition close to homo-geneous is highly dependent on the initial condition.The proof is conceptually simple, first we calculatea interval bounds contained in a fixed point’s basinof attraction, and a piece of unstable manifold of thehomogeneous state, and then we integrate in timethe bounds for unstable manifold. If the bounds aremapped into the basin of attraction our procedure issuccessful in proving the existence of the particularheteroclinic connection. We combine several tech-niques from some of the authors‘ and P. Zgliczynskiworks. We obtain a bounded region within the basinof attraction by using a logarithmic norm calculation,an explicit bound for an unstable manifold by usingso-called cone conditions. We propagate in time aninterval box containing a piece of unstable manifoldby using our e�cient algorithm for rigorous integra-tion of PDEs forward in time, which is able to han-dle large integration times within a reasonable time.Moreover, we present how to use the PDE scaling in-variance to extend those results for arbitrary large "parameter values.

Explicit Error Estimates for a HopfBifurcation in Wright’s Equation

Jonathan JaquetteRutgers University, USAJan Bouwe van den Berg

Wright’s equation y‘(t) = �↵y(t� 1){1+ y(t)} has asupercritical Hopf bifurcation at ↵ = ⇡/2, and a long-standing conjecture claims that there are no periodicorbits when ↵ < ⇡ /2 and there is an unique slowly os-cillating periodic orbit when ↵ > ⇡ /2. Recent workshave used computer assisted proofs to make progresson this conjecture, however these techniques breakdown near the critical value ↵ = ⇡/2.

In this talk, we use a Newton-Kantorovich type ar-gument to develop explicit error estimates on thebranch of periodic orbits described by the Hopf bifur-cation. This bifurcation analysis produces a funda-mentally local result; we can prove the uniqueness ofperiodic solutions with small amplitude when ↵�⇡/2is small and the frequency is close to ⇡/2. Neverthe-less, for a small neighborhood about ↵ = ⇡/2, we areable to extend our bifurcation analysis into globalresult by way of computer assisted proofs.

Set-Based Methods for Global Dy-namics

William KaliesFlorida Atlantic University, USASarah Day, Vincent Naudot, Marcus Fontaine

In this talk we will give a brief overview of set-basedmethods for computing global dynamics using Conleytheory and the current state of algorithms and soft-ware. We will illustrate the techniques with some ex-amples. First we obtain computable, rigorous coverof the global invariant set and an outer approxima-tion of the dynamics for a class of integro-di↵erenceequations with smooth nonlinearities that are well-approximated by polynomials. The dynamics of theKot-Scha↵er map with Ricker nonlinearity is used toillustrate the computational method. Next we com-pute the dynamics of an attractor which includes a“re-injected” cuspidal horseshoe, as well as other ex-amples as time permits.

Periodic Solutions for SomeHyperbolic-Type PDEs

Hans KochThe University of Texas at Austin, USAGianni Arioli

We consider the wave equation and the beam equa-tion, with cubic nonlinearity, on the unit interval.These are infinite dimensional Hamiltonian systems,and there are small denominator issues for peri-odic orbits with irrational frequencies. Nevertheless,solutions for rational frequencies appear to lie on“branches“, and these branches appear to undergobifurcations. We describe some numerical observa-tions and some rigorous results. The latter includethe existence of non-small periodic solutions for thebeam equation for positive-measure sets of frequen-cies.


Computer-Assisted Proofs for LongPeriodic Orbits of Parabolic PDEsand Their Stability

Jean-Philippe LessardLaval University, CanadaJordi-Lluis Figueras, Marcio Gameiro, Rafaelde la Llave

In this talk, we introduce a computer-assisted tech-nique for the analysis of periodic orbits of parabolicPDEs. The idea is to use a Newton-Kantorovich typeargument to obtain rigorous proofs of existence of theperiodic orbits in a weighted ell-infinity Banach spaceof space-time Fourier coe�cients with algebraic de-cay. As an application, our proposed method is ap-plied to prove existence of several periodic orbits inthe Kuramoto-Sivashinsky PDE, which is a popularmodel to study spatiotemporal chaos. We prove theexistence of some periodic orbits with large period.We also present rigorous results about the stability ofthe orbits by solving an associated eigenvalue prob-lem.

Verified Propagation of Large Setsof Initial Conditions and ParameterRanges and Applications for Proofsof chaoticity

Kyoko MakinoMichigan State University, USAMartin Berz

Rigorous high-order methods including the Taylormodel approach allow the propagation of initial con-ditions with an error that scales as a high power of thediameter. This allows for propagation of large rangesof domains and parameter ranges by virtue of auto-matic domain decomposition tools, not unlike meth-ods for mesh generation in non-verified PDE solvers.Using these methods, it is often possible to propagatevery large domains in an economical manner. Weshow various applications of the method, includingproofs of chaoticity of the well-known Lorenz systemfor large ranges of parameters, as well as applica-tions for the commonly used method of discretizingthe motion into a finite but large graph and studyingglobal dynamics based on that representation.

Rigorous Numerics for Fast-SlowSystems with an Explicit Multi-Scale Parameter Range Via theCovering-Exchange

Kaname MatsueThe Institute of Statistical Mathematics, Japan

We provide a rigorous numerical computationmethod to validate periodic, homoclinic and hete-roclinic orbits as the continuation of singular limitorbits for the fast-slow system x = f(x, y,✏ ), y =✏g(x, y,✏ ). Our validation procedure is based ontopological tools called isolating blocks, cone con-

dition and covering relations. Such tools provideus with existence theorems of global orbits whichshadow singular orbits in terms of a new concept, thecovering-exchange. The covering-exchange consistsof three parts; slow shadowing, drop and jump, whichgive us not only generalized topological verificationtheorems, but also easy implementations for validat-ing trajectories near slow manifolds in a wide range,via rigorous numerics. Our procedure is available tovalidate global orbits not only for su�ciently small✏ > 0 but all ✏ in a given half-open interval (0, ✏

0

].Several sample verification examples are shown as ademonstration of applicability.

Parameterization Method for In-variant Manifolds: E�cient Numer-ics and Validated Computatoin

Jason Mireles JamesFlorida Atlantic University, USA

I will discuss some aspects of the parameteriza-tion method for invariant manifolds, focusing onnumerical computations with mathematically rigor-ous a-posteriori error bounds. The parameterizationmethod is a functional analytic framework for study-ing invariant manifolds such as invariant circles andtori, and also stable/unstable manifolds for equilib-ria and periodic orbits. One feature of the methodis that it provides a natural notion of a-posteriorierror which can be exploited in computer assistedproofs. Parameterized manifolds are also useful forstudying connecting orbits in dynamical systems andI will mention some applications.

Validated Computation of UnstableManifolds for Parabolic PDEs

Christian ReinhardtVU Amsterdam, NetherlandsJason Mireles James

This talk presents a method for computing polyno-mial approximations together with a-posteriori errorbounds of unstable manifolds associated with equilib-rium solutions to parabolic PDEs posed on compactdomains with suitable boundary conditions. Thesepolynomials have a finite number of variables, eventhough they map into an infinite dimensional statespace. The approach is based on the parametrizationmethod and builds on explicit knowledge of the spec-tral stability data at the equilibrium solution that wealso obtain via validated numerical methods. We im-plement the method numerically, and develop explicita-posteriori error bounds using the notion of radiipolynomials. By combining the a-posteriori error es-timates with careful management of floating pointround-o↵ errors we obtain mathematically rigorousbounds on the truncation and discretization errorsassociated with our polynomial approximation. Oneof the motivations for this method is the validatedcomputation of connecting orbits for PDEs. We illus-trate the method with applications to Fisher’s equa-tion. This is joint work with Jason Mireles-James.


Validation of the Bifurcation Di-agram in the 2D Ohta-KawasakiProblem

Jan Bouwe van den BergVU Amsterdam, NetherlandsJF Williams

In this talk we discuss a rigorous numerical method tocompare local minimizers of the Ohta-Kawasaki func-tional in two dimensions. In particular, we validatethe phase diagram identifying regions of parameter

space where rolls are favorable, where hexagonallypacked spots have lowest energy and finally wherethe constant mixed state does. More generally, wepresent a computational method to rigorously deter-mine such features in problems where optimal do-main sizes are not known a priori. In terms of thepractical realities of applying the computer assistedtheorems ideas, this work represents a step forwardpast clean-cut test problems to more elaborate vari-ational problems in pattern formation.


Special Session 82: Numerical Simulations and Computations forStochastic Dynamics

Yao Li, University of Massachusetts Amherst, USAMolei Tao, Georgia Institute of Technology, USA

Numerical methods for stochastic dynamics have a wide range of applications in many disciplines, such aschemical physics, statistical mechanics, biological sciences, and fluid dynamics. In this special session, wefocus on recent developments of numerical simulations and computations for stochastic dynamics arisingfrom various applications. Problems to be addressed include both improvements of algorithms and novelapplications to other scientific fields.

Markov Chain ApproximationMethods for the Numerical So-lution of Stochastic Di↵erentialEquations

Nawaf Bou-RabeeRutgers University – Camden, USA

Stochastic di↵erential equations (SDEs) model ran-dom fluctuations in applications as diverse as: molec-ular dynamics, mathematical finance, population dy-namics, epidemiology, laser dynamics and atmo-sphere/ocean sciences. For the most part, theseSDEs cannot be solved exactly and numerical meth-ods are used to approximate their solution. Themain goal of these approximations is to estimatestatistics associated to the SDE solution like meanfirst passage times, exit probabilities, and multi-time expectations of observables. This talk demon-strates that the go-to method for numerically solv-ing SDEs – Euler-Maruyama – is impractical to usefor: SDEs with numerical sti↵ness, long-time sim-ulation of ergodic SDEs, SDEs with unattainableboundaries, SDEs with internal discontinuities, SDEswith boundary conditions (e.g. Dirichlet, Neumann,and oblique derivative boundary conditions), andSDEs with interface conditions (for these SDEs exis-tence/uniqueness of weak solutions are active areasof research). These issues motivate a shift in perspec-tive that naturally leads to Markov Chain Approxi-mation Methods. We will show how to use numericalPDE methods to construct these types of approxi-mations, and stochastic methods to simulate/analyzethem.

Analysis of Long-Time Behaviorof Large Continuous-Time MarkovChains with Exponentially SmallTransition rates

Maria CameronUniversity of Maryland, USATingyue Gan

Large stochastic networks (continuous-time Markovchains) with exponentially small transition rates arisein modeling of complex physical, chemical, and bio-logical systems. Time-reversible networks of this kindrepresent, e.g., energy landscapes of atomic or molec-ular clusters, while time-irreversible ones can modelthe dynamics of e.g. biological cell cycle or walks ofmolecular motors. I will present a novel algorithm for

the analysis of long-time behavior of such networksconsisting of two steps. It is a greedy graph algorithmthat builds the hierarchies of optimal W-graphs andFreidlin’s cycles. The output of this algorithm al-lows us to find asymptotic approximations for eigen-values, extract Freidlin’s cycles, identify metastableand quasi-invariant subsets of states, and o↵er severalways to cluster the network. The proposed method-ology will be illustrated on examples coming fromnatural sciences.

Uncertainty Quantification forStochastic Systems with Mem-ory

Eric HallUniversity of Massachusetts Amherst, USAMarkos Katsoulakis, Luc Rey-Bellet

We give a minimal variance method for estimatingparametric sensitivities of observables of stochasticsystems with memory.

Numerical Dynamics of RandomOrdinary Di↵erential Equations

Xiaoying HanAuburn University, USAPeter E. Kloeden

Numerical dynamics is concerned with the relation-ship between the dynamical behaviour of the so-lutions of a random ordinary di↵erential equation(RODE) and that of the solutions of a numeri-cal scheme. Two major issues, the preservation ofan attractor and a hyperbolic neighborhood underdiscretization will be considered in the context ofRODEs and the random dynamical systems.

Capturing Rare Events with theHeterogeneous Multiscale Method

David KellyNew York University, USAEric Vanden-Eijnden

We discuss heterogeneous multiscale methods(HMM) and their ability to capture fluctuationsacting on the slow variables in fast-slow systems. Inparticular, it is shown via analysis of central limittheorems (CLT) and large deviation principles (LDP)that the standard version of HMM artificially am-


plifies these fluctuations. A simple modification ofHMM, termed parallel HMM, is introduced and isshown to remedy this problem, capturing fluctua-tions correctly both at the level of the CLT and theLDP.

Coarse-Graining of Stochastic Dy-namics

Frederic LegollEcole des Ponts, France

The question of coarse-graining is ubiquitous in manyapplied sciences, including molecular dynamics. Inthis work, we are interested in deriving e↵ective prop-erties for the dynamics of a coarse-grained variable⇠(X), where X describes the configuration of the sys-tem, and ⇠ is a smooth scalar function. Typically, Xis a high-dimensional variable (representing for in-stance the positions of all the particles in the sys-tem), whereas ⇠(X) is a coarse-grained information(e.g. a particular angle between some atoms of themolecule).We assume that the configuration Xt of the com-plete system evolves according to the overdampedLangevin stochastic di↵erential equation, and we pro-pose an e↵ective closed dynamics that approximates(under time-scale separation assumptions) the evolu-tion of ⇠(Xt). Such an e↵ective dynamics may be use-ful to compute more e�ciently e.g. transition ratesfrom one configuration of the system to another one.Several estimations of the accuracy of the e↵ectivedynamics will be given, using various mathematicaltools.Based on joint works with T. Lelievre and S. Olla.

A Fast Exact Simulation Methodfor Markov Jump Processes

Yao LiUniversity of Massachusetts Amherst, USA

In this talk I will present a new method for the simu-lation of Markov jump processes, called the Hashing-Leaping method (HLM). As a novel method of thestochastic simulation algorithm(SSA), the HLM hasconditional constant computational cost per event,which is independent of the number of exponentialclocks in the Markov process. In particular, it is com-patible with parallel machines in many ways. In thetalk, I will introduce the method, demonstrate its im-plementation and parameter tuning, and discuss itsparallelization.

A New Model for Realistic RandomPerturbations of Stochastic Oscilla-tors

Wuchen LiGeorgia Institute of Technology, USA

Classical theories predict that solutions of di↵eren-tial equations will leave any neighborhood of a stablelimit cycle, if white noise is added to the system. Inreality, many engineering systems modeled by second

order di↵erential equations, like the van der Pol os-cillator, show incredible robustness against noise per-turbations, and the perturbed trajectories remain inthe neighborhood of a stable limit cycle for all timesof practical interest. In this talk, we propose a newmodel of noise to bridge this apparent discrepancybetween theory and practice. Restricting to pertur-bations from within this new class of noise, we con-sider stochastic perturbations of second order di↵er-ential systems that -in the unperturbed case- admitasymptotically stable limit cycles. We show that theperturbed solutions are globally bounded and remainin a tubular neighborhood of the underlying deter-ministic periodic orbit. This is a joint work withProf Luca Dieci and Prof Haomin Zhou in Georgiatech.

An Analysis of Implicit Sampling inthe Small-Noise Limit

Kevin LinUniversity of Arizona, USAJonathan Goodman, Matthias Morzfeld

Weighted direct samplers, also known as importancesamplers, are Monte Carlo algorithms for generat-ing independent, weighted samples from a given tar-get probability distribution. Such algorithms havea variety of applications in, e.g., data assimilationand state estimation problems involving stochasticand chaotic dynamics. One challenge in designingand implementing weighted samplers is to ensure thevariance of the weights, and that of the resulting es-timator, are well-behaved. In recent work, Chorin,Tu, Morzfeld, and coworkers have introduced a classof novel weighted samplers called implcit samplers,which have been shown to possess a number of de-sirable properties. In this talk, I will report on ananalysis of the variance of implicit samplers in thesmall-noise limit, and describe a simple method sug-gested by the analysis for obtaining higher-order im-plicit samplers. The algorithms are compared on con-crete test problems.

Analysis and Simulation of Ntracel-lular Bio-Chemical Reacting Net-works with Multiple Time Scales

Di LiuMichigan State University, USA

Intracellular reacting networks involving gene regu-lation often exhibits multiscale properties. That in-cludes multiple reacting rates, multiple populationmagnitudes and multi-stability. Direct StochasticSimulation Algorithm (SSA) would turn out to beine�cient dealing with such systems. Schemes suchas Nested SSA and Tau-leaping method have provedto be e↵ective for certain asymptotic regimes. I willpresent recent results on the convergence analysis andapplications of the algorithms.


Path-Space Information Metricsand Coarse-Graining of Non-Equilibrium Stochastic Systems

Petr PlechacUniversity of Delaware, USA

We discuss information-theoretic tools for obtainingoptimized coarse-grained molecular models for bothequilibrium and non-equilibrium molecular dynam-ics. The presented approach compares microscopicbehavior of molecular systems to parametric or non-parametric coarse-grained systems using the relativeentropy between distributions on the path space. Itallows us to formulate a corresponding path spacevariational inference problem. The methods becomeentirely data-driven when the microscopic dynamicsare replaced with corresponding correlated data inthe form of time series.

Some Numerical Methods for Hy-perbolic Periodic Orbits and RareEvents in Nongradient Systems

Molei TaoGeorgia Tech, USA

We consider di↵erential equations perturbed by smallnoises. The goal is to quantify what noises can do andpossibly also utilize them. Noise-induced transitionsare understood by optimizing probabilities character-ized by Freidlin-Wentzell large deviation theory. Ingradient systems, metastable transitions were knownto cross separatrices at saddle points. We investigatenongradient systems, and show a very di↵erent typeof transitions that cross hyperbolic periodic orbits in-stead. Numerical tools for both identifying such pe-riodic orbits and computing transition paths are de-scribed. If time permits, I will also discuss how theseresults may help design control strategies. Joint workwith Eric Vanden-Eijnden.


Special Session 83: Dynamical Systems and their Applications

Maria Correia, University of Evora, PortugalScott Cook, Tarleton State University, USA

This special session will cover a variety of applications of dynamical systems, with special focus on billiardsystems and application to statistical mechanics. However, we invite talks from all areas of dynamicalsystems. The aim of this session is to bring together researchers who work in dynamical systems to exchangeand discuss recent developments in this field. Young researchers working in dynamical systems are especiallywelcome to present their recent results.

Entropy and Its Variational Princi-ple for Locally Compact MetrizableSystems

Andre CaldasUniversity of Brasilia (UnB), BrazilMauro Patrao

For a given topological dynamical system (X,T ) overa compact setX with a metric d, the variational prin-ciple states that

supµ

hµ(T ) = h(T ) = hd(T ),

where hµ(T ) is the Kolmogorov-Sinai entropy, withthe supremum taken over every T -invariant probabil-ity measure, hd(T ) is the Bowen entropy, and h(T ) isthe topological entropy as defined by Adler, Konheimand McAndrew.In the present work, we extend the definition of topo-logical entropy h(T ) for the non compact case in auseful manner. Instead of arbitrary open covers, werestrict our attention to what we have called ”admis-sible covers”. Then, we extend the above result toinclude any continuous map T : X ! X, when Xis locally compact separable and metrizable. In thiscase, the variational principle reads

supµ

hµ(T ) = h(T ) = mind

hd(T ),

where the minimum is taken over every distance com-patible with the topology of X, and is attained whend comes from the one-point compactification of X.

The Periodic Lorentz Gas withRandom Billiard Microstructure

Timothy ChumleyIowa State University, USA

The periodic Lorentz gas model of mathematicalphysics consists of a point particle moving in spaceand colliding with a periodic arrangement of fixedrigid scatterers. In this talk we introduce a variationon the periodic, finite horizon Lorentz gas where col-lisions are not modeled by the usual law of specularreflection, but rather by a novel class of Markov oper-ators P derived to model a microscopic structure onthe surface of the scatterers. The resulting Markovchain on the state space of positions and velocities,which we call the periodic Lorentz gas with randombilliard microstructure, is shown to be geometrically

ergodic with a rate that depends on parameters spec-ified by the microscopic structure. We also show thata similar result holds in the presence of a weak ho-mogeneous electric field, and prove that the resultingsteady state velocity of the particle is proportionalto the strength of the field in agreement with theclassical Ohm’s law. Our proof of geometric ergod-icity requires the construction of a Lyapunov func-tion, which we accomplish by approximating P bya di↵erential operator whose spectral theory is wellunderstood.

Heat Flow, Entropy Production,and Thermophoresis in RandomBilliard Systems

Scott CookTexas A&M - Tarleton State University, USARenato Feres

We discuss a class of random billiard systems withsmall scale structure as a framework to study thermo-dynamics in probabilistic, microscopic models. Wefirst introduce a notion of temperature via a proba-bilistic reflection law. This allows for heat reservoirsalong the boundary and the study of thermodynami-cal properties like heat flow and entropy production.Though we begin with a simple one dimensional, sin-gle particle system, we will focus on three dimen-sional systems with many interacting particles. Mo-tivated by the study of thermophoresis in aerosols, wedistinguish one large Brownian particle and study itsdrift velocity under various conditions. If time per-mits, we will discuss techniques for simulating suchsystems using parallel algorithms on a GPU.

Elliptic Periodic Orbits and Ergod-icity of Moon Billiards

Maria CorreiaUniversity of Massachusetts Amherst, USAHongkun Zhang

We construct a two-parameter family of moon-shaped billiard tables with boundary made of twocircular arcs. We analytically study the stability ofsome periodic orbits and prove there is a class ofbilliards in this family with elliptic periodic orbits.These moon billiards can be viewed as generalizationof annular billiards which all have KAM islands. Wealso numerically observe a subclass of moon-shapedbilliards with a single ergodic component.


The Dynamics of No-Slip Billiards

Chris CoxWashington University in St. Louis, USA

No-slip billiards are based on a collision model inwhich small rotating disks may exchange linear andangular momentum at collisions with the boundary,unlike the standard billiard model. We present re-sults on periodicity and boundedness of orbits whichdemonstrate the marked di↵erence in the dynamicsfrom those of standard billiards. Computer generatedphase portraits demonstrate non-ergodic features,suggesting chaotic no-slip billiards cannot readily beconstructed using the common techniques for gener-ating chaos in standard billiards.

Parametric Study of Novel Modelof Friction

Krzysztof JankowskiLodz University of Technology, PolandAndrzej Stefanski

Friction is one of the most essential factors influenc-ing dynamical response of engineering system. Thismechanical phenomenon shows strongly non-linearand complex behavior, taking into consideration in-teractions on the surfaces and in their strict vicin-ity. Due to this, friction stands a vital problem fortribologists and engineers, as it leads to the energyloss, wear and deterioration of materials. Much hasbeen done to reduce and overcome its detrimental ef-fects, yet to date no generic solution has been found.The first step, leading to deeper understanding offriction, is modeling. In this paper, we introduce anovel dynamic friction model and perform a detailedparametric study to analyze the influence of frictionparameters on the system responses and friction forcerelations.

General KAM Theorems and TheirApplications to Invariant Tori withPrescribed Frequencies

Xu JunxiangSoutheast University, Peoples Rep of ChinaXuezhu Lu

In this paper we develop some new KAM techniqueto prove two general KAM theorems for nearly in-tegrable Hamiltonian systems without assuming anynon-degeneracy condition. Many of KAM-type re-sults (including the classical KAM theorem) are spe-cial cases of our theorems under some non-degeneracycondition and some smoothness condition. Moreover,we can obtain some interesting results about KAMtori with prescribed frequencies.

A Trichotomy of the Singularities of2-Dimensional Bounded InvertiblePiecewise Isometric Dynamics

Byungik KahngUniversity of North Texas at Dallas, USA

The iterative dynamics of planar piecewise isometriesis a 2-dimensional analogue of the interval exchangedynamics in 1-dimensional space. Its applicationsinclude billiard and dual billiard dynamics, digitalsignal processing in electric engineering and kickedoscillators in nonlinear physics. The complexity of 2-dimensional piecewise isometric dynamics comes ex-clusively from the singularity, and therefore, the char-acterization of the singularity is an important steptoward better understanding of the system. We be-gin our talk with some known results on the classi-fication of the singularities. However, the aforemen-tioned classification is somewhat incomplete in thatclear distinctions between some types of the singu-larities and practical criteria to test them are un-available. Through this talk, we aim to resolve thisdi�culty and complete the trichotomy. We also dis-cuss some of the dynamical properties that appear tobe related to this trichotomy.

Dynamics of Quasiperiodic Cocyclesin T ⇥ SU(2)

Nikolaos KaralioliosUFF, Brazil

We present a full classification of the dynamics ofquasiperiodic cocycles in T ⇥ SU(2), under a fullmeasure Diophantine assumption on the frequency ofthe cocycle. The phenomena encountered compriseunique ergodicity in the space of Distributions, weakmixing, preservation of a, measurable or smoother,foliation in invariant tori, countable Lebesgue spec-trum. The proof of the classification uses Renor-malization of the Dynamics, methods inspired fromK.A.M. theory, and an Anosov-Katok-like construc-tion. Our work follows that of H. Eliasson and R.Krikorian, among others.

On the Reducibility of a NonlinearPeriodic System with DegenerateEquilibrium

Jianli LiangHuaqiao University, Peoples Rep of China

In this paper, we prove the reducibility of a class ofnonlinear periodic di↵erential equation with degener-ate equilibrium point under small perturbation, andobtain a periodic solution near the equilibrium point.The result is new and more general, including thatof the other paper.


Lyapunov Functions and StronglyHomogeneous Sets for SingularFlows

Luciana SalgadoUniversity of Bahia, Brazil

Consider a compact finite dimensional riemannianmanifold M . A set ⇤ is said to be strongly homo-geneous of index I for a C1 vector field X over M ifthere exist neighborhoods U of ⇤and V of X suchthat all (hyperbolic) periodic orbits in U with re-spect to any vector field in V have the same indexI. We relate the notions of infinitesimal Lyapunovfunctions (J-algebra of Potapov) and strongly homo-geneous sets for singular flows, and we present someapplications of this theory to singular hyperbolicity.

On Reducibility of 2-DimensionalLinear Quasi-Periodic System withSmall Parameter

Kun WangSoutheast University, Peoples Rep of ChinaJunxiang Xu, Min Zhu

In this paper we consider a real linear analytic quasi-periodic system of 2-dimension, whose coe�cient ma-trix depends on a small parameter Cm-smoothly andcloses to constant. Under some non-resonance con-

ditions about the basic frequencies and the eigen-values of the constant matrix and without any non-degeneracy assumption with respect to the small pa-rameter, we prove that the system is reducible formany of the su�ciently small parameters.

Some Topological Properties andEntropy for Partially HyperbolicDi↵eomorphisms

Yujun ZhuHebei Normal University, Peoples Rep of ChinaHuyi Hu, Lin Wang, Yunhua Zhou

In this talk, some topological properties, suchas quasi-stability, quasi-shadowing property, cen-ter spectral decomposition and center specification,for partially hyperbolic di↵eomorphisms are inves-tigated. As applications, some results on the en-tropy are obtained for partially hyperbolic di↵eomor-phisms. The results are from the works joint withHuyi Hu, Lin Wang and Yunhua Zhou.


Special Session 84: Recent Advances and Challenges in Coastal Dynamics

Antoine Rousseau, INRIA, FranceEmmanuel Frenod, Universite de Bretagne-Sud, France

Coastal areas are more and more threatened by the sea level rise caused by global warning, and yet 60%of the world population lives in a 100km wide coastal strip (80% within 30km in French Brittany). Thisis why coastlines are concerned with many issues, of various types: economical, ecological, social, political,etc. To address some of these very important questions, this session will cover various modeling issuesrelated to coastal environments, such as coastal oceanography (including wave modeling), morphodynamicsor biology of coastal ecosystems. This session will be an opportunity to present a state of the art of variousmathematical fields applied to coastal environment (modeling, statistics, scientific computing, numericalanalysis, etc.) and to introduce challenges brought by natural and social sciences.

Numerical Simulation for a TwoDimensional Dispersive ShallowWater System.

Nora AissioueneInria Paris, FranceMarie-Odile Bristeau, Edwige Godlewski,Jacques Sainte-Marie

We propose a numerical method for a two dimen-sional dispersive shallow water system with topogra-phy. This model is derived from a depth averaged Eu-ler incompressible system with free surface and takesinto account a non-hydrostatic pressure which im-plies to solve an elliptic equation. The contribution ofthe dispersive terms leads to having a di�cult prob-lem which requires to develop new numerical meth-ods. The approach is based on an operator splittingprocedure in the spirit of the prediction-correctionmethod initially introduced by Chorin-Temam. Theprediction part leads to solving a shallow water sys-tem for which we use finite volume methods, whilethe correction part leads to solving a mixed problemin velocity and pressure. Then, from the variationalformulation of the mixed problem proposed, we ap-ply a finite element method with compatible spaces tothe two dimensional problem on unstructured grids.Comparisons with analytical solutions and classicaltest cases are performed to evaluate the e�ciency ofour method.

Discrete Asymptotic Equations forLong-Wave Propagation

Mathieu ColinIPB, INRIA CARDAMOM, FranceS. Bellec, M. Ricchuito

In this talk, we present a new systematic methodto obtain some discrete numerical models for incom-pressible free-surface flows. The method consists infirst discretizing the Euler equations with respect toone variable, keeping the other ones unchanged andthen performing an asymptotic analysis on the re-sulting system. For the sake of simplicity, we chooseto illustrate this method in the context of the Pere-

grine asymptotic regime, that is we propose an alter-native numerical scheme for the so-called Peregrineequations. We then study the linear dispersion char-acteristics of our new scheme and present several nu-merical experiments.

Multi Scale Approach for CoastalPhenomena

Emmanuel FrenodUniversite Bretagne Sud, France

In this talk, we will present situations in whichasymptotic analysis allows us to tackle phenomenaarising in coastal areas. In particular, we will con-sider dune dynamics for which tide is a high fre-quency phenomenon and confinement in which thereare interlocked areas with various scales.

Recent Tsunamis in Chile: Learningfrom Observations, Hydrodynamics,and Numerical Modeling

Pontificia Universidad Catolica de Chile, Chile

Since 2010, three destructive tsunamis have been gen-erated by large subduction earthquakes causing casu-alties and important economic damages in coastal ar-eas. We have characterized the impact of these eventsthrough post tsunami surveys, tidal gage recordsanalysis, and numerical modeling. The applicationof Nonlinear Shallow Water Equations (NSWE) tomodel tsunami hydrodynamics and inundation pro-cesses has been a valuable source of information,which in combination with field observations and wa-ter wave hydrodynamic theories, allowed us to char-acterize the physics of these events and to understandthe importance of the bathymetric control on waveamplification and runups. In this talk, we will sum-marize this new knowledge, highlighting the useful-ness of state-of-the art NSWE models for interpretingtsunami hydrodynamics, and particularly, some re-sults with respect to their validity and performance.The experience gained in recent years through thestudy of these events has also enabled us to transferthis knowledge into Decision Support Systems (DSS)for tsunami Early Warning Systems (EWS) that arenow been tested in the national service in charge oftsunamis in Chile.


A Numerical Study of NonlinearWave-Currents Interactions in Shal-low Water

Fabien MarcheUniversity of Montpellier and Inria, France

In this talk, we focus on the recently introducedGreen-Naghdi equations with vorticity. In a firstpart, after describing this new set of equations,we study the existence of solitary waves which re-sult from a complex balance between non-linear ef-fects, dispersion and vorticity. We highlight someinteresting behaviors, like a dissymmetry in thewaves‘profiles and the occurrence of a critical waveheight ruling the existence of the right going waveand of special peaked waves with a singularity atthe crest. In a second part, we introduce a newdiscrete formulation for these equations, based on ahybrid Finite-Volume/Finite-Di↵erence strategy. Anew Finite-Volume scheme is developed, based on anapproximate Riemann solver of the HLLC type, and adiscrete preservation of the set of admissible states isensured. Some numerical validations are performed,

highlighting in the meantime some of the specialsolitary waves exhibited in the first part. In a lastpart, we describe a new numerical strategy allowingto compute the full velocity field, including its ver-tical variation, based on an asymptotic descriptionand the introduction of level lines. This is a joinedwork with D.Lannes.

Towards Coupling Coastal andLarge Ocean Models

Antoine RousseauInria Chile, FranceEric Blayo

In this work we are interested in the search of in-terface conditions to couple hydrostatic and nonhy-drostatic ocean models. To this aim, we considersimplified systems and use a time discretization tohandle linear equations. We recall the links betweenthe two models (with the particular role of the as-pect ratio � = H/L ⌧ 1) and introduce an iterativemethod based on the Schwarz algorithm (widely usedin domain decomposition methods).


Special Session 85: Di↵erential Equation Modeling and Analysis forBrain and Other Complex Bio-systems

Jianzhong Su, University of Texas at Arlington, USALixia Duan, North China University of Technology, Peoples Rep of China

Pengcheng Xiao, University of Evansville, USA

Many biological systems, such as neuronal systems, genomic systems, and immune systems, are featuredby nonlinear and complex patterns in spatial and temporal dimensions. These phenomena carry significantbiological information and regulate down-stream biological mechanisms. Understanding the mechanismsunderlying such events by quantitative modeling represents a mathematical challenge of current interest.Yet all these systems share the similar dynamical system issues in ordinary/partial di↵erent equation suchas bifurcation, stability, oscillations, stochastic noise as well as issues in determining model parameters fromexperimental data sets and computational errors of the models. This special session o↵ers a forum to ex-change the state of the art theoretical advances related to this promising area as well as computational tools.It will foster and encourage communication and interaction between researchers in these directions. Thecommon themes include mathematical models and data analysis, theoretical analysis, computational andstatistical methods of dynamical systems and di↵erential equations for the bio-system based models, as wellas applications in brain research. The topics may include but not restrict to:

1. Dynamics and computation of neuronal systems– Modeling and dynamical analysis of biological neurons and neuronal networks, – Generation, encoding andtransduction of neuronal signals and patterns. – Modeling and analysis of cognitive information processingmechanisms – Dynamic abnormality in neruronal systems due to diseases.

2. Dynamics of immune systems– Modeling biomedical processes, including tumor growth, cardio-vascular diseases, infection, and healing,mediated by immunologic mechanisms. – Analysis of mathematical models for dynamics features such asinstabilities, bifurcations that provide insight into the nature of the underlying bio-physical mechanisms. –Modeling wound healing and inflammatory responses, including cell to cell interactions, foreign body reac-tions and quantitative as well as qualitative comparison with experimental data.

3. Data analysis and modeling of brain activities– Complexity theory applied to brain – Perception, learning and memory functions in brain. – Computationalevolutionary biology. – Models, analysis and algorithms in Bioinformatics.

Stability and Bifurcations ofRhythms in Neuronal Circuits

Deniz AlacamGeorgia State University, USAJarod Collens, Aaron Kelley, Krishna Pusu-luri, Drake Knapper, Justus Schawabedal

We are interested in exploring repetitive dynamicsgenerated by constituent building blocks, or “motifs“that make up more complex central pattern generator(CPG) circuits, and the dynamic principles underly-ing more general multi-stable rhythmic patterns. Weconsider basic 3-cell motifs, as well as biologicallyplausible circuits determining locomotion behaviorsin sea slugs.

The roles of asymmetric and unique connections, andthe intrinsic properties of their associated cells in gen-erating a set of coexisting synchronous patterns ofbursting are studied. The particular kinds of networkstructures reflecting the known physiology of vari-ous CPG networks in real animals are described. Wecharacterize how observed multi-stable states arisefrom coupling, and how real circuits may take advan-tage of multi-stability to dynamically switch betweenthe corresponding polyrhythmic outputs.

References

[1] Alacam D. and Shilnikov AL. Making a swimcentral pattern generator out of latent parabolicbursters. Bifurcations and Chaos 25(7),1540003, 2015, doi: 10.1142/s0218127415400039

[2] Wojcik J., Clewley R., Schwabedal J. andShilnikov AL. Key bifurcations of burst-ing polyrhythms in 3-cell central pat-tern generators. PLoS ONE 9(4): e92918.doi:10.1371/journal.pone.0092918


Using a Mathematical Model toAssess the Roles of T Cells andCytokines in Transplant Rejection

Julia ArcieroIndiana University-Purdue University Indianapolis,USAAnirudh Arun, Andrew Maturo, GiorgioRaimondi

Organ transplantation is a life-saving surgical pro-cedure through which the functionality of a failingorgan system can be restored. However, without thelife-long administration of immunosuppressive drugs,the recipient’s immune system will launch a mas-sive immune attack that will ultimately destroy thegraft. Long-term use of immunosuppressive drugsleads to an increased risk of infection, cardiovascu-lar disease, and cancer, and thus there is currently agreat medical need to identify new strategies of in-tervention to induce transplant acceptance while pre-serving the functionality of the immune system. Thisstudy introduces an experimentally-based mathemat-ical model to examine the complexities of the dy-namic interactions between key elements of the im-mune system and the transplant and to predict howalterations in the immune response influence the re-jection of the transplant. The assumptions of theODE model are based on mouse models of hearttransplant rejection. The model predicts that de-creasing the translocation rate of e↵ector cells fromthe lymph node to the graft generally delays trans-plant rejection. Adoptive transfer of naive regula-tory T cells also delays rejection. Ultimately, thismodel will be used to identify new strategies thatpreserve the protective role of the immune responsewhile maintaining the functional role of the trans-plant.

Chemical and Mechanical Mech-anisms Making Arterial PlaquesVulnerable to Rupture

Jonathan BellUMBC, USAAnimikh Biswas

Atherosclerosis is an inflammatory disease leadingto high risk arterial plaques. A high risk plaque isvulnerable to rupture, causing a stroke, a heart at-tack, or liver damage, depending on the plaque’s lo-cation. Most arterial plaques are stable, that is, arenot high risk. Our interest is why, mechanistically,some plaques become vulnerable, while others remainstable. We first model the cellular and chemical dy-namics in a maturing plaque, where a fibrous capis developing and chemotaxis plays a significant role.We explain cross-chemotaxis, presenting some theoryand simulations. As time permits, we then brieflydiscuss the role of blood shear stress on the endothe-lial cell layer, its e↵ects on the chemical pathwayswithin the endothelial cells, and how to incorporatethis mechanism into our plaque model.

Volume Transmission and Home-ostasis of Neurotransmitters

Janet BestOhio State University, USAMike Reed, Fred Nijhout, Sean Lawley

In volume transmission, neurons in one brain nucleussend their axons to a second nucleus where neuro-transmitter is released into the extracellular space,modulating the activity of the electrophysiologicalcircuitry. Here we present a number of mathemat-ical issues that arise in calculating the concentrationof neurotransmitter in the extracellular space and itshomeostatic regulation.

The E↵ects of Sodium on the Burst-ing Transitions in the Pre-BotzingerComplex

Lixia DuanNorth China University of Technology, Peoples Repof ChinaXi Chen, Qishao Lu, Mingjun Ji

Activity of neurons in the pre-Botzinger complexwithin the mammalian brain stem has an impor-tant role in the generation of respiratory rhythms.Neurons within the pre-Botzinger complex have beenfound experimentally to yields typical bursting ac-tivities. Previous experimental results have shownthat the dynamics of sodium and calcium within eachcell may be responsible for various bursting mecha-nisms. In this talk, we study the bursting activi-ties related to the respiratory rhythms in the pre-Botzinger complex based on a mathematical modelproposed by Butera. Using the one-dimensional firstrecurrence map induced by dynamics, we investigatethe di↵erent bursting patterns and their transitionof the pre-Botzinger complex neurons based on theButera model. After we derived a one-dimensionalmap from the dynamical characters of the di↵erentialequations, and we obtained analytical conditions forthe transition of di↵erent bursting patterns. Theseanalytical results were verified through numericallysimulations. We conclude that the one-dimensionalmap contains similar rhythmic patterns as the Buteramodel and can be used as a simpler modeling toolto study fast-slow models like pre-Botzinger complexneural circuit.

Rhythmic Oscillations of Un-weighted and Weighted BurstingHodkin-Huxley Neuronal Networks

Fang HanDonghua University, Peoples Rep of ChinaQi Shi, Zhijie Wang

Rhythmic oscillations of neuronal networks are actu-ally synchronous behaviors, which play an importantrole in neural systems. In this paper, the propertiesof oscillation frequency of unweighted and weightedbursting Hodkin-Huxley neuronal networks are stud-


ied, respectively. For the unweighted neuronal net-work, neurons are coupled by inhibitory synapses andit is found that the oscillation frequency of the wholenetwork is sensitive to the parameter values of synap-tic delay and synaptic decay time, but is less sensi-tive to the parameter value of synaptic conductance.For the weighted network, neurons are coupled byexcitatory synapses with a synaptic learning rule ap-plied, and it is found that the oscillation frequencyof the network decreases monotonically along withthe increase of the synaptic learning rate, the cou-pling strength, the delay time and the decay timeconstant.

Evaluation of Bifurcation Phe-nomena in a Modified Shen-LarterModel for Intracellular Ca2+ Burst-ing Oscillations

Quanbao JiHuainan Normal University, Peoples Rep of ChinaYi Zhou, Zhuoqin Yang, Xiangying Meng

The present work describes an evaluation of the bifur-cation phenomena in a modified Shen-Larter modelbased on calcium-induced calcium release and inosi-tol triphosphate crosscoupling for calcium ion (Ca2+)bursting oscillations. A time delay for negative Ca2+

feedback on the inositol triphosphate (IP3

) receptoris added to the original Shen-Larter model, by intro-ducing the proportion of receptors not inactivated byCa2+ as a new variable. Compared with the originalmodel, the number of chaotic regions for a stimula-tion level r is significantly reduced, and regions ofCa2+ oscillations (particularly bursting) appear tobecome slightly enlarged. In addition to the freeCa2+ concentration in the endoplasmic reticulum,the IP

3

concentration in the cytosol is also consid-ered as a slow variable. Di↵erent topological typesof bursting oscillations in this modified model arepresented and classified, based on fast/slow dynam-ical analysis and codimension-2 bifurcations. Fur-thermore, classification and transition mechanisms ofCa2+ oscillations could help to understand or detectmore distinctive oscillatory behaviors of real cells inresponse to di↵erent levels of stimulation.

Spatiotemporal Patterns in Energy-E�cient Cortical Hodgkin-HuxleyNetwork Model

Jiajia LiXi‘an Jiaotong University, Peoples Rep of ChinaYing Wu

Studies of electrical activities in neurons got a break-through when the Hodgkin-Huxley neuron model wasestablished in 1952. However, the model used squidgiant axon, which cannot adapt to the mammal corti-cal neuron with stable temperature about 37 celsiusdegree. In recent studies, there has been a corti-cal Hodgkin-Huxley model of homeothermal mam-mals established, which successfully includes mam-mal’s energy-e�cient properties when single cortical

neuron cell was considered. But the population elec-trical activities of this type neuron model have beenstudied sparsely. Here, based on this model of a two-dimensional regular neuron network, we studied theformation of special spatiotemporal patterns in thenetwork, target and spiral waves, and their inter-conversion due to temperature changes.

A Concise and Exhaustive ODEModel for Characterizing the Ca2+

Channel/BKCa Channel Complexat Local and Whole-Cell Levels

Francesco MontefuscoUniversity of Padova, ItalyMorten G. Pedersen

Large-conductance Ca2+ -activated K+ channels(BKCa channels) are co-localized with voltage-gatedCa2+ channels (CaV), and are thus regulated bylocal Ca2+ levels. First, we show that an ODEmodel of single-channel gating with two states (closedand open) is able to reproduce the BKCa channelcharacteristics and dynamics by fitting experimen-tal data. Then, we propose a concise ODE modelof the CaV/BKCa channel complex by coupling thetwo state model for the BKCa channel with a threestate (closed, open and inactivated) model for theCa2+ channel. We also perform Monte Carlo sim-ulations for the devised complex in order to modelits stochastic gating; our ODE model is able to es-timate the number of open BKCa channels as func-tion of the amplitude and duration of the action po-tential, thus providing an analytical explanation forthe Monte Carlo results. Finally, we show that it ispossible to reduce the ODE model of the developedCaV/BKCa complex by exploiting an approximatedone state model for the activation of the CaV/BKCacomplex and coupling it with the inactivation of theCa2+ channel. This approximated model can provideinsights into the main features of the complex and iseasily used for whole-cell models.

Bursting and Synchronization ina Two-Compartment Model withCurrent Feedback Control

Meng PanGuangdong Pharmaceutical University, Peoples Repof ChinaQuanbao Ji, Haixia Wang, Qishao Lu

We investigate bursting patterns in the single mod-ified two-compartment model and synchronizationtransition path of two coupled bursters. It isfound that with proper parameters, the single cellcan produce two type of bursting, that is, “sub-Hopf/homoclinic” via “fold/homoclinic” hysteresisloop and “circle/fold cycle” via “circle/subHopf” hys-teresis loop. By coupling two bursters electrically, atransition process from non-synchronization to burstsynchronization, then to spike synchronization, andfinally toward nearly complete synchronization is ob-


served with the increase of the coupling strength.Moreover, when two neurons achieve nearly completesynchronization, it is discovered that the fast vari-ables are easier to get synchronization than the slowvariables, which is opposite to the ordinary situation.The reason is worthy of further research. In addi-tion, the generation mechanism of nearly completesynchronization phenomenon is given by extendingfast-slow analysis from the single to the coupled sys-tem. The results can help us better understand thesynchronization dynamics in the coupled system withmulti-time-scale. It is known that synchronizationplays an important role in the transfer of informa-tion, thus our results have theoretical and physiolog-ical significance.

A Mathematical Model ofHematopoietic Stem Cell Treat-ments in Patients with Lymphoma

Angela ReynoldsVirginia Commonwealth University, USARacheal Cooper, AllisonScalora, Elaine Wang,John McCarty, Jennifer Anderson, CatherineRoberts, Troy Lund, Amir Toor

Hematopoietic stem cells (HSCs) present in the bonemarrow are responsible for maintaining peripheralblood cell counts. Patients may need HSCs mobi-lized and collected for transplantation after high dosechemotherapy. Additionally, healthy individuals candonate HSCs. There is considerable variability in theyield in patients and normal donors despite using thesame collection protocol. Understanding the mecha-nisms governing HSC growth will allow optimizationof this treatment. To explore these mechanisms, wecreated a mathematical model, which accounts forHSC proliferation and most of the cell types that dif-ferentiated from HSCs in both the bone marrow andperipheral blood. Also, we model treatment with Fil-grastim and HSC collection. The e↵ect of the drugPlerixafor was also simulated in individuals with poorcollection.

Using Optimal Control Theory toAnalyze the Treatment of a Bac-terial Infection in a Wound withOxygen Therapy

Richard SchugartWestern Kentucky University, USASuzanne Lenhart, K. Renee Fister, StephenGu↵ey

A mathematical model describing the interactions ofbacteria, inflammatory cells, and oxygen in a wounddescribing the treatment of a bacterial infection us-ing oxygen therapy will be presented and analyzed.A second variation of the model will be presented inan optimal control setting with the control variablebeing the input of supplemental oxygen. Numericalresults of the optimal control model will be presentedand future directions will be discussed.

Interaction Function of CoupledBursting Neurons

Xia ShiBeijing University of Posts and Telecommunications,Peoples Rep of ChinaJiadong Zhang

The interaction functions of electrically coupledHindmarsh-Rose (HR) neurons for di↵erent firingpatterns are investigated in this paper. By applyingthe phase reduction technique, the phase responsecurve (PRC) of the spiking neuron and burst phaseresponse curve (BPRC) of the bursting neuron arederived. Then the interaction function of two cou-pled neurons can be calculated numerically accordingto the PRC (or BPRC) and the voltage time courseof the neuron. Results show that the BPRC is moreand more complicated with the increase of the spik-ing number within a burst, and the curve of the inter-action function oscillates more and more frequentlywith it. However, two certain things are unchanged:� = 0, which corresponds to the in-phase synchro-nization state, is always the stable equilibrium, whilethe anti-phase synchronization state with � = 0.5 isan unstable equilibrium.

Influence of Coupling on OscillatoryProperties of Bursting Solutions inNeuron Models

Jianzhong SuUniversity of Texas - Arlington, USAAlice Lubbe

Neurons often exhibit bursting oscillations, as amechanism to modulate and set pace for other brainfunctionalities. These bursting oscillations are dis-tinctly characterized by a silent phase of slowly evolv-ing steady states and an active phase of rapid firingoscillations. These bursting neurons are modeled byfast-slow systems consisting of several ordinary di↵er-ential equations. In a network of neurons, their col-lective oscillatory behavior may di↵er from individualneurons due to coupling and inputs from other neu-rons. We analyze the transition mechanisms betweenperiodic and chaotic/random behavior in a coupledsystem of neurons. Using geometric and bifurcationanalysis, we provide insight how coupling can reg-ularize chaotic trajectories using the flow inducedPoincare map.


Fast Regular Firing Induced byInter and Intra Time Delays in TwoClustered Neuronal Networks

Xiaojuan SunBeijing University of Posts and Communications,Peoples Rep of China

In this paper, we consider two clustered neuronalnetworks with dense intra synaptic links within eachcluster and sparse inter synaptic links between them.We focus on the e↵ects of intra and inter time de-lays on the spiking regularity and timing in the twoclusters. With the aid of simulation results, we showthat intermediate intra and inter time delays are ableto separately induce fast regular firing spiking activ-ity with a high firing rate as well as a high spikingregularity. Moreover, when both intra and inter timedelays are present, we find that fast regular firings areinduced much more frequently than if only a singletype of delay is present in the system. The presentedresults indicate that appropriately adjusted inter andintra time delays can significantly facilitate fast reg-ular firing in neuronal networks. Based on a detailedanalysis, we conjecture that this is most likely whenthe greatest common divisor of the intra and intertime delay falls into a range where fast regular fir-ings are induced by suitable intra or inter time delaysalone.

Modeling of Immune Response toViral and Bacterial Infections

David SwigonUniversity of Pittsburgh, USAE. Mochan, B. Ermentrout, G. Clermont

Immune response to responds to infection begins byinitiating a spectrum of immune responses regulatedby an intricate network of signaling interactions thathave not yet been completely characterized. In thetalk I will outline a series of models that provide qual-itative and quantitative prediction of the time courseof a disease, aid in understanding of the mechanismsof the immune response, and can be utilized in thestudy the e↵ects of an anti-viral or anti-bacterial drugtreatment. Our latest e↵ort has been focused on en-semble models that reflect the uncertainty about pa-rameter values, data sparseness, and the likely varia-tion of the disease outcome across a population. Thetechnique is useful when the model contains many un-known parameters, such as reaction rate constants ofbiochemical processes, which are poorly constrainedand their direct measurement in vivo is not feasible.

Map Reduction of a 3D Hodgkin-Huxley Model

Kelly ToppinDrexel University, USADennis Yang, Yixin Guo

We study the dynamics of a 3D Hodgkin-Huxleymodel for neurons in the basal ganglia. The modelswitches between two di↵erent states that representneurons with or without external input. To inves-tigate how these neurons respond to input signals,we develop a projection method to reduce the 3DHodgkin-Huxley model to a 2D map. Since each ofthe two di↵erent states has a stable limit cycle withweak and strong stable bundles, we project trajec-tories onto a weak stable manifold of the limit cyclealong an invariant fiber bundle over the manifold, re-ducing the dynamics to a rotation along the limit cy-cle and an exponential decay toward the limit cycle.By this projection, we reduce the 3D Poincare mapof the Hodgkin-Huxley system to a 2D map. Thenwe analyze the existence and bifurcations of periodicorbits of the 2D map as parameters vary.

An Immuno-Chemotherapy AgainstColorectal Cancer: Modeling andAnalysis

Qing WangShepherd University, USAZhijun Wang, David J. Klinke

Recently, a chemotherapy agent oxaliplatin (OXP) incombination with interleukin-12 (IL-12) was used toeliminate pre-existing liver metastatic colorectal can-cer in a murine model. We developed a multi-scaleimpulsive ODE model to describe the interaction be-tween the immune system and tumor in response tothe combined IL-12 and OXP therapy. Model param-eters were calibrated to published experimental datausing a genetic algorithm. Sensitivity of parameters,local stability analysis, and treatment strategies tocontrol tumor growth were discussed. This researchwas supported by the NIGMS of the NIH grant aspart of the WV-INBRE (P20GM103434).

Applications of Ordinal Longitu-dinal Data Analysis with MultiplePredictors

Xiaohui WangUniversity of Texas-Rio Grande Valley, USA

Ordinal longitudinal data is often seen in behavioralsciences, public health and medical studies. For ex-ample, in Alliance for a Healthy Border program(2006-2008), a chronic disease prevention programthrough twelve federally qualified community healthcenters serving primarily Hispanics in communitiesalong the U.S.-Mexico border, survey and healthmeasurements were obtained at three time points ac-cording to the pre-post-post study design. Successes


of the program were evaluated with dichotomous ortrichotomous ordinal outcomes of weight reduction,glycemic control, and physical activity improvement.In this study, we apply several modeling methodsto evaluate program successes that are ordinal lon-gitudinal data in nature. The ordinal response mod-eling methods include generalized estimating equa-tion (GEE) approach for cumulative logit models andtransitional ordinal modeling with multiple predic-tors.

Extending Levelt’s Propositions toPerceptual Multistability InvolvingInterocular Grouping

Yunjiao WangTexas Southern University, USAAlain Jacot-Guillarmod, Yunjiao Wang, Clau-dia Pedroza, Haluk Ogmen, Kresimir Josic,Zachary Kilpatrick

Levelt’s Propositions have been a touchstone for ex-perimental and modeling studies of perceptual mul-tistability. We asked whether Levelt’s Propositionsextend to perceptual multistability involving inte-rocular grouping. To address this question we usedsplit-grating stimuli with complementary halves ofthe same color. As in previous studies, subjects re-ported four percepts in alternation: the two stim-uli presented to each eye, as well as two interocu-larly grouped, single color percepts. Most subjectsresponded to increased color saturation by more fre-quently reporting a single color image, thus increas-ing the predominance of grouped percepts (Levelt’sProposition I). In these subjects increased predom-inance was due to a decrease in the average domi-nance duration of single-eye percepts, while that ofgrouped percepts remained largely una↵ected. Thisis in accordance with Levelt’s Proposition II, whichposits that the average dominance duration of thestronger (in this case single-eye) percept is primar-ily a↵ected by changes in stimulus strength. To ex-plain these observations, we introduced a hierarchicalmodel consisting of four low-level neural populationsresponding to input to each visual hemifield, andhigher-level populations representing the four per-cepts. The model explains perceptual multistabilityinvolving interocular grouping via competition acrossmultiple visual system layers.

Bifurcation Analysis in theHypothalamic-Pituitary-AdrenalAxis Including Gulcocorticoid Re-ceptor Complex

Pengcheng XiaoUniversity of Evansville, USAJianzhong Su

The hypothalamic-pituiary-adrenal (HPA) axis playsan important role in the regulation of neuroendocrineand sympathetic nervous systems. Emerging evi-dence has shown that glucocorticoid act on gluta-mate neuro-transmission system and consequently in-

fluences neuronal activities cognitive function. Inthis paper, we numerically derive the two parame-ter bifurcation analysis on one HPA model includingGulcocorticoid Receptor (GR) to explore the gluco-corticoid bistability.

Moment-Flux Models for BacterialChemotaxis in Large Signal Gradi-ents

Chuan XueOhio State University, USAXige Yang

Chemotaxis is a fundamental process in the life ofmany prokaryotic and eukaryotic cells. Chemotaxisof bacterial populations has been modeled by bothindividual-based stochastic models that take into ac-count the biochemistry of intracellular signaling, andcontinuum PDE models that track the evolution ofthe cell density in space and time. Continuum mod-els have been derived from individual-based modelsthat describe intracellular signaling by a system ofODEs. The derivations rely on quasi-steady stateapproximations of the internal ODE system. Whilethis assumption is valid if cell movement is subject toslowly changing signals, it is violated if cells are ex-posed to rapidly changing signals. In the latter casecurrent continuum models break down and do notmatch the underlying individual-based model quan-titatively. We derive new PDE models for bacterialchemotaxis in large signal gradients that involve notonly the cell density and flux, but also moments ofthe intracellular signals as a measure of the devia-tion of cell’s internal state from its steady state. Thederivation is based on a new moment closure methodwithout calling the quasi-steady state assumption ofintracellular signaling. Numerical simulations sug-gest that the resulting model matches the popula-tion dynamics quantitatively for a much larger rangeof signals.

Stochasticity and Bifurcations ina Reduced Model with InterlinkedPositive and Negative FeedbackLoops of CREB1 and CREB2 Stim-ulated by 5-HT

Zhuoqin YangBeihang University, Peoples Rep of China

The cyclic AMP (cAMP) response element bindingprotein (CREB) family of transcription factors is cru-cial in regulating gene expression required for long-term memory (LTM) formation. Song et al. pro-posed a minimal model with only interlinked posi-tive and negative feedback loops of transcriptionalregulation by the activator CREB1 and the repressorCREB2. Without considering feedbacks between theCREB proteins, Pettigrew et al. developed a com-putational model characterizing complex dynamicsof biochemical pathways downstream of 5HT recep-tors. In this work, to describe more simply the bio-chemical pathways and gene regulation underlying


5HT induced LTM, we add the important extracel-lular sensitizing stimulus 5HT as well as the prod-uct Apuch into the minimal model. Di↵erent dy-namics including monostability, bistability and mul-tistability is investigated by means of codimension-2 bifurcation analysis. Comparative analysis of de-terministic and stochastic dynamics reveals diversestochastic behaviors resulted from the finite numberof molecules.

Modeling and Analysis of NeonatalSeizures with Synaptic Plasticityfor Real Infants’ EEG Signals

Honghui ZhangNorthwestern Polytechenical University, PeoplesRep of ChinaJianzhong Su

This paper aims to explore the internal dynamicalmechanisms of seizure through modeling and analysisof EEG signals, which can help to do early seizure de-tection and treatment for epileptic patients. We stud-

ied the nonlinear dynamics of neonatal seizures by amathematical model established with synaptic plas-ticity considered. First, according to several neona-tal electroencephalogram with epilepsy, we present adynamics seizure model that accounts for the basicexperimental observations of neonatal seizure moti-vated by previous model. The great ability of the pro-posed model to produce qualitatively relevant behav-iors was shown by numerical simulations. Meanwhile,the rationality of the connecting structure hypothe-sis in the modeling process was verified. Further,through adjusting the threshold condition and exci-tation strength of synaptic plasticity, we elucidatethe e↵ect of synaptic plasticity on neonatal seizure.Our results show that synaptic plasticity has greate↵ect on the duration of seizure activities, which cansupport the hormonal therapy of synaptic plasticityfor seizure control.


Special Session 86: Pattern Formation and Recognition in StructuredInformation and Biological Systems with External Forcing

Jianhong Wu, York University, CanadaKunquan Lan, Ryerson University, Canada

Xi Huo, Ryerson University, Canada

This session will bring together participants working on nonlinear dynamics using di↵erential equationsand real data to detect and describe spatio-temporal patterns in dynamical systems from life sciences andinformation management. Emphasis will be on how structural complexity and external forcing includingharvesting impact system dynamics, and how complex data and parameters can be summarized via relativesimple indices, and how these are relevant to bifurcation and scenario analysis and optimal control.

Absenteeism, Presenteeism andInfectious Diseases in a Local Econ-omy

Monica CojocaruUniversity of Guelph, Guelph ON, CanadaE. Thommes, S. Athar

In this paper we study the cost of absenteeism andpresenteeism (going to work while sick) during a pan-demic in a local economy with several geographicallydistinct locations, and with work force populationsconsisting of individuals who live and work in thesame city, and individuals who live and work in dif-ferent locations (daily commuters). We run simula-tions to study the e↵ects of the fear factor and of theseverity of disease on the number of missed work daysin the region, which we translate into loss of produc-tivity costs. We find that higher values of the fearparameter lead to high absenteeism and lower infec-tion levels. However, , we also show that for severepandemics (such as when the number of secondary in-fections is higher) there are scenarios where there ex-ists a unique value of the fear parameter which leadsto minimum economic costs for the regional economy.This indicates that “staying at home” policies dur-ing an epidemic could be implemented for the workforce, without reaching a state of emergency.

Instability and Non-Synchronization in Delayed Dif-ferential Equations Induced byUsing Fast and Random Switching

Yao GuoYork University, CanadaWei Lin, Yuming Chen, Jianhong Wu

This talk first presents a counterintuitive examplewhere even though all the time-delayed subsystemsare exponentially stable, the behaviours of the ran-domly switched system change from stable dynamicsto unstable dynamics with a decrease of the dwelltime. Then by using the theories of stochastic pro-cesses and delay di↵erential equations, we present ageneral theoretical result on when the fast and ran-dom switching induced instability should occur. Andwe extend this to the case of nonlinear time-delayedswitched systems, which is eventually used to dealwith the non-synchronization problem of switched

networks. Many simulations are given to illustrateour theory as well. In addition, further numericalsimulations also show that even switched systemswithout time-delays can be destabilized or desynchro-nized by using fast and random switching.

Linear Stability of DelayedReaction-Di↵usion Systems

Peter HinowUniversity of Wisconsin - Milwaukee, USAMaya Mincheva

A common feature of pattern formation in both spaceand time is the destabilization of a stable equilib-rium solution of an ordinary di↵erential equation byadding di↵usion, delay, or both. Here we study linearstability of general reaction-di↵usion systems witho↵-diagonal time delays. We show that a delay-stablesystem cannot be destabilized by di↵usion, and thata di↵usion stable system is also stable with respectto delay, if the di↵usion is su�ciently fast. A systemwith direct negative feedback which is strongly sta-ble with respect to di↵usion can be destabilized byo↵-diagonal delay.

We acknowledge support from the Simons Founda-tion through the grant “Collaboration on Mathemat-ical Biology“ to Peter Hinow.

Application of Lasalle’s InvariancePrinciple in a 3-Dimensional DelayDi↵erential Equation System

Xi HuoRyerson University, CanadaXiaodan Sun, Yanni Xiao, Kunquan Lan,Jianhong Wu

We will present our mathematical results, proofs,and open problems arise from the convalescent bloodtransfusion system. This system models the dona-tion, treatment, and storage dynamics for large-scaleblood therapy - an issue originated from the 2014-2015 West Africa Ebola outbreaks.


Identification of Time Delay andIts Role in Biological DynamicalSystems

Wei LinFudan University, Peoples Rep of China

Time delays are omnipresently observed in many na-ture and artificial systems including physical, bio-logical, and chemical systems. Naturally, two kindsof questions arise: ”How to identify the time delayswhen a certain amount of datasets are obtained fromthe experiments or real world systems” and ”How tocharacterize the intrinsic roles of time delays that areplayed in coupled network systems?” In this talk, wewill introduce recent works that address the previoustwo questions, and show the significance of time de-lays in dealing with various representative systems ofbiological significance.

The Dynamics of HIV Spreading inthe Network of Lymphocyte Recir-culation in Vivo

Jie LouShanghai University, Peoples Rep of ChinaYing Huang

Although antiretroviral therapy (ART) can e↵ec-tively inhibit replication of human immunodeficiencyvirus (HIV), the virus is able to persist in cellular andanatomical viral reservoirs. The aim of this model isto gain better understanding of HIV persistence, inregards to the lymphocyte recirculation network ofimmune system as well as the central nervous sys-tem. Our findings probably can reflect the inabil-ity of some drugs to penetrate the blood-brain (orblood-testis) barrier and emphasize the ability of aninfected individual to transmit HIV even under theART. We also find that level of HIV free virus inperipheral blood may not give a correct reflection ofwhat is occurring within other organs in vivo.

Detecting Unknown Unstable Statesin High-Dimensional Nonlinear Dy-namical Systems

Huanfei MaSoochow University, Peoples Rep of China

There are various kinds of unstable states in non-linear dynamical systems, such as unstable periodicorbits (UPOs) and unstable steady states (USSs).Detecting such unstable states of a high-dimensionalnonlinear dynamical systems is generally believed tobe a challenging problem. Particularly, when the un-stable states are unknown, the detecting or controlmethods should be designed in a reference-free way.

We propose some non-invasive and adaptive methodsto deal with such problems, without requiring any apriori information of the unstable states. Moreover,we will further discuss how to detect UPOs and USSsof a system only with the output time series.

Global Analysis on Viral Dynamics

Hongying ShuTongji University, Peoples Rep of ChinaLin Wang, James Watmough

Determining sharp conditions for the global stabil-ity of equilibria remains one of the most challengingproblems in the analysis of models for the manage-ment and control of biological systems. Yet such re-sults are necessary for derivation of parameter thresh-olds for eradication of pests or clearing infections.This applies particularly to models involving non-linearity and delays. In this talk, we provide somegeneral results applicable to immune system dynam-ics. This general model admits three types of equi-libria: infection-free equilibria, CTL-inactivated in-fection equilibria, and CTL-activated infection equi-libria, and two critical values: the basic reproductionnumber for viral infection and the viral reproduc-tion number at the CTL-inactivated infection equi-librium. Our results cover and improve many existingones and include the case when the nonlinear func-tions are nonmonotone.

Using a System of Reaction-Di↵usion Equations with NonlocalE↵ects to Model Spatial Patterningin the York River Tidal marshes

Sofya ZaytsevaThe College of William and Mary, USALeah Shaw, Junping Shi, Rom Lipcius

Spatial patterning in multi-species communities canbe critical to ensuring their proper function and sur-vival. Therefore, studying the formation of self-organized patterns in ecology is crucial for under-standing the underlying interactions in the commu-nity and its ability to adapt to various environmentalchanges. A pattern of finger-like projections has re-cently been observed on the shore of the York Riverfor salt marsh cordgrass, mussels and sediment. Wepropose a system of reaction-di↵usion equations withnonlocal e↵ects to explain the formation of this pat-tern through interactions between grass, mussels andsediment. The nonlocal e↵ects are modeled throughan influence kernel, specifying the strength of inter-actions between individuals as a function of the dis-tance between them. We numerically integrate thePDE model in MATLAB and find that the model dis-plays stable patterns reminiscent of those observed inthe field. To achieve a better understanding of theunderlying dynamics, we analyze the full system aswell as the corresponding grass-sediment subsystem.We investigate the stability of various steady states,the bifurcations they undergo and their ecological in-terpretation.


Special Session 87: Direct, Inverse and Control Problems for Di↵erentialEquations

Angelo Favini, Universita di Bologna, ItalyDavide Guidetti, Universita di Bologna, Italy

New results on direct , inverse and control problems related to abstract di↵erential equations and theirapplications to PDEs are concerned. Particular attention is devoted to recent developments on Sobolev typeequations.

Inverse Problems for Neuronal Ca-ble Models on Graphs

Jonathan BellUMBC, USASergei Avdonin

For a parabolic equation defined on a tree graph do-main, a dynamic Neumann-to-Dirichlet map associ-ated with the boundary vertices can be used to re-cover the topology of the graph, length of the edges,and unknown coe�cients and source terms in theequation. The motivation for this investigation isthat the parabolic equation comes from a neuronalcable equation defined on the dendritic tree of aneuron, and the inverse problem concerns parameteridentification of k unknown distributed conductanceparameters.

Optimal Control Problems forLeontief Type Systems: NumericalMethods and Applications

Alevtina KellerSouth Ural State University, RussiaAleksandr Shestakov

The report provides an overview of results of a nu-merical research of optimal control problems for theLeontief type systems. Such systems arise in mod-elling of di↵erent processes and objects, for examplemeasuring transducers, economic systems of an en-terprise, and dynamics of a cell cycle. Leontief typesystem is a finite-dimensional analogue of the Sobolevtype equation therefore our research is based on themethods of the theory of degenerate groups of opera-tors. The report presents an algorithm for finding ofapproximate solutions to a variety of optimal controlproblems for Leontief type systems with the Showal-ter - Sidorov initial condition which is more conve-nient in numerical research. The proof of the con-vergence of the approximate solutions to the preciseone is an important result. The issues of improving ofthe e�ciency of numerical algorithms and their mod-ifications in the numerical study of applications arediscussed. Special attention is given to the numericalalgorithms for solving of the optimal measurementproblems which are the problems of restoration ofsignals dynamically distorted both by inertia of themeasuring device, and resonances in its circuits. Theresults of computational experiments are presented.

Analytical and Numerical Inves-tigations of the Optimal ControlProblem for Semilinear SobolevType Equations

Natalia ManakovaSouth Ural State University, Russia

Generally, the processes applied in mechanics, engi-neering and production are controllable, therefore,within respective applied problems it is usually es-sential to control the external actions e�cient enoughto achieve required results in such processes. De-spite the fact that the research field of optimal con-trol problems for distributed systems is rather large,the solutions control matters for confluent semi-linearequations, unresolved for derivative with time, arestudied insu�ciently. Such equations are called semi-linear Sobolev type equations. Author proposed suf-ficient conditions for the existence of a solution ofthe optimal control problems for semilinear Sobolevtype equations with s-monotone and p-coercive op-erators. Theorems of existence and uniqueness ofweak generalized solution to the Cauchy or theShowalter - Sidorov problem for a class of degeneratenon-classical equations of mathematical physics arestated. The theory is based on the phase space andthe Galerkin methods. The developed scheme of anumerical method allows one to find an approximatesolution to optimal control problems for consideredmodels. On the basis of abstract results the exis-tence of optimal control of processes of filtration anddeformation are obtained. The necessary conditionsfor optimal control are provided.

On the Solvability of DegenerateOperator-Di↵erential Equations ofthe First Order in the Spaces ofDi↵erentiable Noises

Minzilia SagadeevaSouth Ural State University, Russia

The equations unsolved with respect to the deriva-tive was began to study systematically in the mid-dle of the last century. Often such equations arecalled the Sobolev type equations. The report ex-amines the solvability of such equations in the spacesof ”noises”. These specific spaces di↵erent from thestandard spaces of random processes by that thederivative of the process can be determined in thesespaces. Namely, the derivative of the process is de-fined as a Nelson - Gliklih derivative.


The concepts previously introduced for the spaces ofdi↵erentiable real-valued ”noises” using the Nelson- Gliklikh derivative are carried over to the case ofcomplex-valued ”noises”. We consider the Sobolevtype equations in the spaces of ”noises” on the con-dition that a degenerate resolving semigroup of classC0 exists. So we consider the relatively radial oper-ators in spaces of complex-valued ”noises”. We con-struct a solution to the weakened Showalter - Sidorovproblem for Sobolev type equation with relativelyp-radial operator in a space of complex-valued pro-cesses. We study the solvability of Chen - Gurtin inspaces of complex-valued ”noises” as the applicationof the main results of this report.

The Sobolev Type Equations andDegenerate Operator (Semi)groups.Theory and Applications

Georgy SviridyukSouth Ural State University, Russia

Sobolev type equations were firstly studied in theworks of A. Poincare. Then they appeared in theworks of S.V. Oseen, J.V. Boussinesq, S.G. Rossbyand other researchers, that were dedicated to the in-vestigation of some hydrodynamics problems. Theirsystematical study started in the middle of theXX century with the works of S.L. Sobolev. Thefirst monograph devoted to the study of such equa-tions appeared in 1999. Nowadays the number ofworks devoted to such equations is increasing exten-sively. Sometimes such equations are called equationsthat are not of Cauchy - Kovalevskaya type, pseu-doparabolic equations, degenerate equations or equa-tions unsolvable with respect to the highest deriva-tive. The term Sobolev type equations was firstlyproposed in the works of R. Showalter. NowadaysSobolev type equations constitute the vast area innonclassical equations of mathematical physics. Theproposed theory of degenerate semigroups of opera-tors is a suitable mathematical tool for the study ofsuch equations. The theory is developing in di↵er-ent directions: optimal control problems, initial-finalvalue problems, equations of high order, and findsapplications in elasticity theory, fluid dynamics, oilproduction, economics, biology, and in the solutionof many technical problems, for example in the the-ory of dynamic measurements.

The Multipoint Initial-Final ValueProblem for Sobolev-Type Equa-tions

Sophiya ZagrebinaSouth Ural State University, Russia

The models of mathematical physics, whose represen-tation in the form of equations or systems of partialdi↵erential equations do not fit one of the classicaltypes such as elliptic, parabolic or hyperbolic, arecalled nonclassical and can be regarded as Sobolev-type equations. The report provides an overview ofthe author’s results in the field of nonclassical de-terminate and stochastic equations of mathemati-cal physics for which the initial-final value problems,generalizing the Cauchy and Showalter - Sidorov con-ditions, are considered. Basic method for the re-search is the Sviridyuk relative spectrum theory. Inaddition, we use a generalized theorem of splitting ofthe space and operators actions in the case of rela-tively bounded operator. Abstract results are illus-trated by the specific initial-final value problem forthe equations in partial derivatives occurring in ap-plications, namely, the filtration theory, fluid dynam-ics theory and deformation theory, considered on thesets of di↵erent geometrical structure.

The Sobolev Type Equations ofHigher Order

Alyona ZamyshlyaevaSouth Ural State University, Russia

This report surveys the author’s results concerningmathematical models based on Sobolev-type equa-tions of higher order. The theory is constructed usingthe available facts on the solvability of initial (initial-final) problems for the first-order Sobolev-type equa-tions. The main idea is a generalization of the theoryof degenerate (semi)groups of operators to the case ofhigher-order equations: decomposition of spaces andactions of the operators, construction of the propaga-tors and the phase space for the homogeneous equa-tion, as well as the set of valid initial values for theinhomogeneous equation. We use the phase spacemethod, which is quite useful for solving the Sobolev-type equations and consists in a reduction of a singu-lar equation to a regular one defined on a certain sub-space of the original space. We reduce mathematicalmodels to initial (initial-final) problems for abstractSobolev-type equations of higher order. The resultsmay find further applications in the study of opti-mal control problems, nonlinear mathematical mod-els, and in the construction of the theory of Sobolev-type equations of higher order in quasi-Banach spacesand stochastic spaces of noises.


Special Session 88: Data Assimilation and Nonlinear Filtering

Kody Law, Oak Ridge National Laboratory, USADavid Kelly, Courant Institute of Mathematical Sciences, USA

Filtering describes the solution of a sequence inverse problems, in which the data arrives in an online fashion.The subject of filtering has enjoyed a long standing symbiosis between classical and probabilistic approaches.Data assimilation can be viewed as a bridge between these approaches, built out of the necessity to obtainsolutions to the filtering problem quickly for very high dimensional, turbulent, nonlinear forecast models,with notable applications in atmospheric and oceanographic science. This mini-symposium aims to bringtogether experts interested in nonlinear filtering, data assimilation and applications, to share their latestresearch.

A Basis for Improving NumericalWeather Prediction in the Gulf Areaby Assimilating Doppler Radar Ra-dial Winds

Mohamed-Naim AnwarUnited Arab Emirates University, United ArabEmiratesFathalla A. Rihan, Chris Collier

This contribution presents a theoretical frameworkto Data Assimilation of Doppler Radial Winds intoa high resolution NWP model using 3D-Var system.NWP is considered as an initial-boundary value prob-lem: given an estimate of the present state of theatmosphere, the model simulates (forecasts) its evo-lution. Specification of proper initial conditions andboundary conditions for numerical dynamical modelsis essential in order to have a well-posed problem andsubsequently a good forecast model (A well-posedinitial/boundary problem has a unique solution thatdepends continuously on the initial/boundary con-ditions). The goal of data assimilation is to con-struct the best possible initial and boundary con-ditions, known as the analysis, from which to inte-grate the NWP model forward in time. We discussthe types of errors that occur in radar radial winds.Some related problems such as nonlinearity and sen-sitivity of the forecast to possible small errors in ini-tial conditions, random observation errors, and thebackground states are also considered. The techniquecan be used to improve the model forecasts, in theGulf area, at the local scale and under high aerosol(dust/sand/pollution) conditions.

Nonlinear Filtering Without aModel and with a Partial Model

Tyrus BerryGeorge Mason University, USAJohn Harlim, Franz Hamilton, Tim Sauer

We first introduce and compare two methods of fil-tering without a model, which we call nonparametricfilters. The first method is based on the di↵usionforecasting algorithm combined with a Bayesian up-date. The second is the Kalman-Takens approachwhich combines a Kalman update with a local lin-ear forecast in the Takens embedding space. Thelimitation of these approaches is that the amount ofhistorical data required for learning the model grows

exponentially in the intrinsic dimension of the un-derlying dynamics. To overcome this limitation, weassume that an incomplete or imperfect parametricmodel is available, and we show how to use the non-parametric approach to correct the model error. Thismeans that we only assume hat the model error islow dimensional. We demonstrate this semiparamet-ric approach to data assimilation and forecasting ona Lorenz-96 system with model error governed bystochastic and chaotic dynamics.

Well-Posedness and Accuracy of aClass of Nonlinear Mini-Max Filters

Michal BranickiMathematics, University of Edinburgh, UK, Scot-landS. Zhuk

We consider the continuous and discrete time filter-ing problem for quasi-linear and quadratic truth dy-namics with linear observations where the optimalestimate is given w.r.t. the quadratic mini-max costfunction. The filter equations are derived by exploit-ing a certain duality between the filtering problemand the optimal stochastic control problem. In par-ticular, this approach provides a framework for asystematic derivation of suboptimal Bayesian filterswhich, unlike approximate Gaussian filters includingthe Extended Kalman filter or the Ensemble Kalmanfilter, does not use ad hoc assumptions on the systemcovariance. For quadratic dissipative truth dynamicswe show that a class of approximate filters derivedwithin this framework is well-posed and accurate ifthe linear observation operator has a su�ciently highrank.

A Stable Particle Filter in High-Dimensions

Dan CrisanImperial College London, EnglandAlex Beskos, Ajay Jasra, Kengo Kamatani,Yan Zhou

We consider the numerical approximation of the fil-tering problem in high dimensions, that is, when thehidden state lies in Rd with d large. For low di-mensional problems, one of the most popular numer-ical procedures for consistent inference is the class ofapproximations termed particle filters or sequentialMonte Carlo methods. However, in high dimensions,


standard particle filters (e.g. the bootstrap particlefilter) can have a cost that is exponential in d forthe algorithm to be stable in an appropriate sense.We develop a new particle filter, called the space-time particle filter, for a specific family of state-spacemodels in discrete time. This new class of particlefilters provide consistent Monte Carlo estimates forany fixed d, as do standard particle filters. More-over, we expect that the state-space particle filterwill scale much better with d than the standard filter.We illustrate this analytically for a model of a sim-ple i.i.d. structure and one of a Markovian structurein the d-dimensional space-direction, when we showthat the algorithm exhibits certain stability proper-ties as d increases at a cost O(nNd2), where n is thetime parameter and N is the number of Monte Carlosamples, that are fixed and independent of d. Simi-lar results are expected to hold, under a more gen-eral structure than the i.i.d. case, independently ofthe dimension. Our theoretical results are also sup-ported by numerical simulations on practical modelsof complex structures. The results suggest that it isindeed possible to tackle some high dimensional fil-tering problems using the space-time particle filterthat standard particle filters cannot handle.

Kernel Methods for NonparametricAnalog Forecasting

Dimitrios GiannakisNew York University, USA

Analog forecasting is a nonparametric technique in-troduced by Lorenz in 1969 which predicts the evo-lution of observables of dynamical systems by follow-ing the evolution of samples in a historical recordof observations of the system which most closely re-semble the observations at forecast initialization. Inthis talk, we discuss a family of forecasting methodswhich improve traditional analog forecasting by com-bining ideas from kernel methods for machine learn-ing and state-space reconstruction for dynamical sys-tems. A key ingredient of our approach is to replacesingle-analog forecasting with weighted ensembles ofanalogs constructed using local similarity kernels.The kernels used here employ a number of dynamics-dependent features designed to improve forecast skill,including Takens‘ delay-coordinate maps (to recoverinformation in the initial data lost through partial ob-servations) and a directional dependence on the dy-namical vector field generating the data. Mathemat-ically, the approach is closely related to kernel meth-ods for out-of-sample extension of functions, and wediscuss alternative strategies based on the Nystrommethod and the multiscale Laplacian pyramids tech-nique. We illustrate these techniques in forecasts ofNorth Pacific sea surface temperature and arctic seaice cover.

Stability of the Ensemble KalmanFilter

David KellyNew York University, USAAndrew Majda, Xin Tong

The Ensemble Kalman Filter (EnKF) is one of thecornerstone filtering methods in geoscience. It al-lows for computationally e�cient assimilation of ob-servational data with high dimensional physical mod-els, such as those found in numerical weather predic-tion. Despite its success, the dynamical behavior ofEnKF is poorly understood, particularly in the real-istic regime of small ensemble sizes. In this talk wewill address two key questions regarding the stabilityof EnKF : 1) Is EnKF stable to perturbations in en-semble initializations ? (The answer is no, in general)and 2) Are there simple modifications that can helpenhance stability? (The answer is yes, thankfully).These questions are addressed in a simple frameworkof ergodicity for Markov processes.

Data Assimilation Algorithm for 3DBenard Convection in Porous Me-dia Employing Only TemperatureMeasurements

Evelyn LunasinUnited States Naval Academy, USAAseel Farhat, Edriss S. Titi

In this paper we propose a continuous data assimila-tion (downscaling) algorithm for the Benard convec-tion in porous media using only discrete spatial-meshmeasurements of the temperature. In this algorithm,we incorporate the observables as a feedback (nudg-ing) term in the evolution equation of the temper-ature. We show that under an appropriate choiceof the nudging parameter and the size of the mesh,and under the assumption that the observed data iserror free, the solution of the proposed algorithm con-verges at an exponential rate, asymptotically in time,to the unique exact unknown reference solution of theoriginal system, associated with the observed (finitedimensional projection of) temperature data. More-over, we note that in the case where the observationalmeasurements are not error free, one can estimate theerror between the solution of the algorithm and theexact reference solution of the system in terms of theerror in the measurements.


Importance Sampling: Computa-tional Complexity and IntrinsicDimension

Daniel Sanz-AlonsoBrown University, SpainSergios Agapiou, Omiros Papaspiliopoulos,Andrew Stuart

We aim to give unity and provide new mathematicalinsight into the growing published literature address-ing the curse of dimensionality of importance sam-pling. We focus on the use of importance samplingin Bayesian learning problems. We highlight, follow-ing the pioneering work of Bickel and co-authors, theimportance of defining suitable notions of dimensionfor these problems. We establish precise connectionsbetween the intrinsic dimension used by Bickel, andthe notion of e↵ective number of parameters, used instatistics an machine learning. We show that bothare finite as long as there is absolute continuity ofposterior with respect to prior. This suggests a uni-fying idea: importance sampling degenerates as lossof absolute continuity is approached. We study therates at which this degeneracy occurs as the dimen-sion of the data and the unknown grow, but also asthe prior becomes less informative or the noise in theobservations smaller. The relevance of these ideas fordata assimilation and filtering problems will becomeapparent.

A Reduced-Basis Approach for Pa-rameterized Backward StochasticDi↵erential Equations

Guannan ZhangOak Ridge National Laboratory, USAWeidong Zhao

This e↵ort is motivated by the relationship betweenbackward SDEs and a class of quasilinear PDEs, de-scribed by the nonlinear Feynman-Kac theory, suchthat our approach can be applied to solving quasi-linear parabolic PDEs with deterministic or randomparameters, where we aim at approximating the pa-rameterized viscosity solution of the PDEs. In theSDE setting, the temporal-spatial di↵erential opera-tor in the PDE is described by the dynamics of theunderlying stochastic process, and the key task in

developing numerical schemes is to approximate theconditional mathematical expectation of the solutionand the forcing term with respect to the stochasticprocess. In this e↵ort, we utilize the empirical inter-polation method (EIM) to approximate the involvedexpectation operator, such that we can obtain the o↵-line online decomposition of computational cost. Themain feature of the SDE approach is that the approx-imate solution can be computed independently ateach spatial grid point without solving linear systemswhen using implicit time-stepping schemes. This fea-ture will lead to a significantly reduction by avoidingthe o↵-line cost of approximating the Jacobian of thenonlinear operator using the EIM. In addition, ourapproach can provide a reduced-basis approximationto not only the solution of the PDE, but also its gradi-ent. Various numerical examples on backward SDEsand the corresponding quasilinear PDEs with param-eterized coe�cients are presented to demonstrate ef-fectiveness of our approach.

Kernel Methods for NonparametricAnalog Forecasting

Zhizhen ZhaoNew York University, USADimitrios Giannakis

Analog forecasting is a nonparametric technique in-troduced by Lorenz in 1969 which predicts the evolu-tion of observables of dynamical systems by followingthe evolution of samples in a historical record of ob-servations of the system which most closely resemblethe observations at forecast initialization. We discussa family of forecasting methods which improve tra-ditional analog forecasting by combining ideas fromkernel methods for machine learning and state-spacereconstruction for dynamical systems. A key in-gredient of our approach is to replace single-analogforecasting with weighted ensembles of analogs con-structed using local similarity kernels. The kernelsused here employ a number of dynamics-dependentfeatures designed to improve forecast skill, includingTakens‘ delay-coordinate maps (to recover informa-tion in the initial data lost through partial observa-tions) and a directional dependence on the dynamicalvector field generating the data. We illustrate thesetechniques in atmosphere ocean science applications.


Special Session 89: Dynamics and Computation

William D. Kalies, Florida Atlantic University, USAVincent Naudot, Florida Atlantic University, USA

Jason Mireles-James, Florida Atlantic University, USA

This session will focus on computational methods for analyzing dynamical systems, such as extractinginvariant structures, symbolic dynamics, and bifurcations. The aim of this session is to bring togetherexperienced and young researchers to exchange ideas and explore recent developments and applications.

Computing Invariant Circles of theArea Preserving Henon Map

David BlessingFlorida Atlantic University, USAJason Mireles-James

We examine the area preserving Henon map, wherenumerical experiments suggest the existence of con-tractable invariant circles. We use a weighted aver-aging scheme due to Yorke, Das, Dock, Saiki, Wu,Flores, and Sander in order to approximate the rota-tion number and Fourier coe�cients of the parame-terization of the circle. Then we apply a Newton-likemethod due to Haro and de la Llave in order to refinethe parameterization. This method is also adaptedto study periodic circles of the area preserving Henonmap (invariant sets where orbits fill several disjointcontractible circles densely). In this case we adaptthe techniques of Haro and de la Llave to simultane-ously compute parameterizations all the circles in away which does not require computing compositionsof the map.

Computational Aspects in the Re-stricted Four Body Problem

Jaime Burgos-GarciaInstituto Tecnologico Autonomo de Mexico, Mexico

The restricted four body problem (r4bp) studies thedynamics of a massless particle under the gravita-tional influence of three point masses that lie in anequilateral configuration provided by the well-knownhomographic solution of the general three body prob-lem. Recently there have been several preliminarystudies on the main objects of the dynamics: equilib-rium points, periodic orbits, invariant manifolds etc.However, the structure of these invariant objects isstill far to be well understood for all the admissi-ble values of the masses. In this talk we will showsome explorations of the above mentioned invariantobjects and we will discuss some perspectives on fur-ther explorations and their possible applications forthe solar system.

KAM Tori in Self-Consistent MapModels

Renato CallejaIIMAS-UNAM, MexicoDavid Martinez, Diego del Castillo, ArturoOlvera

I will present a Hamiltonian mean-field model. Themodel provides a simplified description of transportin marginally stable systems including vorticity mix-ing in strong shear flow and electron dynamics inplasmas. Self-consistency is incorporated through amean-field that couples all the degrees of freedom.The model is formulated as a large set of N coupledstandard-like twist maps. Invariant tori and theirbreakup play a central role in the study of globaltransport in these self-consistent map examples. Iwill present an algorithm to compute, continue andapproximate the breakdown of analyticity of invari-ant tori in a simplified version of a self-consistentmodel. This is joint work with Diego del Castillo,David Martinez and Arturo Olvera.

Inertial Manifolds Computations

Yu-Min ChungThe College of William and Mary, USA

An inertial manifold, first introduced by Foias, Sell,and Temam in 1988, is a finite-dimensional, exponen-tially attracting, and positively invariant Lipschitzmanifold. If a dynamical system possess an inertialmanifold, it is known that all long time behaviors,such as fixed points, limit cycles, and more impor-tantly, the global attractor, are contained in the in-ertial manifold. Moreover, when restricted dynamicson the inertial manifold, such system not only be-comes finite dimensional but also shares the samelong time behavior of the original system. Althoughits theory is well developed, the computation remainsa challenge problem. In this talk, we present re-cent progress on inertial manifolds computations, in-cluding algorithms, convergent analysis, implementa-tions, and some open questions.


Parametrization Method for Sta-ble/Unstable Manifolds of PeriodicPoints for Maps

Jorge GonzalezFlorida Atlantic University, USAJ.D Mireles James

The Parameterization Method is a general functionalanalytic framework for studying invariant manifoldsof dynamical systems. We develop a version ofthe method for stable/unstable manifolds associatedwith periodic points of discrete time dynamical sys-tems. The novelty of our approach is that by intro-ducing new variables we are able to avoid computingcompositions of the map. We describe the methodin general and implement the method for some oneand two dimensional manifolds in some two and threedimensional dynamical systems.The rigorous validations of our numerical computa-tions are established using functional analysis tech-niques and relying on the Radii Polynomial Theorem.This computer-assisted tool is a Newton-Kantorovichtype argument tailored to the field.

CAP-KAM Part II: on the Ap-plication of an A Posteriori KAMTheorem

Alex HaroUniversitat de Barcelona, SpainJordi-Lluıs Figueras, Alejandro Luque

We present a methodology to rigorously validate agiven approximation of a quasi-periodic Lagrangiantorus of an exact symplectic map. The approach con-sists in verifying the hypotheses of an a-posterioriKAM theorem based of the parameterization method(following Rafael de la Llave and collaborators). Acrucial point of our imprementation is an Approxi-mation Lemma that allows us to control the norm ofperiodic functions using their discrete Fourier trans-form. This an other technical aspects that are of in-dependent interest are presented by Alejandro Luquein this session. An outstanding consequence of thisapproach is that the computational cost of the vali-dation is assymptotically equivalent of the cost of thenumerical computation of invariant tori using the pa-rameterization method. We illustrate the methodol-ogy with several examples, as the standard map, thenontwist standard map and the Froeschle map.

Rabies SEIR with a Kick and Com-puting Chaos

Stephen IppolitoASTA, USA

In the field of epidemiology the standard dynamictechniques involve discrete maps, and continuousmodels such as ODEs. The intent of this work is topresent the mathematics necessary to study hybridsof these two models in the cases where complexity isbelieved to exist. In particular we study the spread

of rabies with the introduction of a birth pulse tothe system. Believing the resulting model to exhibitcomplex dynamics, we then consider techniques forcomputing the stable and unstable manifold of a sad-dle point of this hybrid map resulting in a transverseintersection.

Use of Lattice Structures of Attrac-tors and Other Techniques for anE�cient Computation of LyapunovFunctions for Morse Decomposi-tions

Dinesh KastiFlorida Atlantic University, USAArnaud Goullet, Shaun Harker, KonstantinMischaikow, William D. Kalies

Some recently developed techniques that lead us toform an e�cient algorithm to construct a piecewiseconstant Lyapunov functions for dynamics generatedby a continuous nonlinear map will be discussed. Thealgorithm uses a memory e�cient data structure forstoring nonuniform grids. It utilizes dijkstra algo-rithm with a dikstra distance that approximates themanhattan distance to compute distance potentialfunction, which is utilized to compute the Lyapunovfunction. We further prove that if the diametersof the grid elements go to zero, then the sequenceof piecewise constant Lyapunov functions generatedby our algorithm converge to a continuous Lyapunovfunction for the dynamics generated by the nonlin-ear map. We illustrate these techniques via the ap-plications on two problems from population biology.Finally, we will elaborate the use and importance oflattice structures of attractors for these techniques.

Rigorous Integration of MaterialSurfaces

Shane KepleyFlorida Atlantic University, USAWilliam Kalies, Jay Mireles-James

The evolution of a particle advected by an analyticvector field can be expressed as a Taylor series onsome interval in time. It is reasonable to expect thata higher dimensional smooth manifold of initial con-ditions should be analytic in both space and time.We show how the evolution of such a surface can becomputed as a sequence of multi-variate Taylor coef-ficients embedded in an appropriate Banach algebra.The computation is made rigorous by defining a suit-able Newton-like operator on this algebra and prov-ing it is a contraction. We will illustrate the methodwith an example and discuss applications which em-phasize two important features of our method: Inte-gration of non-autonomous flows and computation of“Lagrangian“ features such as material derivatives.


CAP-KAM Part I: RigorousComputer-Assisted Estimates inSmall Divisors Problems

Alejandro LuqueInstituto de Ciencias Matem‘aticas, SpainJordi Lluis Figueras, Alex Haro

The aim of this talk is to discuss several problemsthat arise when applying an a posteriori KAM the-orem in particular problems. We will consider thethree following situations:1) Obtaining rigorous and sharp estimates of analiticnorms for functions depending on multiple angles.To this end, we resolt to an Approximation Lemmabased on DFT.2) Assigning Diophantine constants to a frequencyvector. This vector may be given with finite preci-sion.3) Obtaining sharp Russmann’s estimates that im-prove the applicability of the KAM theorem.The above problems are part of a validation algo-rithm that will be presented and illustrated by AlexHaro in a subsequent talk of this session SS89. Thisis a joint work with Jordi-Lluis Figueras and AlexHaro.

Rigorous Numerics of Blow-Up So-lutions for ODEs

Kaname MatsueThe Institute of Statistical Mathematics, JapanAkitoshi Takayasu, Takiko Sasaki, KazuakiTanaka, Makoto Mizuguchi, Shin-ichi Oishi

This talk is concerned with blow-up solutions ofan autonomous system of ordinary di↵erential equa-tions. We propose a numerical verification methodfor constructing blow-up solutions with their blow-up times on the basis of the compactification andthe Lyapunov function validation. Blow-up solutionsare regarded as connecting orbits for the desingular-ized dynamics on the compactified space via com-pactification. Under appropriate assumptions, blow-up time can be estimated by Lyapunov tracing, are-parameterization technique of the time variable.The necessary criteria for the construction are en-sured with verified numerical computations.

Connecting Orbits Between Peri-odic Orbits for the Lorenz Equation

Maxime MurrayFlorida Atlantic University, USAJason Mireles-James, Jean-Philippe Lessard

In this talk I discuss the use Taylor/Fourier ex-pansions to compute parametrizations of the stableand unstable manifolds for periodic orbits. Thesesparametrisations are then used to recast the studyof connecting orbits into boundary value problems

that are solved with Chebyshev series. Finally, a-posteriori analysis is done to verify existence andtransversality of the manifolds and their intersec-tions. The Lorenz equation is used to give a concreteillustration of application for this method.

Invariant Manifold for Hybrid Maps

Vincent NaudotFlorida Atlantic University, USAJayson Mireles James, Qiuying Lu

We study, from a numerical point of view, the(un)stable manifolds for hybrid maps. This latteris the composition of a linear di↵eomorphism andthe time T of smooth vector field. Such systemsmodel physical processes where a di↵erential equa-tion is occasionally kicked by a strong disturbance.We propose a numerical method for computing theseinvariant manifold which leads to high order polyno-mial parameterization of the immersion. We obtaina representation of the dynamics on the manifold interms of a simple conjugacy relation. We illustratethe utility of the method by studying a planar exam-ple system, that is an Hamiltonian base hybrid map.

Computation of Lyapunov Con-stants in Switching Systems

Yun TianShanghai Normal University, Peoples Rep of ChinaPei Yu

In this talk, a new method with an e�cient algorithmis developed for computing the Lyapunov constantsof planar switching systems, and then applied tostudy bifurcation of limit cycles in a switching Bautinsystem. A complete classification on the conditionsof a singular point being a center in this Bautin sys-tem is obtained. Further, an example of switchingsystems is constructed to show the existence of 10small-amplitude limit cycles bifurcating from a cen-ter. This is a joint work with Pei Yu.

Periodically PerturbedHamiltonian-Hopf Bifurcation.

Arturo VieiroUniversitat de Barcelona, SpainE. Fontich, C. Simo

We shall consider the e↵ect of a periodic perturba-tion on a 2-dof autonomous Hamiltonian system un-dergoing a Hamiltonian-Hopf bifurcation. The sys-tem considered is obtained as a suitable truncationof the normal form plus a concrete perturbation.We will describe the asymptotic behaviour of the in-variant 2-dimensional stable/unstable manifolds andtheir splitting. The theoretical results will be com-pared with direct computations of the invariant man-ifolds. A careful analysis of the associated Poincare-Melnikov integral will provide a description of thesequence of parameters corresponding to changes onthe dominant harmonics of the splitting function.


Special Session 91: Harmonic Analysis and Partial Di↵erential Equations

William Bray, Missouri State University, USAMark A. Pinsky, Northwestern University, USA

The focus of this special session is to bring together experts to present and discuss the various deep facetsof the connections between harmonic analysis and partial di↵erential equations. Subtopics could include:asymptotics of Fourier transforms, Fourier integral operators, regularity theorems in PDE, and connectionswith geometric analysis.

Integrability of Fourier Integrals onEuclidean Space

William BrayMissouri State University, USA

Under what conditions (smoothness, regularity, de-cay) on a function defined on Euclidean space ofd dimension is it the Fourier transform of an inte-grable function? This problem cast in the realm ofclassical trigonometric series has a history spanning100+ years beginning with the work of W. Young(1913). Results on this problem for Fourier integralsand d = 1 are of more recent origin (e.g. Moricz(1992), Liflyand (1993)). In this talk recent resultsin the case d > 1 will be discussed and related toother known results.

Fractional Operators with SingularDrift: Smoothing Properties andMorrey-Campanato Spaces

Diego ChamorroUniversite d’Evry, FranceStephane Menozzi

We investigate some smoothness properties for atransport-di↵usion equation involving a class of non-degenerate Levy type operators with singular drift.Our main argument is based on a duality methodusing the molecular decomposition of Hardy spacesthrough which we derive some Holder continuity forthe associated parabolic PDE. This property will befulfilled as far as the singular drift belongs to a suit-able Morrey-Campanato space for which the regular-izing properties of the Levy operator su�ce to obtainglobal Holder continuity.

Lp Bounds for Wave Operatorsfor the Schrodinger Equation withThreshold Eigenvalue

William GreenRose-Hulman Institute of Technology, USAMichael Goldberg

The wave operators W± are a valuable tool for link-ing properties of the perturbed Schrodinger evolutioneitHPac(H) to properties of the corresponding freeevolution e�it�. We consider operators of the formH = ��+V on Rn, n � 5 which have an eigenvalueat zero.

It was recently proven by Yajima that, under theseconditions, the wave operators are bounded onLp(Rn) for all p 2 (1, n

2

). We recover this re-sult, including the p = 1 endpoint, and should thatthe upper end of the range can be expanded if theeigenspace satisfies certain orthogonality conditions.

Fourier Transform Estimates for 2DFunctions Defined Through Real-Analytic Functions and AssociatedPDE and PDE-Like Problems

Michael GreenblattUniversity of Illinois at Chicago, USA

We describe some Fourier transform decay estimatesfor a reasonably general class of functions definedthrough real-analytic functions in two dimensions.These functions are allowed to have singularities.The estimates are proved with the help of an ap-propriate resolution of singularities algorithm. Theyimprove on earlier work in that, among other things,they apply to globally defined functions and have anatural geometric interpretation which we will de-scribe. The estimates are sharp for a certain range ofindices in the theorems. We will also describe someapplications to associated PDE and PDE-like prob-lems.

A Remark on Geometric Separationwith Shearlets

Kanghui GuoMissouri State University, USADemetrio Labate

Shearlets may be viewed as directional wavelets.Shearlets is well known for its ability to representoptimally the cartoon-like images both in R2 and R3.In the literature, this fact helped to prove that Shear-lets can separate geometric objects. In these resultsof geometric separation, one could only handle theimages with piecewise linear edges. In this talk, wewill show that in the setting of the above results, thegeometric objects can be separated only if the theedges of the images are piecewise linear.


Deep Wavelet Scattering for Quan-tum Energy Regression

Matthew HirnMichigan State University, USAStephane Mallat, Nicolas Poilvert

Physical functionals are usually computed as solu-tions of variational problems or from solutions of par-tial di↵erential equations, which may require hugecomputations for complex systems. Quantum chem-istry calculations of molecular energies is such an ex-ample. Machine learning algorithms do not simulatethe physical system but estimate solutions by inter-polating values provided by a training set of knownexamples. However, precise interpolations may re-quire a number of examples that is exponential inthe system dimension, and are thus intractable. Thiscurse of dimensionality may be avoided by comput-ing interpolations in smaller approximation spaces,which take advantage of physical invariants. Wepresent a novel approach for the regression of quan-tum mechanical energies based on the scatteringtransform of an intermediate electron density repre-sentation. The scattering transform is composed ofiterated wavelet transforms and modulus operators,and possesses the appropriate invariant and stabilityproperties for molecular energy regression. Numer-ical experiments give state of the art accuracy overdata bases of organic molecules, while theoretical re-sults guarantee performance for the component of theenergy resulting from Coulombic interactions.

General Types of Spherical MeanOperator and K-Functionals ofFractional Orders

Thais JordaoUniversidade de Sao Paulo, BrazilXingping Sun

We design a general type of spherical mean oper-ators, depending on a real number as parameter,and employ them to approximate Lp class functions.We show that optimal orders of approximation areachieved via appropriately defined K-functionals offractional orders. Asymptotic relations between therate of approximation of the new operator and the K-functional of fractional order were stablished. Whenthe parameter we work with is taken as a naturalnumber the general type of spherical mean operator,the K-funcional and also the result relating such ob-jects turn out the same as in Dai & Ditzian (2004)which introduces a class of “multi-layered sphericalmean operators. More details can be found in Jordao& Sun (2015).

*Universidade de Sao Paulo, Brasil. Partially sup-ported by FAPESP #2014/06209-1.

Higher Dimensional ScatteringTheory and Integral RepresentationFormulas

Dorina MitreaUniversity of Missouri, USAE. Marmolejo-Olea, I. Mitrea, M. Mitrea

In this talk I will answer the following basic question:What are the optimal assumptions, of geometric andanalytic nature, which guarantee that a null-solutionu of the Helmholtz operator �+ k2 in an exteriordomain ⌦ can be represented in terms of layer po-tentials naturally associated with the said Helmholtzoperator and given domain? This work, at the inter-face between Geometric Measure Theory, HarmonicAnalysis, Scattering Theory, and Cli↵ord Analysis,generalizes and unifies classical results of Sommer-feld, Weyl, Muller, and Calderon.

The Role of Infinitesimal Flatness inthe Solvability of Elliptic BoundaryProblems in Uniformly RectifiableDomains

Marius MitreaUniversity of Missouri, USA

The goal of this talk is to illustrate the phenomenonthat uniform rectifiability together with infinitesimalflatness (understood as the demand that the outwardunit normal is close to having vanishing mean oscilla-tions) typically implies solvability results for ellipticboundary value problems formulated in such a geo-metric setting.

Action of a Scattering Map onWeighted Sobolev Spaces in thePlane

Katharine OttBates College, USARussell Brown, Peter Perry

We consider a scattering map that arises in the @approach to the scattering theory for the Davey-Stewartson II equation and show that the map is aninvertible map between certain weighted L2 Sobolevspaces.

Well-Distributed Points on Annuli.

Steven SengerMissouri State University, USAAlex Iosevich, Xingping Sun, Shelby Kilmer

We consider several classes of well-distributed pointsets in Rd, including lattice points, and present esti-mates on how many points from such sets could beincident with annuli.


Quasi-Monte Carlo Approximaitonof Continuous Functions

Xingping SunMissouri State University, USASteven Senger, Zongmin Wu

We design a class of quasi-Monte Carlo operators,and employ them to approximate continuous func-tions defined on domains with the interior cone con-dition. We show that such an approximation schemeachieve optimal orders in terms of the modulus ofcontinuity.


Special Session 92: Variational, Topological and Set-Valued Methods forNonlinear Problems

Pasquale Candito, Universita di Reggio Calabria, ItalyGiuseppina D’Agua, Universita di Messina, Italy

Roberto Livrea, Universita di Reggio Calabria, ItalySalvatore A. Marano, Universita di Catania, Italy

The aim of this session is to focus on the qualitative analysis of nonlinear problems, e.g., ordinary and partialdi↵erential equations, variational-hemivariational inequalities, di↵erence and algebraic systems. Emphasiswill be given to the results obtained by exploiting the synergy between the classical nonlinear analysismethods like the critical point theory, the fixed point theorems, the topological degree, the Morse theory, theset-valued analysis and so on. In particular, the existence, non-existence, multiplicity and sign informationof the solutions of a wide range of nonlinear problems will be studied.

High Multiplicity of Positive Peri-odic Solutions for Super-SublinearIndefinite Problems: a TopologicalApproach

Guglielmo FeltrinSISSA (Trieste), ItalyAlberto Boscaggin, Fabio Zanolin

We study the periodic boundary value problem as-sociated with the second order nonlinear di↵erentialequation

u00 +��a+(t)� µa�(t)

�g(u) = 0,

where g(u) has superlinear growth at zero and sub-linear growth at infinity. For �, µ positive and large,we prove the existence of 3m � 1 positive T -periodicsolutions when the weight function a(t) has m pos-itive humps separated by m negative ones (in a T -periodicity interval). As a byproduct of our approachwe also provide abundance of positive subharmonicsolutions and symbolic dynamics. The proof is basedon coincidence degree theory for locally compact op-erators on open unbounded sets and also applies toNeumann and Dirichlet boundary conditions.This is a joint work with Alberto Boscaggin (Univer-sity of Torino, Italy) and Fabio Zanolin (Universityof Udine, Italy).

On Polya’s Inequality for TorsionalRigidity and First Dirichlet Eigen-value

Vincenzo FeroneUniversita di Napoli Federico II, ItalyCarlo Nitsch, Cristina Trombetti

An inequality proved by Polya establishes a boundfrom above of the product between the torsionalrigidity and the first Dirichlet eigenvalue for thelaplacian in un domain with fixed measure. We willdiscuss the sharpness of the bound and the possibilityof improving it in suitable classes of domains.

Twist Conditions for a HigherDimensional Poincare–Birkho↵Theorem: an Avoiding Cones For-mulation

Paolo GidoniSISSA (Trieste), ItalyAlessandro Fonda

In this talk we discuss how the concept of twist can beextended to higher dimensional systems, in particu-lar when considering generalizations of the Poincare–Birkho↵ Theorem for 2N -dimensional Hamiltoniansystems. Following the spirit of similar results ob-tained for Poincare–Miranda-like fixed point theo-rems, I present a new kind of boundary condition,called avoiding cones condition, that unifies and ex-tend the higher dimensional twist conditions previ-ously proposed for the Poincare–Birkho↵ Theorem.

Existence Theorems for Ellipticand Evolutionary Variational andQuasi-Variational Inequalities

Akhtar KhanRochester Institute of Technology, USADumitru Motreanu

This talk gives new existence results for ellipticand evolutionary variational and quasi-variational in-equalities. Specifically, we give an existence theo-rem for evolutionary variational inequalities involv-ing di↵erent types of pseudo-monotone operators.Another existence result embarks on elliptic varia-tional inequalities driven by maximal monotone oper-ators. We propose a new recessivity assumption thatextends all the classical coercivity conditions. Wealso obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way el-liptic and evolutionary problems.


Critical Groups Under Saddle PointReduction and Applications to El-liptic Resonant Problems

Shibo LiuXiamen University, Peoples Rep of China

Infinite dimensional Morse theory is very useful instudying nonlinear equations. The basic conceptsin the theory are critical groups at isolated criticalpoints and critical groups at infinity. On the otherhand, saddle point reduction is a powerful tool in crit-ical point theory. With this reduction, the problemof finding critical points of a functional reduces tofinding critical points for a reduced functional defin-ing in s subspace. To combine the Morse theory andthe reduction method, a natural problem is the rela-tion between the critical groups of the original func-tional and the reduced functional. In this talk, wewill present our results on this topic. It turns out thatthe critical groups are almost isomorphic. As appli-cation, we study some elliptic resonant problems withvariable coe�cients, where the energy functional maynot satisfy the PS condition. We obtain multiple so-lutions for such problems.

Symmetry Breaking for a Problemin Optimal Insulation

Carlo NitschUniversity of Napoli Federico II, ItalyDorin Bucur, Giuseppe Buttazzo

We consider the problem of optimally insulating agiven domain; this amounts to solve a nonlinear vari-ational problem, where the optimal thickness of theinsulator is obtained as the boundary trace of thesolution. We deal with two di↵erent criteria of opti-mization: the first one consists in the minimization ofthe total energy of the system, while the second oneinvolves the first eigenvalue of the related di↵eren-tial operator. Surprisingly, the second optimizationproblem presents a symmetry breaking in the sensethat for a ball the optimal thickness is nonsymmetricwhen the total amount of insulator is small enough.

N-Laplacian Problems with CriticalTrudinger-Moser Nonlinearities

Kanishka PereraFlorida Institute of Technology, USAYang Yang

We prove existence and multiplicity results for a N -Laplacian problem with a critical exponential nonlin-earity that is a natural analog of the Brezis-Nirenbergproblem for the borderline case of the Sobolev in-equality. This extends results in the literature forthe semilinear case N = 2 to all N � 2. WhenN > 2 the nonlinear operator ��N has no linear

eigenspaces and hence this extension requires newabstract critical point theorems that are not basedon linear subspaces. We prove new abstract resultsbased on the Z

2

-cohomological index and a relatedpseudo-index that are applicable here.

Some Existence Results of InfinitelyMany Solutions to Elliptic Problemswith p(x)�Laplacian and Nonhomo-geneous Neumann Conditions

Angela SciammettaUniversita degli Studi di Messina, ItalyGiuseppina D’Agui

The aim of this talk is to establish the existence of anunbounded sequence of weak solutions for a class ofdi↵erential equations with p(x)�Laplacian and sub-ject to small perturbations of nonhomogeneous Neu-mann conditions. The approach is based on varia-tional methods.

Remarks on the Ambrosetti-ProdiPeriodic Problem

Elisa SovranoUniversity of Udine, ItalyFabio Zanolin

In 2011 a very interesting note of Antonio Ambrosettiin honor of Giovanni Prodi appeared along with alist of open questions about global inversion the-orems and their applications. One of these ques-tions regards the study of the periodic Ambrosetti-Prodi problem for an ordinary di↵erential second or-der equation. Our contribution to this problem con-cerns the study of the equation: u00 + f(u) = p(t)where p(t) is a T -periodic stepwise forcing term andthe nonlinearity f is a locally Lipschitz continuousfunction such that f(0) = 0, f(s) > 0 for all s 6= 0,lim

s!±1f(s) = +1 and it is strictly decreasing for

s 0 and strictly increasing for s � 0. Assumingthat the nonlinear term is a positive function withglobal minimum at zero which satisfies the previousgrowth conditions, we prove under suitable condi-tions on p(t) the existence of infinitely many periodicsolution. Moreover, we show the presence of chaotic-like dynamics via topological methods.

Fractional Inclusions with Impulsesand Nonlocal Boundary Conditionsin a Banach Space

Valentina TaddeiUniversity of Modena and Reggio Emilia, ItalyI. Benedetti, V. Obukhovskii

As it is well known, fractional equations are usedto describe anomalous di↵usion with long-range ef-fects as well as memory or hereditary properties ofvarious processes. On the other hand, there aremany phenomena characterized by parameters sub-ject to short-term perturbations, which are repre-


sented through impulses. In this talk we give anexistence result for fractional inclusions in abstractspaces with impulses and non-local boundary condi-tions. We apply a technique based on weak topol-ogy to avoid any compactness assumption, usuallyrequired to use topological tools in abstract spaces.Applications to integro-partial-di↵erential inclusionscoming from population dynamic are given.

The Neumann Eigenvalue Problemfor the 1-Laplacian

Cristina TrombettiUniversity of Napoli Federico II, ItalyLuca Esposito, Bernd Kawohl, Carlo Nitsch

The first nontrivial eigenfunction of the Neumanneigenvalue problem for the p-Laplacian converges, asp goes to1, to a viscosity solution of a suitable eigen-value problem for the 1-Laplacian. We show amongother things that the limiting eigenvalue is in fact thefirst nonzero eigenvalue, and derive a number conse-quences, which are nonlinear analogues of well-knowninequalities for the linear (2-)Laplacian.

Global a Priori Bounds for WeakSolutions to Quasilinear ParabolicEquations with NonstandardGrowth

Patrick WinkertUniversity of Technology Berlin, GermanyRico Zacher

In this talk we study a rather wide class of quasilinearparabolic problems with nonlinear boundary condi-tion and nonstandard growth terms. It includes theimportant case of equations with a p(t, x)-Laplacian.By means of the localization method and De Giorgi’siteration technique we derive global a priori boundsfor weak solutions of such problems. Our results seemto be new even in the constant exponent case.


Special Session 93: Nonlinear Dispersive Equations and IntegrableSystems

Joachim Escher, Leibniz Universitat Hannover, GermanyZhaoyang Yin, Sun Yat-Sen University and Macau University of Science and Technology,

Peoples Rep of ChinaZhijun Qiao, University of Texas - Rio Grande Valley, USA

The session is devoted to recent developments of the analysis of water waves, with the particular focus onoceanic waves, solitons, integrable systems and related PDEs. Topics include the qualitative mathematicalanalysis, such as local and global well-posedness, regularity, stability, asymptotic behavior of solutions,integrability and solitary waves.

Classification of Integrable 2-Component Peakon Equations fromLax Pairs

Stephen AncoBrock University, CanadaFatane Moberashermini

A classification is presented for integrable 2-component peakon equations arising from 2x2 Laxpairs

Multipeakons of a Two-ComponentModified Camassa-Holm Equationand The Relation with the FiniteKac-Van Moerbeke Lattice

Xiangke ChangUniversity of Saskatchewan, CanadaXingbiao Hu, Jacek Szmigielski

This talk is concerned about a two-component modi-fied Camassa-Holm equation, which was proposed bySong, Qu and Qiao. A spectral and the inverse spec-tral problem are studied for the two-component mod-ified Camassa-Holm type for measures associated tointerlacing peaks. It is shown that the spectral prob-lem is equivalent to an inhomogenous string problemwith Dirichlet/Neumann boundary conditions. Theinverse problem is solved by Stieltjes‘ continued frac-tion expansion, leading to an explicit construction ofpeakon solutions. Su�cient conditions for the globalexistence in t are given. The large time asymptoticsreveals that, asymptotically, peakons break into two-peakon bound-states moving with constant speeds.The peakon flow is shown to project to one of theisospectral flows of the finite Kac-van Moerbeke lat-tice (or called the finite Lotka-Volterra lattice or theLangmuir lattice).

Analyticity, Gevrey Regularity andUnique Continuation for an Inte-grable Multi-Component PeakonSystem with an Arbitrary Polyno-mial Function

Qiaoyi HuSouth China Agricultural University, Peoples Rep ofChinaZhijun Qiao

In this paper, we study the Cauchy problem for anintegrable 2N-component peakon system which is in-volved in an arbitrary polynomial function. Basedon a generalized Ovsyannikov type theorem, we firstprove the existence and uniqueness of solutions forthe system in the Gevrey-Sobolev spaces with thelower bound of the lifespan. Then we show the con-tinuity of the data-to-solution map for the system.Furthermore, by introducing a family of continuousdi↵eomorphisms of a line and utilizing the fine struc-ture of the system, we demonstrate the system ex-hibits unique continuation.

High-Order Approximations ofTraveling Water Waves

Konstantinos KalimerisRICAM, Austrian Academy of Sciences, Austria

In this talk we consider the classical water wave prob-lem described by the Euler equations with a free sur-face under the influence of gravity over a flat bot-tom. We restrict our attention to two-dimensional,finite-depth periodic water waves with general vortic-ity. We formulate this problem as nonlinear (fixed)boundary value problem, through a semi-hodographtransformation. An asymptotic technique is appliedto approximate the solutions of this problem that cor-respond to non-laminar flows. We provide high-orderapproximations to periodic traveling wave profiles,depending on the total mechanical energy of the wa-ter wave. Moreover, we provide the velocity field andthe pressure beneath the waves, in flows with con-stant vorticity over a flat bed.


Control Theory of Time-VaryingLinear Systems in the Frame Workof Nest Algebra

Liu LiuDalian University of Technology, Peoples Rep ofChinaYufeng Lu

As the development of H1 control theory, a lot ofinsight has been obtained by considering its time-varying analogue on an appropriate complex Hilbertspace of input-output signals. In the context of op-erator theory, the algebra of stable, causal, discrete(continuous) time, time-varying linear systems is infact the discrete (continuous) nest algebra. In thistalk, we introduce the connection between controltheory and nest algebra theory, some properties ofthe system representations, factorizations and givesome new stabilizability criteria for the time-varyinglinear systems in the framework of nest algebra.

Well-Posedness and Global Solu-tions in Fluid Models with Vorticity

Tony LyonsWaterford Institute of Technology, IrelandJoachim Escher, David Henry, Boris Kolev

In this talk we present a two-component model forshallow-water waves incorporating vorticity. It willbe outlined how the model may be interpreted as ageodesic flow on a right invariant metric of the groupof smooth di↵eomorphisms of the circle. The geomet-ric interpretation of the fluid model also ensures thewell-posedness of the model, while the existence ofglobal solutions can be shown to follow from a prioriestimates.

Extremal Norms of the PotentialsRecovered from Inverse DirichletProblems

Jiangang QiShandong University at Weihai, Peoples Rep ofChinaShaozhu Chen

Consider the Sturm-Liouville eigenvalue problem�y“(x) + q(x)y(x) =n y(x), x 2 [0, 1], y(0) =y(1) = 0, where q 2 L1[0, 1], and denote its spectrumby �(q). For a real number �, define ⌦ (�) = {q 2L1[0, 1] : � 2 �(q)} and E(�) = inf{kqk : q 2 ⌦(�)}.We will set up a formula for E(�) explicitly in termsof � and specify where the infimum can be attained.As an application, we will give extremal values of thenth eigenvalue of the Dirichlet problem for poten-tials on a sphere in L1[0, 1], n � 1. The proofs arebased on a new Lyapunov-type inequality for Sturm-Liouville equations with potentials.

Steady Periodic Water Waves forFixed-Depth Rotational Flows withDiscontinuous Vorticity

Silvia Sastre-GomezUniversity College Cork, IrelandD. Henry

In this work we study steady two-dimensional peri-odic water waves problems over a fixed depth withthe vorticity discontinuous. We consider a modifiedheight function, which explicitly introduces the meandepth into the rotational water wave problem. Sincethe vorticity is discontinuous, the equations are ex-pressed in a weak form and the solutions are consid-ered in the sense of distributions. We use Crandall-Rabinowitz local bifurcation to prove the existenceof weak solutions.

Blow-Up Results and Soliton So-lutions for a Generalized VariableCoe�cient Nonlinear SchroedingerEquation

Erwin SuazoUniversity of Texas, RGV, USAJose Escorcia

In this talk, by means of similarity transformationswe study exact analytical solutions for a generalizednonlinear Schroedinger equation with variable coe�-cients. This equation appears in literature describingthe evolution of coherent light in a nonlinear Kerrmedium, Bose-Einstein condensates phenomena andhigh intensity pulse propagation in optical fibers. Byrestricting the coe�cients to satisfy Ermakov-Riccatisystems with multiparameter solutions, we presentconditions for existence of explicit solutions with sin-gularities and a family of oscillating periodic soliton-type solutions. Also, we show the existence of bright-, dark- and Peregrine-type soliton solutions, and bymeans of a computer algebra system we exemplifythe nontrivial dynamics of the solitary wave centerof these solutions produced by our multiparameterapproach.

Spectra of a Class of Non-Symmetric Operators in HilbertSpaces with Applications to Singu-lar Di↵erential Operators

Huaqing SunShandong University at Weihai, Peoples Rep ofChinaBing Xie

This paper is concerned with a class of non-symmetric operators. A su�cient condition forpoints of the essential spectrum of a J -symmetricoperator is given in terms of the numbers of linearlyindependent solutions of certain homogeneous equa-tion, and a characterization for points of the essen-tial spectrum plus the set of all eigenvalues of a J -


symmetric operator is obtained in terms of the num-bers of linearly independent solutions of certain inho-mogeneous equation. These results are established ina purely operator-theoretic setting. Therefore, theycan be used for any J -symmetric di↵erential expres-sions for whatever J is taken. They are appliedto singular J -symmetric Hamiltonian systems andits special form of singular Sturm-Liouville equationswith complex-valued coe�cients, where J is specifiedas the usual operation of complex conjugation in thecorresponding spaces. Furthermore, three concreteexamples are provided to illustrate these results.

Riemann-Hilbert Approach forthe FQXL Model: a GeneralizedCamassa-Holm Equation with Cu-bic and Quadratic Nonlinearity

Zhen WangDalian University of Technology, Peoples Rep ofChinaQiao Zhijun

In this paper, the inverse scattering transform as-sociated with a Riemann-Hilbert problem is formu-lated for the FQXL model: a generalized Camassa-Holm equation mt = 1

2

k1

[m(u2�u2

x)]x+1

2

k2

(2mux+mxu), m = u � uxx, which was originally includedin the work of Fokas and recently shown integrablein the sense of Lax pair, bi-Hamilton structure, andconservation laws by Qiao, Xia, and Li. Moreover,the parametric multi-soliton solutions are presentedfor the case of reflectionless potentials.

Qualitative Analysis of a Time-Delayed Free Boundary Problem forTumor Growth Under the Action ofExternal Inhibitors

Shihe XuZhaoqing University, Peoples Rep of ChinaQinghua Zhou, Meng Bai

In this paper we consider a time delayed free bound-ary problem for tumor growth under the action ofexternal inhibitors. It is assumed that the processof proliferation is delayed compared to apoptosis.The existence and uniqueness of a global solution isproved. Morever, the asymptotic behavior of the so-lution is studied.

Submodule of Hardy Space Overthe Bidisc

Yixin YangDalian University of Technology, Peoples Rep ofChina

Submodules of H2(D2) are well known to be verycomplicated. We can construct submodules from thepoint of view of operator model theory of B. Sz.-Nagyand C. Foias. For a B(H2(w))-valued inner function⇥(z), M = ⇥ (z)H2

H2(w)

(D) is a Tz-invariant subspace

of H2(D2), but in general M is not a submodule ofH2(D2). For some⇥( z), it may produce some newsubmodules and have very interesting properties. Byusing this type of submodules, we show that multi-variable analogue of the Berger-Shaw theorem is notlikely to be possible.

Remarks on the Well-Posedness ofCamassa-Holm Type Equations inBesov Spaces

Zhaoyang YinMacau University of Science and Technology, MacauJinlu Li

In this paper, we prove the solution map of theCauchy problem of Camassa-Holm type equations de-pends continuously on the initial data in nonhomo-geneous Besov spaces in the sense of Hadamard byusing the Littlewood-Paley theory and the methodintroduced by Kato and Danchin.

From the Solutions to Constructthe Schrodinger-Like Equation withSource Term and Its NumericalSimulations

Fajun YuSchool of Mathematics and Systematic Sciences,Shenyang Normal University, Peoples Rep of China

We present a brand new method to construct theSchrodinger-like equations from a solution in this pa-per. Some Schrodinger-like equations with sourcesare derived by using a generalized solution.And weprove that the equation with source term has a weaksolution. At last, the numerical simulations on theevolution and solitons collision of rogue wave solu-tions are performed to verify the prediction of theanalytical formulations.

Somes Explicit Solutions of a Finite-Dimensional Integrable System

Jinshun ZhangHuaqiao University, Peoples Rep of China

A finite-dimensional Hamiltonian system associatedwith CKdV equation is considered. It’s an integrablesystem in Liouville sense. We present an e↵ectivemethod to construct explicit solutions of the system.Some explicit solutions are obtained.


On the Cauchy Problem for a Gen-eralized Cross-Coupled Camassa-Holm System with N-Peakons andHigher-Order Nonlinearities

Shouming ZhouChongqing Normal University, Peoples Rep of ChinaZhijun Qiao and Chunlai Mu

In present talk, we study the Cauchy problemfor a generalized cross-coupled Camassa-Holm sys-tem with peakons and higher-order nonlinearities.By the transport equations theory and the classi-cal Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomoge-neous Besov spaces Bs

p,r ⇥ Bsp,r with 1 p, r +1

and s > max{2+ 1

p, 5

2

} is obtained. Moreover, the lo-

cal well-posedness in critical Besov space B5/22,1 ⇥B5/2

2,1

and the blow-up criteria are also established. Ourother purpose is to consider the well-posedness inthe sense of Hadamard, non-uniform dependence andHolder continuity of the data-to-solution map for thissystem on both the periodic and the nonperiodiccase. Using a Galerkin-type approximation scheme,

it is shown that this equation is well-posed in Sobolevspaces Hs ⇥ Hs, s > 5/2 in the sense of Hadamard,that is, the data-to-solution map is continuous. Fur-thermore, in conjunction with the well-posedness es-timate, it is also proved that this dependence is sharpby showing that the solution map is not uniformlycontinuous. Finally, the Holder continuous in theHr ⇥ Hr topology when 0 r < s with Holder ex-ponent ↵ depending on both s and r are shown.

Global Existence of Solutions andSteady States of the Nonlinear Size-Structured Population Models withDistributed Delay in the Recruit-ment

Qinghua ZhouZhaoqing University, Peoples Rep of ChinaMeng Bai, Minhai Huang

We study the nonlinear size-structured populationmodels with distributed delay in the recruitment.The global existence of solutions and the non-trivialsteady states of the models are obtained by using thesemigroup techniques in Banach spaces.


Special Session 94: Infinite Dimensional Dynamics in Analysis

Cho-Ho Chu, Queen Mary University of London, England

In recent years, there has been active research in complex and functional analysis relating to linear and non-linear dynamics, in particular, in infinite dimensional holomorphic dynamics and C*-dynamical systems.The aim of the special session is to bring together researchers in these fields and related areas, for instance,di↵erential equations and operator theory, to inform recent progress and explore potential links betweentheir research areas.

Viscous Dynamics in RelativisticFluids

Shabnam BeheshtiQueen Mary University London, England

It is known that viscous e↵ects lead to nontrivialdynamical behaviour in homogeneous cosmologicalmodels such as FLRW and Bianchi spacetimes. Re-cent progress by in well-posedness of certain Einstein-Navier-Stokes systems motivates revisiting modelsinvolving dynamic velocities, first proposed by Lich-nerowicz in 1967. Using a dynamical systems ap-proach, we investigate the role of dynamic velocityin a cosmological background; we demonstrate thatthe additional degree of freedom a↵orded by an as-sociated index plays a key, geometric role in the evo-lution of relativistic fluids. An important open ques-tion involves finding admissibility conditions for theequation of state of the system, possibly via thermo-dynamic methods for holomorphic maps.

Dynamics of Maps

Cho-Ho ChuQueen Mary, University of London, England

The dynamics of a map f : X ! X studies the be-haviour of its iterates (fn). The main question iswhether one can predict the fate of all orbits of f ,where an orbit of a point p in X is the sequence(fn(p)). This is a di�cult but interesting question.We will discuss some examples in which f is a trans-lation map or a smooth map.

Young Diagram Di↵erential Opera-tors

Arran HammWinthrop University, USAShabnam Beheshti

A Young diagram is a collection of cells arranged inleft-justified rows with row length weakly decreasingdown the rows. In this talk I will discuss a classof di↵erential operators defined in connection withYoung diagrams. Using these operators, equationsfrom the KP hierarchy take on surprisingly simpleforms. Additional properties of these operators willbe discussed including a “mixed partial derivative“formula which, when applied repeatedly, gives riseto a natural combinatorial identity. Joint work withShabnam Beheshti.

Holomorphic Mappings on a Com-plex Banach Space

Tatsuhiro HondaHiroshima Institute of Technology, Japan

Let X be a complex Banach space with the unit ballB. The family M is a natural generalization to com-plex Banach spaces of the well known Caratheodoryfamily of functions with positive real part on the unitdisc U in C. We consider subfamilies of M, anddiscuss some properties for holomorphic mappingsf : B ! X which belongs to the subfamilies.

Extremal Problems, LoewnerChains and The Loewner Di↵er-ential Equation in Cn and ComplexBanach Spaces

Gabriela KohrBabes-Bolyai University, Cluj-Napoca, RomaniaIan Graham, Hidetaka Hamada, Mirela Kohr

In this talk we survey classical and also modern re-sults in the theory of Loewner chains and the gener-alized Loewner di↵erential equation on the unit ballin Cn and complex Banach spaces. We also presentrecent applications in the study of extremal prob-lems associated with compact families of normalizedbiholomorphic mappings, which have parametric rep-resentation on the unit ball in Cn. Finally, we pointout certain open problems and conjectures relatedto Loewner chains, Herglotz vector fields, and thegeneralized Loewner di↵erential equation in Cn andreflexive complex Banach spaces.

A Note on Kirwan’s Conjecture

Yuk LeungUniversity of Delaware, USA

Using Schi↵er’s modification of Loewner’s chain, Kir-wan and Schober obtained a new set of extremal co-e�cient problems on the class ⌃ of functions

z +1X

n=0

bnzn

of functions that are univalent and analytic in |z| > 1.Based on their results, William Kirwan proposed thatthe real part of nb

1

�bn n must hold for every func-tion in ⌃ with equality holding for the Koebe functionK(z) = z + 2 + 1/z.


In this talk, we develop a second variational for-mula of the Koebe function using the classicalLoewner’s di↵erential equation. Together with vari-ous known identities of the hyper-geometric polyno-mials P (0,1)

n (x), we prove that Kirwan’s conjecture istrue for functions in ⌃ that are close to the Koebefunction.

Inverse of Disjointness PreservingOperators

Lei LiNankai University, Peoples Rep of China

A linear operator between function spaces is disjoint-ness preserving if it maps disjoint functions to dis-joint functions. Here, two functions are said to bedisjoint if at least one of them vanishes at each point.I will talk about the linear disjointness preservingoperators between various types of function spaces,including spaces of (little) Lipschitz functions, uni-formly continuous functions and di↵erentiable func-tions. It is shown that a disjointness preserving linearisomorphism whose domain is one of these types ofspaces (scalar-valued) has a disjointness preservinginverse.

Continuous Orbit Equivalence

Xin LiQueen Mary University of London, England

We discuss the notion of continuous orbit equiva-lence, and its relationship to Geometric Group The-ory as well as Cartan subalgebras of C*-algebras.

Picard Type Theorems and Appli-cations to Certain PDEs

Bao Qin LiFlorida International University, USA

A connection between Picard’s theorem and entire so-lutions of certain functional equations will be given,from which Picard type theorems are then derivedand applied to certain nonlinear partial di↵erentialequations.

Complex Oscillation and ExplicitDetermination of Semi-Finite BandGap Solutions of the Whittaker-HillEquation

Xudan LuoThe Hong Kong University of Science and Technol-ogy, Hong KongYik-Man Chiang

In this talk, we discuss the general solution ofWhittaker-Hill equation under confluent hypergeo-metric basis. According to certain initial conditionsand boundary conditions, we get four special solu-tions which are either even or odd and either peri-odic or semi-periodic. We have extended Ince’s work

in 1923. Moreover, we show that the solutions ofWhittaker-Hill equation with finite exponent of con-vergence of zero-sequence correspond to the termi-nating solutions of Whittaker-Hill equation. Finally,the semi-finite band gap problem of Whittaker-Hillpotential is considered also.

Lie Structures in Operator Algebras

Lina OliveiraUniversidade de Lisboa, PortugalM. Santos

In this talk we consider reflexive operator algebrashaving subspace lattices of fixed types and their Liemodules. It will be shown how the interplay betweenthe subspace lattice of the algebra and its Lie alge-braic structure leads to establishing a decompositiontheorem for Lie modules along the lines of that ob-tained in the 90’s by Hudson–Marcoux–Sourour forLie ideals.

The Inverse Linearization Problem

Maxim OlshaniiUMass Boston, USA

We investigate the relationship between the nonlinearpartial di↵erential equations (PDEs) of mathematicalphysics and the their linearizations around stationarylocalized solutions. It turns out that for some classesof PDEs, it is possible to solve the Inverse Lineariza-tion Problem, i.e. given the linearization, to restorethe original PDE. Of a particular interest are the in-stances of transparency of the former that are shownto hint on the possible integrability of the latter.

On the Structure of C0-Semigroupsof Holomorphic Carath‘eodoryIsometries

Laszlo StachoUniversity of Szeged, Hungary

We extend Vesentini’s description of the infinitesi-mal generators of strongly continuous one-parametersemigroups of holomorphic Caratoodory isometriesof the unit ball of a complex Hilbert space to thesetting of reflexive Cartan factors. Our treatmentis based on intensive use of joint fixed points alongwith Kaup type ideas with partial vector fields ofsecond degree. In particular we establish closed for-mulas for the Hilbert space case in terms of spec-tral resolutions of skew self-adjoint dilations relatedto the Reich-Shoikhet non-linear infinitesimal gen-erator. We also provide partial results toward aHille-Yosida type theory for holomorphic self-mapsof bounded domains in Banach spaces.


Symmetries of the Darboux Equa-tion

Chiu Yin TsangThe Hong Kong University of Science and Technol-ogy, Hong KongYik-Man Chiang, Avery Ching

The Darboux equation (1882) was a generalizationof both Picard’s and Hermite’s equations. All theseequations are generalizations of the well-known Lameequation (1837). The equation was rediscoveredby Treibich and Verdier in the 1980s concerning ithaving finite-gap property in an algebraic geometriccharacterization. The equation is a (doubly periodic)torus version of the Heun equation which lives onthe Riemann sphere. It turns out that the Darbouxequation has a better symmetry structure comparedto that of the Heun equation.In this talk, we will describe the symmetry of theDarboux equation via the study of the transforma-tions which induce the automorphisms of the Dar-boux equation. We show how to apply the automor-phisms to generate the 192 local solutions of the Dar-boux equation. This is a joint work with Yik-ManChiang and Avery Ching.

Evolution Algebras of ArbitraryDimension

M. Victoria VelascoUniversity of Granada, Spain

In this talk we discuss the structure of the evo-lution algebras or arbitrary dimension, a type ofnon-associative algebras (which are nor even power-associative) that, dynamically, represent discrete dy-namical systems. These algebras have deep connec-tions with the graph theory, the stochastic processes,the mathematical physics and some other branchs ofthe Science. They emerged to enlighten the study ofnon-Mendelian inheritance in genetics.

Shift Operators on ContinuousFunctions

Ngai-Ching WongNational Sun Yat-sen University, TaiwanLi-Shu Chen, Jyh-Shyang Jeang

We study (quasi-)n-shift operators on C0

(X), whereX is a locally compact and Hausdor↵ space. WhenC

0

(X) admits a disjointness preserving quasi-n-shiftT , there is a countable subset of X1 = X [ {1 }

equipped with a tree-like structure, called '-tree,with exactly n joints such that the action of T onC

0

(X) can be implemented as a shift on the '-tree.If T is an n-shift, then the '-tree is dense in X andthus X is separable. By analyzing the structure ofthe '-tree, we show that every (quasi-)n-shift on c

0

can always be written as a product of n (quasi-)1-shifts. Although it is not the case for general C

0

(X)as shown by our counter examples, we can do so afterdilation. We also study the case when the shift is anisometry instead. In this case, we will work on a shifton a tree structure of the dual space M(X) of C

0

(X)instead.

On Hausdor↵ Dimension of theJulia Set of Entire Functions withFast Growth

Zhuan YeNorthern Illinois Univeristy, USAJie Ding, Jun Wang

We construct a family of transcendental entire func-tions which lie outside the Eremenko-Lyubich classin general and are of infinity growth order. Mostimportantly, we show that the intersection of Juliaset and escaping set of these entire functions has fullHausdor↵ dimension. We also obtain some theoremson Hausdor↵ measures of the intersection of Julia setand escaping set.

A Galoisian Approach to ComplexOscillation Theory

Chiang Yik ManHong Kong University of Science and Technology,Hong KongGuofu Yu

We demonstrate that complex non-oscillatory solu-tions (in the sense of Nevanlinna theory) of certainclass of Hill equations are among the Liouvillian so-lutions of associated di↵erential equations. We shallestablish a full equivalence between the two view-points when the Hill potential is a linear combina-tion of four exponential functions. This equation isclosely related to the classical Lame and Mathieuequations. We shall also discuss new orthogonalityfound for these non-oscillatory solutions.


Special Session 95: PDEs from Gauge Field Theories and MathematicalPhysics

Jongmin Han, Kyung Hee University, KoreaNamkwon Kim, Chosun University, Korea

This special session deals with recent advances in PDEs from mathematical physics. Special focuses arebrought into PDEs from (2+1) dimensional gauge field theories including Maxwell-Chern-Simons models,SU(2), SU(3), O(3) models, etc. Popular topics are existence and properties of solutions to self-dual equa-tions, non-self-dual equations, and time-dependent problems. Related topics for self-dual equations includeelliptic systems involving exponential nonlinearity in two dimension. This session also deals with other PDEsfrom various fields of mathematical physics such as fluid mechanics.

Short Time Regularity to the Equa-tions of Unsteady Motion of ShearThickening Incompressible Fluids

Hyeong-Ohk BaeAjou University, KoreaJorg Wolf

We address the existence of strong solutions to asystem of equations of motion of an incompressiblenon-Newtonian fluid. Our aim is to prove the short-time existence of strong solutions for the case of shearthickening viscosity, which corresponds to the powerlaw ⌫(D) = |D|q�2 (2 < q2.23 · · · . The results areobtained by flattening the boundary and by using thedi↵erence quotient method. Near the boundary, weuse weighted estimates in the normal direction.

Asymptotic Self-Similarity of EntireSolutions for Quasilinear Equationswith Exponential Nonlinearity

Soohyun BaeHanbat National University, Korea

We consider the asymptotic self-similarity of entiresolutions for quasilinear equations with exponentialnonlinearity. The main result is described for p�Laplace equation with exponential nonlinearity.

Small Data Global Existence andDecay for Relativistic Chern–Simons Equations

Myeongju ChaeHankyong Natinal University, KoreaSung-Jin Oh

We establish a general small data global existenceand decay theorem for Chern–Simons theories witha general gauge group, coupled with a massive rela-tivistic field of spin 0 or 1/2. A key idea is to developand employ a gauge invariant vector field method forrelativistic Chern–Simons theories, which allows usto avoid the long range e↵ect of charge.

Uniqueness of Positive Solution toa Coupled Cooperative VariationalElliptic System on an Interval

Jann-Long ChernNational Central University, TaiwanJunping Shi, Yulian An

Oscillatory behavior of solutions of linearized equa-tions for cooperative semilinear elliptic systemsof two equations on one-dimensional domains areproved, and it is shown that the stability of the pos-itive solutions for such semilinear system is closelyrelated to the oscillatory behavior. These proper-ties are used to prove the uniqueness of positive so-lutions to some semilinear elliptic systems with non-linearities satisfying certain variational structure andgrowth conditions.This talk is based on the joint work with Profs. Jun-ping Shi and Yulian An.

Mixed Type Solutions in the Self-Dual SU(3) Chern-Simons Theory

Kwangseok ChoeInha University, KoreaNamkwon Kim, Chang-Shou Lin

We consider an elliptic system arising from thenonabelian relativistic self-dual SU(3) Chern-Simonstheory. This system is given on R2, and it consistsof two elliptic equations with exponential nonlinear-ities. We review recent progress on the existence ofradially symmetric solutions (u

1

, u2

) satisfying themixed-type boundary condition near1; u

1

convergeswhile u

1

tends to �1 near 1.


Radial Solutions for the Gravita-tional Maxwell Gauged O(3) SigmaModel

Na Ri ChoiJongmin Han

In this paper, we study an elliptic equation arisingfrom the self-dual equations for the Maxwell gaugedO(3) sigma model coupled with gravitation. We com-pletely classify all radial solutions for single vortexcase according to values of aN where a is a scaledgravitational constant and N is the total vortex num-ber.

Existence for Mixed Type Solutionsin Chern-Simons Theory

Namkwon KimChosun University, KoreaKwangseok Choe, C-S Lin

We present some of recent progress on the existenceof mixed type solutions in the gauged Chern-Simonstheories in the whole space. In the radial symmet-ric setting, mixed type solutions correspond to boun-day points in the phase space and their informa-tion is closely related to the nontopological solutions.Hence, in the point of view of shooting, such solutionsare very subtle and one needs di↵erent approach. Wepresent degree theoretic and variational approach.

Dynamic Transitions of GeneralizedBurgers Equation

Kiah Wah OngIndiana University, USALimei Li

Phase transitions and bifurcations are of central im-portance in nonlinear sciences. Some typical exam-ples include the solid/liquid/gas transitions, segrega-tion of block copolymers in a polymer melt, the onsetof Rayleigh-Benard convection, etc. In this talk, thedynamic transition of the one dimensional general-ized Burgers equation with periodic boundary condi-tion will be discussed.

Standing Waves for the NonlinearSemi-Relativistic Schrodinger Equa-tions

Jinmyoung SeokKyonggi University, KoreaWoocheol Choi

The nonlinear Schrodinger equations arises from thequatum physics to describe the dynamics of a largeset of identical quantum particles interacting eachother. During several decades, their standing wavesolutions, one of their solitons, have been extensivelystudied under the aid of variational methods and,nowadays, we have fine knowledge about the exis-tence, regularity and qualitative properties such asthe radial symmetry or the decay rate at infinity.In this talk, we are concerned with the nonlinearsemi-relativistic Schrodinger equations, one of rela-tivistic counterparts of nonilnear Schrodinger eqa-tions. We introduce some results about the ex-istence, nonexistence and qualitative properties oftheir standing waves on the entire space. We alsodeal with the Dirichlet problems on bounded do-mains. In this setting, we are mainly interested inthe problems involving the nonlinearity with criticalexponent, under which Brezis-Nirenberg type resultsare discussed.

Radial Solutions for the Einstein-Maxwell-Higgs Model

Ju Hee SonKyung Hee University, KoreaJongmin Han

In this paper, we are concerned with an elliptic sys-tem arising from the Einstein-Maxwell-Higgs modelwhich describes electromagnetic dynamics coupledwith gravitational fields in space-time. Reducing thissystem to a single equation and setting up the ra-dial ansatz, we prove the existence of the radial so-lutions of topological and nontopological boundaryconditions of type I and II. There are two importantpositive constants: a representing the gravitationalconstant and N representing the total vortex num-ber. We establish the solutions for any a and N andgive a complete classification of all solutions, whichimproves previous known results.

On the Gravitational MaxwellGauged O(3) Sigma Model

Kyungwoo SongKyung Hee University, Korea

We improve the existence result ofnite energy solutions to the self-dual equations of thegravitational Maxwell gauged O(3) sigma model in(2+1)-dimensional Minkowski space for arbitrary lo-cation of strings for a small gravitational constantusing the standard super- and subsolution method.We construct an explicit supersolution and use theweighted Sobolev space for a subsolution.


Di↵usions with Singular Drift:Convergence to Robin BoundaryCondition

Minha YooNational Institute for Mathematical Sciences, KoreaInwon Kim

In this talk, we consider parabolic equations with sin-gular drift, where the drift penalizes di↵usion outsideof a given space-time domain. Using only PDE ar-guments we show that the corresponding solutions

converge to solutions of boundary value problems. Inthe case of divergence-form equations we show, by ex-plicit formula, that the limiting boundary conditiondepends not only on di↵usion operator but also onthe space-time geometry of the confining domain. Inparticular if the domain is time-dependent we showthat robin boundary data appears in the limit evenfor the most generic choice of di↵usion and drift.


Special Session 96: Complex Biological and Ecological Systems

Yun Kang, Arizona State University, USAKomi Messan, Arizona State University, USA

Marisabel Rodriguez, Arizona State University, USA

The emergence and evolution of complex interaction among biological organisms have trigger researchersaiming to understand structural, functionality, and relational patterns that are imperative to species sur-vivability. The interdisciplinary study of the dynamics of ecological and evolutionary processes in adaptiveenvironments present both challenges and opportunities for research and education. We bring both biologistsand mathematicians to study dynamics of complex biological systems including topics of disease, ecology,social insects, species interactions, and the link of genetics, environmental changes and social behavior. The-oretical frameworks will include agent-based modeling, network analysis, di↵erential equations, stochasticprocesses, and optimization. The invited speakers will consist of both mathematician and biologists, bothsenior and junior researchers, especially minority population, to promote the maximum collaborations.

Weakly Nonlinear Analysis of Non-local Fisher Equation

Ozgur AydogmusSocial Sciences University of Ankara, Turkey

We consider the Fisher equation with resource com-petition which is modeled by an asymmetric nonlo-cal convolution integral. We introduce two parame-ters for the convolution kernel describing the range ofnonlocality and the extent of asymmetry. In previousstudies it was shown that the spatially homogenousequilibrium of this model becomes unstable for suf-ficiently small di↵usion rates and traveling and sta-tionary wave type patterns was observed near thestability boundary. We further analyze the behaviorof solutions to this model near the stability bound-ary using the techniques of weakly nonlinear analysis.We first obtain a cubic Stuart-Landau type equationand give its parameters in terms of Fourier transformsof the kernels. This analysis allows us to study thechange in amplitudes of the solutions with respect torange of nonlocality and extent of asymmetry of thekernel function. We show that both continuous anddiscontinuous transitions from disordered behavior toordered one are possible. When discontinuous tran-sitions are observed, cubic equations are not enoughto get quantitave behavior of the amplitudes of thepatterns. We then obtain and study quartic Stuart-Landau equations to get a complete picture of theamplitudes near stability boundary. We also verifythese results numerically.

Comparison of Pollination Networks

David ChanVCU, USAJames Lee, Rodney Dyer

From the agent-based, correlated random walk modelpresented, we observe the e↵ects of varying the max-imum turning angle, �

max

, tree density, !, pollen car-ryover, max, and the probability of fertilization, P,on the distribution of pollen within a tree popula-tion. We see that varying �

max

and max changesthe dispersal distance of pollen, which greatly a↵ectsmany measures of connectivity. The clustering coef-ficient of fathers is maximized when �

max

is between

60� and 90�. Varying ! does not have a major e↵ecton the clustering coe�cient of fathers, but it doeshave a greater e↵ect on other measures of geneticdiversity. In particular we compare our simulationswith randomly-placed trees with that of actual treeplacement of C. florida at the VCU Rice Center, andshow that knowing the tree locations is critical inthe understanding how pollen is distributed within aspecific ecosystem.

Modeling the Geographic Spread ofRabies in China

Jing ChenUniversity of Miami, USALan Zou, Zhen Jin, Shigui Ruan

In the last 20 years or so, rural communities and areasin Mainland China invaded by rabies are graduallyand significantly enlarged. Some provinces such asShaanxi and Shanxi, used to be rabies free, have in-creasing numbers of human infections cases now. Re-cent phylogeographical analyses of rabies virus cladesindicate that the human rabies cases in di↵erent andgeographically unconnected provinces in China areepidemiologically related. In order to investigate howthe movement of dogs changes the geographicallyinter-provincial spread of rabies in Mainland China,we propose a multi-patch model for the transmis-sion dynamics of rabies between dogs and humans,in which each province is regarded as a patch. To in-vestigate the rabies virus clades lineages observed inthe phylogeographical analyses, the two-patch modelwill be used to simulate the human rabies data tostudy the inter-provincial spread of rabies betweenGuangxi and Guizhou, Fujian and Hebei and Sichuanand Guizhou, respectively. In order to reduce andprevent geographical spread of rabies in China, ourresults suggest that the management of dog marketand trade need to be regulated and transportation ofdogs need to be better monitored and under constantsurveillance.


An Adaptive Feedback Methodol-ogy for Determining InformationContent in Stable Population Stud-ies

Rebecca EverettNorth Carolina State University, USAH. T. Banks, J. E. Banks, J. D. Stark

Entomologists use demographic data to estimate thee↵ects of toxicants on populations. Although demo-graphic data has been shown to give a more accu-rate estimate in many instances than other typesof data, the collection of demographic data can beboth time-consuming and costly. An important ques-tion is whether partial demographic data can replacefull demographic data while still providing an accu-rate picture of the impact of a toxicant on a pop-ulation. We develop statistical and mathematicalbased methodologies for determining (as the experi-ment progresses) the amount of information requiredto complete the estimation of stable population pa-rameters with pre-specified levels of confidence. Thiswill be discussed in the context of life table modelsand data for growth/death for three species of Daph-niids. This represents joint work with H. T. Banksat North Carolina State University, J. E. Banks atCalifornia State University, Monterey Bay, and J. D.Stark at Washington State University, Puyallup.

Dynamics and Pattern Formation ina ModifiedLeslie-Gower Model withAllee E↵ect and Bazykin FunctionalResponse

Peng FengFlorida Gulf Coast University, USA

In this paper, we study the dynamics of a di↵usivemodified LeslieGower model with the multiplicativeAllee e↵ect and Bazykin functional response. We givedetailed study on the stability of equilibria. Nonex-istence of nonconstant positive steady state solutionsare shown to identify the rage of parameters of spa-tial pattern formation. We also give the conditions ofTuring instability and perform a series of numericalsimulations and find that the model exhibits complexpatterns.

Traveling Waves for a BiologicalSystem

Zhaosheng FengUniversity of Texas-Rio Grande Valley, USA

In this talk, we study the case that some speciesmigrate from densely populated areas into sparselypopulated areas to avoid crowding, and investigate amore general reaction-di↵usion system by consideringdensity-dependent dispersion as a regulatory mecha-nism of the cyclic changes. Here the probability thatan animal moves from the point x1 to x2 dependson the density at x1. Under certain conditions, we

apply the higher terms in the Taylor series and thecenter manifold method to obtain the local behavioraround a non-hyperbolic point of codimension one inthe pase plane, and use the Lie symmetry reductionmethod to explore bounded traveling wave solutions.

A Biophysical Model of Contact-Mediated Dormancy of Archaea byViruses

Hayriye GulbudakGeorgia Institute of Technology, USAJoshua S. Weitz

The canonical view of the interactions betweenviruses and their microbial hosts presumes thatchanges in host and virus fate require the initia-tion of infection of a host by a virus. That is, firstvirus particles di↵use randomly outside of host cells,then the virus genome enters the target host cell,and only then do intracellular dynamics and regu-lation of virus and host cell fate unfold. Intracellu-lar dynamics may lead to the death of the host celland release of viruses, to the elimination of the virusgenome through cellular defense mechanisms, or theintegration of the virus genome with the host as achromosomal or extra-chromosomal element. Herewe revisit this canonical view, inspired by recent ex-perimental findings of Bautista and colleagues (mBio,2015) in which the majority of target host cells canbe induced into a dormant state when exposed to ei-ther active or de-activated viruses, even when virusesare present at low relative titer. We propose thatboth the qualitative phenomena and the quantitativetime-scales of dormancy induction can be reconciledgiven the hypothesis that cellular physiology can bealtered by contact on the surface of host cells ratherthan strictly by infection. In order to test this hy-pothesis, we develop and study a biophysical modelof contact-mediated dynamics involving virus parti-cles and target cells. We show how virus particlescan catalyze cellular transformations amongst manycells, even if they ultimately infect only one (or none).

Semi-Kolmogorov’s Predator-PreyModels in Varying Environments

Xiaoying HanAuburn University, USATomas Caraballo, Renato Colluci

In this talk I will introduce several semi-kolmogorovtype of predation models with indirect e↵ect, includ-ing a nonautonomous model, a random model withreal bounded noise, a stochastic model with whitenoise, and a stochastic model with continuous-timeMarkov chain. In particular I will talk about the longterm behavior of these systems.


Evolutionary Games on the Lattice:Death-Birth Updating Process

Nicolas LanchierArizona State University, USAStephen Evilsizor

The death-birth updating process is an example ofspatial evolutionary game where the players, locatedon the infinite square lattice and characterized by oneof two possible strategies, update their strategy atrate one by mimicking one of their neighbors chosenat random with a probability proportional to theirpayo↵. In this talk, the stochastic spatial model iscompared with its deterministic non-spatial counter-part: the replicator equation. Though strategies cancoexist in both models, the inclusion of space sig-nificantly reduces the coexistence parameter region.Moreover, for the prisoner’s dilemma game, at leastin the presence of one-dimensional nearest neighborinteractions, cooperators can out-compete defectors,which contrasts with the replicator equation in whichthe defectors always win. The main ingredients toprove these results are block constructions, couplingarguments and optional stopping for martingales.

Mathematical Assessment ofMethamphetamine EpidemicAmong Men Who Have Sex withMen

Aprillya LanzNorfolk State University, USAAbba Gumel

Methamphetamine is an addictive stimulant that re-leases high levels of neurotransmitter dopamine. Theuse of methamphetamine has shown to increase li-bido and reduces inhibition. As a result, metham-phetamine is commonly used among men who havesex with men to initiate, enhance, and prolong sexualencounters. This, in turns, promotes high risk sex-ual behavior in this community of methamphetamineusers which increases the risk of acquiring an STD.Furthermore, studies have shown that the use ofmethamphetamine is associated with more frequentrisky sexual behaviors among HIV positive men whencompared with HIV negative men. In this presenta-tion, we will present a compartmental model thatrepresents the dynamics of methamphetamine abusein the men who have sex with men community from amathematical perspective. The model considers dif-ferent stages of progression of meth use and individu-als who are temporary or permanently quitters. Theanalysis of the model is presented in terms of themeth generation number, a threshold value typicallyknown as R

0

. It is shown that the model has mul-tiple equilibria for which their stabilities are deter-mined. Furthermore, numerical simulations are per-formed along with sensitivity analysis to determineimportant parameters to the model.

Fighting Mosquito-Borne Diseaseswith Sterile Mosquitoes

Jia LiUniversity of Alabama in Huntsville, USA

To prevent the transmissions of mosquito-borne dis-eases, sterile mosquitoe become an e↵ective weapon.To study the impact of releasing sterile mosquitoesinto the field of wild mosquitoes, we formulatemathematical models of interactive wild and ster-ile mosquitoes, considering di↵erent strategies of re-leases. Density-dependent vital rates are includedand Allee e↵ects are incorporated in the functionalmating rates. With fundamental analysis of the dy-namics of the interactive mosquitoes, we then intro-duce them into simple compartmental malaria trans-mission models. We study the dynamics of the simplemalaria model connecting with the mosquito models,and investigate the impact of the di↵erent strategiesof releases on the disease transmissions.

Dynamics of Low and HighPathogenic Avian Influenza in Wildand Domestic Bird Populations

Maia MartchevaUniversity of Florida, USAN. Tuncer, Juan Torres, M. Barfield, R.D.Holt

Avian influenza H5N1 is at present the most dan-gerous zoonotic disease infecting wild and domesticbirds. Should the virus mutate and become e�cientlyhuman-to-human transmittable, a pandemic will oc-cur with high mortality. Avian influenza H5N1 ex-ists in two forms: Low pathogenic (LPAI) and highpathogenic (HPAI). In this talk we build a model ofLPAI and HPAI in wild and domestic birds. Birds,wild and domestic, who have been priorly infectedwith LPAI are partially protected against HPAI. Wecompute the relevant reproduction numbers and in-vasion reproduction numbers. We find that the sys-tems has a disease-free equilibrium, LPAI-only equi-librium, HPAI-only equilibrium and at least one co-existence equilibrium. Furthermore, the LPAI-onlyequilibrium and HPAI-only equilibrium are locallyasymptotically stable under appropriate conditionson the reproduction numbers. In contrast, the co-existence equilibrium can lose stability and oscilla-tions are possible. We show that the oscillationsare caused by the cross-immunity and can exist inthe wild bird system, separate from the domesticbird system. For a pathogen circulating in a multi-species system, species A is called a sink (source), ifthe pathogen cannot (can) sustain itself in species Awithout the inflow of infectives from other species.We investigate the sink/source status of LPAI andHPAI in wild and domestic birds.


Evolution of Biodiversity Patternsin Evolving Fluvial Landscapes

Rachata MuneepeerakulUniversity of Florida, USAEnrico Bertuzzo, Andrea Rinaldo, IgnacioRodriguez-Iturbe

Biodiversity patterns are governed by landscapestructure and dispersal behaviors of organisms thatlive in it. Landscape, however, evolves, and organ-isms evolve their dispersal behaviors with it. Howdo biodiversity patterns change amid these geomor-phological and biological evolutions? To addressthis question, we implement neutral metacommunitymodels in river network landscapes at di↵erent stagesof the geomorphological evolution. Here, organismscompete for space according to neutral dynamics andat the same time adapt their dispersal behaviors.Preliminary results suggest that the landscape anddispersal evolutions a↵ect various biodiversity pat-

terns, namely relative species abundance, local andglobal richness, and spatial similarity of species com-positions, in di↵erent ways. These points to interest-ing interplay between geomorphological and biologi-cal evolutions.

Seasonality in Disease Modeling

Jin WangUniversity of Tennessee at Chattanooga, USA

The transmission and spread of many infectious dis-eases exhibit seasonal patterns, which can be math-ematically described by non-autonomous di↵erentialequations with periodic coe�cients. We compute thebasic reproduction number, R

0

, associated with suchperiodic epidemic models, using a non-trivial numeri-cal procedure. We then analyze the threshold dynam-ics characterized by the basic reproduction number.Particularly, we establish the uniform persistence ofthe system when R

0

> 1. In addition, we presentseveral examples to illuminate the results.


Special Session 97: Qualitative and Quantitative Techniques forDi↵erential Equations arising in Economics, Finance and Natural

Sciences

Rehana Naz, Centre for Mathematics and Statistical Sciences, Lahore School of Economics Lahore,Pakistan, Pakistan

Mariano Torrisi, Dipartimento di Matematica ed Informatica, Universita di Catania, ItalyImran Naeem, Lahore University of Management Sciences (LUMS), PakistanCelestin Wafo Soh, Mathematics Department, Jackson State University, USA

The di↵erential equations play a vital role in many disciplines from natural to social sciences. Most ofphysical laws in natural sciences are expressed in terms of di↵erential equations. In this session we try tointegrate analysis, models and methods in the scope of natural sciences as well as social sciences framework.The Economists study dynamical systems for sustainable Economic growth. Stochastic di↵erential equationsare the standard models for financial quantities important in financial market. Biologists (Epidemiologists)investigate the determinants of health-related states (including disease) using mathematical tools. Di↵eren-tial equations are mathematically studied from several di↵erent perspectives; this session will focus on theQualitative and Quantitative techniques (including numerical methods) for ordinary di↵erential equations,partial di↵erential equations, fractional di↵erential equations, di↵erence equations, stochastic di↵erentialequations, integro-di↵erential equations. Potential topics, of this session, include but are not limited to:– Economic growth theory – Optimal control – Di↵erential equations modeling natural and economic mod-els – Financial models e.g. Hamilton-Jacobi equation, Hamilton-Jacobi-Bellman equations, Option models,Black-Schole models – Equivalence transformations – Stability analysis – Numerical techniques for specialproblems in modeling – Symmetries, Di↵erential Equations, and Applications – Modeling and Math Biology– Fluid Mechanics

Metachronal Beating of Cilia withthe E↵ects of Di↵erent ShapedNanoparticles in a Pump with Ex-panding Or Contracting Wall

Noreen AkbarNational University of Sciences and Technology,Pakistan

In this talk, metachronal beating of Cilia for Cu-water nanofluid will be considered with the e↵ectsof di↵erent shaped nanoparticles. For the analysisHamilton-Crosser model will be used for the e↵ectivethermal conductivity of the nanofluids. In addition,heat transfer in human body will also be studied.Governing equations modification with long wave-length and low Reynold number approximation casewill be presented. Exact solutions will be presentedfor the simplified equations. Behavior of di↵erentflow parameters for di↵erent shapes of nanosize par-ticles will be discussed through graphs.

Flow and Heat Transfer Analysisfor a Non-Newtonian Past Overa Porous Plate with Partial SlipConditions

Yasir AliNational University of Sciences and Technology,Pakistan

In this talk I will discuss a steady non-Newtonianslip boundary layer condition post over a porous plateembedded in a porous medium. The similarity trans-formations will be presented to transform the gov-erning partial di↵erential equations (PDEs) into asystem of nonlinear ordinary di↵erential equations

(ODEs). The numerical solution will be discussed forthe resulting ODEs. Physical interpretation for thevelocity, temperature, skin friction coe�cient, per-meability, suction/injection parameter, Prandlt num-ber and Nusselt number will be presented throughgraphs and tables.

Convergence of Aggregated Vari-ability in Energy Consumption Data

Ai Ling Amy PohOkayama University, JapanEvgeny Mozgunov, Tan Chin Woo

A variability of energy consumption is the total vari-ance divided by total mean consumption. Real datashows convergence of aggregated variability with thenumber of customers. We investigate the mathemat-ical reasons of this phenomenon, as well as the sub-tleties of convergence rate. We show that the resultsfor convergence on real data are consistent with theprediction of a simple sum of random correlated vari-ables.

Mixed Regimes of Fluids in PorousMedia

Emine CelikTexas Tech University, USALuan Hoang, Akif Ibragimov, Thinh Kieu

In porous media, there are three known regimesof fluid flows, namely, pre-Darcy, Darcy andForchheimer/post-Darcy. Because of their di↵erentnatures, these are usually treated separately in lit-erature. To study complex flows, a single equationof motion is used to describe all three regimes andunify the mathematical treaments. Several scenar-


ios and models are then considered for slightly com-pressible fluids. A nonlinear parabolic equation forthe pressure is derived, which is degenerate when thepressure gradient is either small or large. We esti-mate the pressure and its gradient in terms of initialand boundary data. Moreover, we establish the con-tinuous dependence of the solutions on the initial,boundary data, and other physical parameters. Thisis a joint work with Luan Hoang, Akif Ibragimov andThinh Kieu.

Symmetry Classification andPeakon Solutions for a Homoge-neous Family of Partial Di↵erentialEquations

Priscila da SilvaUniversidade Federal do ABC, BrazilIgor Leite Freire

In this talk we discuss a 4-parameter homogeneousfamily of equations in terms of weak solutions andconservation laws. A Lie point symmetry classifica-tion is performed. We also show that the L2 normis conserved in the solutions of the equation. Fur-thermore, we find relations between the parametersso the equation admits a special kind of solution withpeaks.

Some Conservation Laws and ExactSolutions for a New Class of Non-linear Dispersive Equations

Erica de Mello SilvaFederal University of Mato Grosso, BrazilWescley Luiz de Souza

In this talk we consider a class of highly nonlin-ear dispersive equations with variable coe�cients,vcK(m,n), that was recently derived from thewell known Rosenau-Hyman compacton equation bymeans of the Lie symmetry approach. We discussabout its nonlinear self-adjointness and present someconservation laws and exact solutions for the partic-ular vcK(1, 1), vcK(2, 2), and vcK(3, 3) equations.

Global Stability of a Multistrain SISModel with Superinfection

Attila DenesBolyai Institute, University of Szeged, HungaryYoshiaki Muroya, Gergely Rost

We study the global stability of a multistrain SISmodel with superinfection. We present an iterativeprocedure to calculate a sequence of reproductionnumbers, and we prove that it completely determinesthe global dynamics of the system. We show thatfor any number of strains with di↵erent infectivities,

the stable coexistence of any subset of the strains ispossible, and we completely characterize all scenar-ios. As an example, we apply our method to a threestrains model. We also show a generalization to agroup multistrain model with patch structure.

Discovery of Hysteresis ModelsThrough the Symmetry Principle

Bassirou DiattaJackson State University, USACelestin Wafo Soh

Our goal in this talk is to employ the symmetryprinciple to obtain rate-independent hysteresis op-erators. The model we consider is a di↵usion equa-tion with hysteresis. Since the combination of thedi↵usion equation and the hysteresis constraint is adi↵erential-algebraic equation, we assume that thehysteresis operator is smooth enough so that we mayreplace the hysteresis constraint by the vanishing ofthe di↵erential of the hysteresis operator. We ana-lyze this new form viz. the coupling of the di↵usionmodel and the vanishing of the di↵erential of thehysteresis operator, using Lie symmetry algorithm.Specifically, we seek maximally symmetric hystere-sis models. The direct Lie approach to this problemleads to determining equations for the symmetriesthat are extremely di�cult to solve. In order to mit-igate this problem, we employ the classification oflow-dimensional Lie algebra following the suggestionof Lahno et al. (Journal of Physics A: Mathematicaland General 32.42 (1999): 7405).

Lie Symmetry Treatment for Pric-ing Options with Transactions CostsUnder the Fractional Black-ScholesModel

Bienvenue Feugang NteumagneUniversity of Pretoria, So AfricaB F Nteumagne, E Mare, E Pindza

We apply Lie symmetries analysis to price and hedgeoptions in the fractional Brownian framework. Thereputation of Lie groups is well spread in the area ofMathematical sciences and lately, in Finance. In thepresence of transactions costs and under fractionalBrownian motions, analytical solutions become dif-ficult to find. Lie symmetries analysis allows us tosimplify the problem and obtain new analytical so-lution. In this paper, we investigate the use of sym-metries to reduce the partial di↵erential equation ob-tained and find the analytical solution. We then pro-posed a hedging procedure and calibration techniquefor these types of options, and test the model on realmarket data. We show the robustness of our method-ology by its application to the pricing of digital op-tions.


A Camassa-Holm Type System Ad-mitting Peakon and Kink Solutions

Igor FreireUFABC, BrazilPriscila Leal da Silva

In this talk we discuss a two component Camassa-Holm type system depending on a continuous param-eter, controlling the nonlinearities. A Lie point sym-metry classification is carried out and the existenceof solutions, particularly peakon, multi-peakon, kinkand multi-kink, is investigated.

A Continuous-In-Time FinancialModel.

Emmanuel FrenodUniversite Bretagne Sud, FranceTarik Chakkour

I will present a continuous-in-time model which isdesigned to be used for the finances of public insti-tutions. This model is based on using measures overtime interval to describe loan scheme, reimbursementscheme and interest payment scheme; and, on usingmathematical operators to describe links existing be-tween those quantities. The consistency of the modelwith respect to the real world will be illustrated us-ing simple examples. Then the model will be used onsimplified examples in order to show its capability tobe used to forecast consequences of a decision or toset out a financial strategy

Group Analysis Applied to Chemo-taxis

Kesh GovinderUKZN, So AfricaP M Tchepmo Djomegni

Chemotaxis is the phenomenon of cell movement inresponse to stimuli. This broad concept covers a va-riety of processes in the cell cycle and, in particular,is crucial to the understanding of the evolution oftumours.We undertake a group theoretic approach to findtravelling wave solutions for a hyperbolic chemotaxismodel. We forsake the usual ansatz approach and,using the invariance properties of the equations, showthat more general solutions can be obtained. Thissystematic approach enables us to maintain the dif-fusivity terms in the model, thereby obtaining trulydi↵using solutions.We expand our model by considering di↵usion ofsubstrates, degradation of chemoattractants and cellgrowth (constant and linear growth rate). In all caseswe show that the system is still tractable. By apply-ing realistic boundary conditions we isolate biologi-cally applicable solutions.Chemotaxis provides another example of the utilityof Lie’s method of extended groups applied to di↵er-ential equations.

Parabolic Problems with DynamicOr Wentzell Boundary Conditionsin Spaces of Holder ContinuousFunctions

Davide GuidettiUniversity of Bologna, Italy

It is aim of this talk to illustrate some maximal reg-ularity results in spaces of Holder continuous func-tions for parabolic systems with dynamic or Wentzellboundary conditions, recently obtained by the au-thor. We shall consider, in particular, the case thatin the boundary condition there appears a di↵usionterm consisting of a second order strongly elliptic tan-gential operator.

Estimation of Jump Times in DailyShare Prices of Nikkei 225 Stock In-dex Using a Jump Di↵usion Model

Shuya KanagawaTokyo City University, Japan

We investigate the daily share prices of the Nikkei225 stock index to identify jump times of the stockindex using a jump di↵usion model, which consistsof the Black-Scholes model with stochastic volatilityand a compound Poisson process. Since the data ofdaily share prices of the Nikkei 225 stock index areobserved at discrete times, it is di�cult to find realjump times from the data. In this paper, we con-sider how to separate jump times from the observedtimes. The volatility of the stock index is estimatedby the historical volatility from the observation ofdaily share prices. We also refer to the number ofdaily share prices for historical volatility and showthat the number is essential for the accuracy of iden-tifying of jump times.

Optimal Dosing Strategies in thePresence of Drug Resistant Bacteria

Adnan KhanLahore University of Management Sciences, PakistanMudassar Imran

In this talk we discuss an ODE based model for theappearance of drug resistant bacteria and e�cientantibiotic dosing strategies. We examine both pe-riodic and optimal discrete antibiotic dosing. Drugresistant bacteria are a↵ected to a lesser degree byantibiotic treatments; we are interested in finding ef-ficient and successful antibiotic dosing strategies inthe presence of both susceptible and resistant bac-teria. We propose and analyze a model for the ap-pearance of drug resistant bacteria, in the presence ofantibiotic treatment. Using optimal control theory,we discuss antibiotic treatment strategies that mini-mize the total bacterial population while minimizingthe antibiotic costs at the same time.


Stagnation Point Flow Over aStretching Cylinder with Vari-able Viscosity and Heat Genera-tion/absorption

Farzana KhanQuaid-i-azam University, PakistanM.Y Malik

This article is focused to discuss the behavior ofstretching cylinder of viscous fluid near stagnationpoint with variable viscosity. The e↵ects of heat gen-eration/absorption are also encountered. Compari-son between solutions obtained through HAM andshooting method is presented. The e↵ect of variationin parameters on velocity and temperature fields areshown graphically.

Solitons and Lumps by BilinearTechniques

Wen-Xiu MaUniversity of South Florida, USA

We will talk about bilinear techniques to solve nonlin-ear partial di↵erential equations. Resonant solitonsand lumps are generated through computational al-gorithms implemented in Maple. The key step is torealize applicability of the linear superposition prin-ciple and positivity of real multivariate polynomials.

Properties of First Integrals forScalar Linear Third-Order ODEs ofMaximal Symmetry

Komal MahomedWits University, So Africa

The simplest scalar linear third-order ordinary dif-ferential equation has 7 Lie point symmetries. Wefind the classifying relation between the symmetriesand the first integrals for the simplest equation. It isshown that the maximal Lie algebra of an integral forthe simplest equation is unique and 4-dimensional.Further, we show that the Lie algebra of the sim-plest linear third-order equation is generated by thesymmetries of the 2 basic integrals. We also presentcounting theorems of the symmetry properties of theintegrals for linear third-order ODEs. We provide in-sights as to how one can generate the full Lie algebraof higher-order ODEs of maximal symmetry from twoof their basic integrals.

Numerical Aspects of Carreau FluidOver a Stretching Sheet with SlipConditions

Muhammad Yousaf MalikQuaid-i-Azam University, PakistanTaimoor Salahuddin

In this paper Carreau fluid model over a stretch-ing sheet with slip conditions is discussed. Thenon-linear partial di↵erential equation is convertedinto ordinary di↵erential equation by using similar-ity transformations. The solution of the requiredequation subject to required boundary conditions isobtained with the help of implicit finite di↵erencescheme known as Keller box method. The impactof di↵erent pertinent physical parameters on velocityprofile are plotted through graphs. Skin friction co-e�cient is also tabulated. Moreover, comparison hasbeen made with the previous literature in order tocheck the accuracy of the method.

Dynamics Analysis and OptimalControl in a Coupled Environment-Growth Model

Tufail MalikKhalifa University, United Arab EmiratesDavide La Torre, Danilo Liuzzi, OluwaseunSharomi, Rachad Zaki

We analyse the interaction between economic growthand pollution accumulation, where pollution is a by-product of production. The stock of pollution neg-atively a↵ects production. A share of the invest-ments is devoted to an environmental tax that isused to dampen the accumulation of pollution. Thiseconomic-environmental model is described by a pairof ordinary di↵erential equations whose dynamicsand steady state characteristics are studied. Thenwe look at this ambient environment from the pointof view of a social planner who can act on consump-tion and taxation, taking the dynamics of capital andpollution as constraints.

Entropy Generation Analysis forthe Peristaltic Transport of a NanoFluid in a Curved Channel HavingCompliant Walls

Ehnber MarajHeavy Industries Taxila Education City University,Pakistan

In this talk I will inspects the peristaltic flow of anano fluid in a curved channel. The governing equa-tions of nano fluid model for curved channel with cur-vature e↵ects will be presented. The coupled di↵eren-tial equations in simplified by using long wave lengthand low Reynolds number assumptions will be dis-cussed. Homotopy perturbation solution for the gov-


erning di↵erential equations will be discussed. En-tropy generation analysis with the physical featuresof pertinent parameters have been presented for ve-locity, temperature, concentration, entropy numberand stream functions at the end of the talks.

Conservation Laws and Structure-Preserving Integration Methods fora Special Class of PDEs

Brian MooreUniversity of Central Florida, USA

Conformal Hamiltonian systems, which are charac-terized by contraction of a symplectic 2-form anddissipation of energy, are generalized to nonlinearPDEs with convection, damping, di↵usion, and/ordispersive terms. The structure of these PDEs is veryspecial, but there are several PDE models with thisstructure that arise in applications, and their conser-vation laws have a straightforward interpretation in adiscrete setting, which paves the way for constructionof discretizations that preserve them. Presentationof numerical methods of this type will be followedby a brief exploration of the benefits of using suchdiscretizations.

Practical Implementation of Algo-rithmic Trading

Evgeny MozgunovCaltech, USA

Creating a winning trading algorithm is a multistepprocess. The steps are: (i) get familiar with thetrading platform and its limitations (ii) set up op-timization and crossvalidation routines (iii) come upwith several strategies and backtest them using (ii).Every step has its own tricks of trade that may becounterintuitive for the beginner. Some economistseven think that stock prices fundamentally cannotbe predicted. To be confident in either point of view,one should take a machine learning approach to theproblem. The problem of predicting stock prices canbe compared to image recognition and other machinelearning classics. Di↵erence is that it’s not enough topredict well, one should find out how to make moneyusing a prediction.

Nonstandard Finite Di↵erenceMethods for Weakly and StronglyCoupled Systems of Convection-Di↵usion Equations

Justin B. MunyakaziUniversity of the Western Cape, So Africa

There have been relatively few publications on cou-pled systems of singularly perturbed convection-di↵usion equations in comparison with their scalarcounterparts. In this talk we consider both weaklyand strongly coupled systems. While the former sat-isfy a maximum principle, the latter do not. We pro-

pose two nonstandard finite di↵erence methods to in-tegrate these systems (one for either case) and provetheir uniform convergence. Numerical investigationsare presented to confirm our theoretical findings. Fi-nally, comparison with existing methods is given.

A New Method to Construct Ap-proximate First Integrals of Gener-alized Hamiltonian Systems

Imran NaeemLahore University of Management Sciences, PakistanR. Naz, I. Naeem

The notion of approximate generalized Hamiltoniansystem is presented. The formulas for approximateHamiltonian operators determining equations andapproximate first integrals are provided explicitly.In order to show e↵ectiveness of approach developedhere, it is applied to establish the approximate firstintegrals of two perturbed coupled Du�ng-Van derPol oscillators. Both resonant and non-resonant casesare investigated in detail. Under a certain parameterrestriction, for resonant case we obtain two stable ap-proximate first integrals whereas only one stable firstintegral is attained for the non-resonant case.

A Current Value Hamiltonian Ap-proach for Discrete Time OptimalControl Problems Arising in Eco-nomic Growth Theory

Rehana NazLahore School of Economics, Pakistan

The notion of current value Hamiltonian is presentedfor the discrete time optimal control problems foreconomic growth models. Pontrygin-type maximumprinciple is developed for the current value Hamil-tonian systems of nonlinear di↵erence equations. Anew method termed as a discrete time current valueHamiltonian method is established for the construc-tion of first integrals for current value Hamiltoniansystems of ordinary di↵erence equations arising inEconomic growth theory. The newly developed tech-nique is explained with the help of a simple illustra-tive example. In order to show e↵ectiveness of ap-proach it is applied to the discrete time fundamen-tal models of economic growth theory: the Ramseymodel and the Lucas-Uzawa model.

Dynamic Optimization Problems inEconomic Growth Theory: CurrentValue Hamiltonian and CurrentValue Lagrangian Techniques

Rehana NazLahore School of Economics, PakistanF. M. Mahomed, Azam Chaudhry

The dynamic optimization problems can be solvedby calculus of variation and by optimal control tech-niques. We have developed a new approach termed asa discount free or current value Lagrangian method


for construction of first integrals for dynamical sys-tems of ordinary di↵erential equations arising in Eco-nomic growth theory. It is shown how one can uti-lize the Legendre transformation in a more generalsetting to provide the equivalence between a currentvalue Hamiltonian and a current or discount free La-grangian when it exists. The approach is algorith-mic and applies to many state variables of the La-grangian. The discount free Lagrangian naturallyarises in economic growth theory and many othereconomic models when the control variables can beeliminated at the outset which is not always possiblein optimal control theory applications of economics.We explain our method with the help of few widelyused economic growth models. We point out the dif-ference of this approach and that of the more generalcurrent value Hamiltonian method proposed earlierfor a current value Hamiltonian which is applicablein a general setting involving time, state, costate andcontrol variables. It is worthy to mention here thata discount free or current value Lagrangian methoddeals with calculus of variation problems whereas thecurrent value Hamiltonian method is applied to theoptimal control problems.

Criteria of Existence of Lorenz At-tractors

Ivan OvsyannikovUniversity of Bremen, Germany

The Lorenz attractor is the well-known example of arobust chaotic behavior. Lorenz attractors were ob-served in various models, both theoretical an applied.However, a fully analytic proof of the birth of suchattractors has been not generally given so far. I willdemonstrate how the criteria developed by Shilnikovand his group can help in this.The second part of the talk will be devoted to theproof of the birth of discrete Lorenz attractors in dif-feomorphisms. Such attractors have a much morecomplicated structure than the classical ones. It isexpected that they may allow homoclinic and het-eroclinic tangencies, i.e. demonstrate the so-called“wild“ behavior.

Intrinsic Nonlinear Dynamical Sys-tems in Isotropic Turbulence

Zheng RanShanghai University, Peoples Rep of China

The central problem of fully developed turbulenceis understanding the energy cascading process andmultiscale interaction. Update, there is no deduc-tive theory which leads to a full physical under-standing or mathematical formulation. Based onthe Karman-Howarth equation in 3D incompressiblefluid, a new isotropic turbulence scale evolution equa-tion and its related theory progress, the existence ofnonlinear dynamic system mearsured by turbulenceTaylor microscale was proven. The present results in-dicate that the enery cascading process has remark-able similarities with the deterministic constructionrules of the logistic map. The cascade appears as

an infinite sequence of period-doubling vortex bifur-cations. Based on this new approach, this projecttreats isotropic turbulence from the point of view ofdynamical systems. The exposition centres arounda number of important issues for turbulence behav-ior in the new nonlinear dynamical system we found.Thus, the modern theory of fractal and multi-fractalnow plays a major role in understanding the scalingbehavior in isotropic turblence.

Optimal Control of the DendriteStructure Using Magnetic Field

Amer RasheedLahore University of Management Sciences (LUMS)Pakistan, Pakistan

In this paper, we present the optimal control of aphase field model, recently developed by A. Rasheedand A. Belmiloudi [1], which represents the e↵ect ofmagnetic field on the evolution of dendrite during thesolidification process of a binary alloy in an isother-mal environment. The aim of this study is to con-trol the desired dynamics of the dendrite by usingmagnetic field as a control variable. In the controlproblem, the cost functional measures the distancebetween the calculated and desired dynamics. Wehave established the existence results and optimalityconditions along with the adjoint problem.

References

[1] A. Rasheed, A. Belmiloudi, Mathematical mod-elling of numerical simulations of dendritegrowth using phase-field method with a mag-netic field e↵ect, Communications in Computa-tional Physics, 14(2), pp. 477-508, 2013.

Well-Posedness and Global At-tractors for a Non-IsothermalViscous Relaxation of NonlocalCahn-Hilliard Equations

Joseph ShombergProvidence College, USA

In this preliminary report we will discuss a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem gen-erates a family of solution operators, exhibiting dis-sipation and conservation. The solution operatorsadmit a family of compact global attractors that arebounded in a more regular phase-space. Our aim isto prove that the family of global attractors satis-fies an upper-semicontinuity type estimate, wherebythe di↵erence between trajectories of the relaxationproblem and the limit isothermal non-viscous prob-lem is explicitly controlled, in the topology of therelaxation problem, in terms of the relaxation pa-rameters as well as other structural parameters.


Dynamics of Price and Wealth in aMulti-Group Asset Flow Model

David SwigonUniversity of Pittsburgh, USAM. DeSantis

Recently developed model of asset flow dynamics al-lows one to use the tools of nonlinear dynamics tostudy various market conditions and trading scenar-ios. I will present the results of a study in whichwe analyzed the influence of investor strategies ontheir wealth. We find that the constant rebalancedportfolio (CRP) strategy, in which investor divideshis wealth equally between di↵erent types of assetsand maintains those proportions constant as the pricechanges, is optimal in that it minimizes the potentiallosses incurred during price fluctuations in the mar-ket. We also show that any other trading strategy canbe taken advantage by other investors in the marketand lead to a loss of wealth, via predatory tradingbehavior.

Approximation of Solutions ofMulti-Dimensional Linear Stochas-tic Di↵erential Equations

Hiroshi TakahashiTokyo Gakugei University, Japan

Yoshihara et al. showed that solutions of some lin-ear stochastic di↵erential equations may be approx-imated by the solutions of the corresponding di↵er-ence equations defined by strong mixing sequences, Inthis talk, we generalized the result obtained by Yoshi-hara et al. to the multi-dimensional linear stochasticdi↵erential equations. We also give some examplesconcerning some finance models.

Symmetry Reductions for a Quan-tum Hydrodynamical Model

Rita TracinaUniversity of Catania, Italy

For the description of charge carrier transport insemiconductors, continuum models have interested inthe last years applied mathematicians and engineerson account of their applications in the design of elec-tron devices. With shrinking dimensions of submi-cron semiconductor devices, the quantum e↵ects areno longer negligible. New symmetry reductions andexact solutions are presented for a quantum hydro-dynamical model.

Applications of PDE Methods onNetworks to Data and Image Anal-ysis

Yves van GennipUniversity of Nottingham, England

In this talk we will see some applications in data anal-ysis and image analysis, that use partial di↵erentialequations techniques to solve problems on networks.An example is graph based image segmentation.

Hypecomplexification of Systems ofOrdinary Di↵erential Equations andLagrangian

Celestin Wafo SohJackson State University, USAF. M. Mahomed

Consider a system of even-order ordinary di↵erentialequations that can be hypercomplexified. Can we al-ways construct a Lagrangian of the system from thatof the base equation? We provide a positive answerto this question under mild nondegeneracy assump-tions. Specifically, we establish that under these re-strictions, the components of the Lagrangian of thebase equation are Lagrangians of the hypercomplex-ified system.

Power Spectra Methods LinkingStochastic Reaction-Di↵usion Sys-tems and Their Deterministic Limits

Thomas WoolleyUniversity of Oxford, EnglandRuth E. Baker, Eamonn A. Ga↵ney, PhilipK. Maini

Being able to create and sustain robust, spatial-temporal inhomogeneity is an important concept inmany areas, including developmental biology. Gen-erally, the mathematical treatments of such systemshave used continuum hypotheses of the interactingpopulations to produce reaction-di↵usion partial dif-ferential equations (PDEs). However, these ignoreany sources of intrinsic stochastic e↵ects. We ad-dress this concern by developing analytical Fouriermethods which allow us to rigorously link the deter-ministic PDEs with their stochastic analogues.Further, we include domain growth into our frame-work as it has been recently shown that growth al-lows a deterministic reaction-di↵usion patterning sys-tem to robustly double its pattern mode. However,this robustness feature is lost when stochasticity isincluded and, thus, we seek to use the presentedFourier methods to understand this patterning dou-bling breakdown. Further, we use our insights tosuggest how to regain the robustness property, evenin this stochastic setting.


Statistical Analysis on ModifiedBlack-Scholes Models with Heavy-Tailed Distributions

Xing YangJackson State University, USA

In this paper the Black-Scholes equation is modifiedwith several heavy-tailed distributions. A statisti-cal analysis is conducted on the new modified mod-els and simulations are implemented. Results fromsimulation are compared and discussed to choose themore realistic model for option pricing.

The Exact and Approximate Solu-tions of the Fractional Delay PartialDi↵erential Equations Using theReproducing Kenel Method

Lihong YangHarbin Engineering University, Peoples Rep of ChinaZhong Chen, Zhisheng Shuai

Fractional delay partial di↵erential equations have re-cently been applied in various areas of engineering,finance,bio-engineering and others. In this talk, weuse the reproducing kernel method to solve the frac-tional delay partial di↵erential equations in the Ca-puto derivative sense. Numerical simulation with theexact solutions is presented.

Numerical Investigation of Cou-ple Stress Fluid Over an ElastingStretching Sheet with Pressure De-pendent Viscosity

I↵at ZehraAir University Islamabad Pakistan, PakistanMalik Muhammad Yousaf, Sohail Nadeem

In this paper, we have examined the steady flow ofcouple stress fluid with pressure dependent viscositybetween two parallel plates in which the lower plateis stretched with while the upper plate is moving withconstant velocity. The governing two dimensionalnonlinear partial di↵erential equations are simplifiedby using suitable form of velocities. The simplifiednonlinear ordinary di↵erential equations are solvednumerically. The physical features of pertinent pa-rameters have been investigated through graphs andnumerical data.


Special Session 98: Inverse Problems and Imaging and Their Applications

Kai Huang, Florida International University, USAYuanchang Sun, Florida International University, USA

Inverse problems and imaging have found numerous applications in many areas from science to engineering.The mathematical studies of these problems pose significant analytical and computational challenges. Thisspecial session of AMIS conference aims to bring together researchers to promote exchange of ideas, andpresent recent developments on the mathematical analysis and computational methods in this area.

A Rigorous Mathematical Theoryfor Electromagnetic Field Enhance-ment in Metallic Nanogaps

Junshan LinAuburn University, USA

Subwavelength apertures and gaps on surfaces of no-ble metals (e.g., gold or silver) induce strong electricfield and extraordinary optical transmission. This re-markable phenomenon can lead to novel applicationsin biological and chemical sensing, spectroscopy, andTHz semiconductor devices. In this talk, I willpresent a quantitative analysis for the field enhance-ment when an electromagnetic wave passes throughtiny metallic gaps. Based upon a rigorous studyof the perfect electrical conductor model, we showthat enormous electric field enhancement occurs in-side the nanogap when there is extreme scale dif-ference between the wavelength of radiation and thethickness of metal films. The analysis also leads toe�cient asymptotic numerical method for calculat-ing the wave field in the nanostructure. The ongoingwork along this research direction will also be high-lighted.

The Di↵erence of L1 and L2 forCompressive Sensing and ImageProcessing

Yifei LouUniversity of Texas Dallas, USAPenghang Yin, Jack Xin

A fundamental problem in compressed sensing (CS)is to reconstruct a sparse signal under a few linearmeasurements far less than the physical dimensionof the signal. Currently, CS favors incoherent sys-tems, in which any two measurements are as littlecorrelated as possible. In this talk, I will presenta novel non-convex approach, which is to minimizethe di↵erence of L1 and L2 norms (L1-L2) in orderto promote sparsity. Some theoretical aspects of L1-L2 minimization are discussed and e�cient minimiza-tion algorithms are constructed and analyzed basedon the di↵erence of convex (DC) function methodol-ogy. Experiments demonstrate that L1-L2 improvesL1 consistently and it outperforms Lp for highly co-herent matrices. Finally, a recovery problem of pointsources from a set of low-frequency measurementswill be present, showing advantages of L1-L2 over L1when a necessary condition for perfect reconstructionis not satisfied.

The Degrees of Freedom of PartlySmooth Regularizers

Samuel VaiterCNRS & Univ. Bourgogne, FranceCharles-Alban Deledalle, Jalal M. Fadili,Gabriel Peyre, Charles Dossal

In this talk, we are concerned with regularized regres-sion problems where the prior regularizer is a properlower semicontinuous and convex function which isalso partly smooth relative to a Riemannian sub-manifold. This encompasses as special cases severalknown penalties such as the Lasso (`1-norm), thegroup Lasso (`1 � `2-norm), the `1-norm, and thenuclear norm. This also includes so-called analysis-type priors, i.e. composition of the previously men-tioned penalties with linear operators, typical exam-ples being the total variation or fused Lasso penalties.We study the sensitivity of any regularized minimizerto perturbations of the observations and provide itsprecise local parameterization. Our main sensitiv-ity analysis result shows that the predictor moveslocally stably along the same active submanifold asthe observations undergo small perturbations. Thislocal stability is a consequence of the smoothnessof the regularizer when restricted to the active sub-manifold, which in turn plays a pivotal role to get aclosed form expression for the variations of the pre-dictor w.r.t. observations. We also show that, for avariety of regularizers, including polyhedral ones orthe group Lasso and its analysis counterpart, this di-vergence formula holds Lebesgue almost everywhere.When the perturbation is random (with an appro-priate continuous distribution), this allows us to de-rive an unbiased estimator of the degrees of freedomand of the risk of the estimator prediction. Our re-sults hold true without requiring the design matrix tobe full column rank. They generalize those alreadyknown in the literature such as the Lasso problem,the general Lasso problem (analysis `1-penalty), orthe group Lasso where existing results for the latterassume that the design is full column rank.


A Gesture-Based Instruction andInput Device Using Acoustic Waves

Yuliang WangHong Kong Baptist University, Peoples Rep of ChinaHongyu Liu, Can Yang

A novel method is proposed for the recognition ofgestures using acoustic waves. The gestures are mod-eled by acoustic scatterers whose shapes are drawnfrom a prescribed dictionary and the recoginitionis modeled as an inverse acoustic scattering prob-lem. The incident wave is generated from a fixedpoint exterior of the scatterer and the scattered fieldis measured at a bounded surface containing thesource point. The recognition algorithm consists oftwo steps and requires two incident wave of di↵er-ent wavenumber. The approximate location of thescatterer is firstly determined by using the measureddata at small wavenumber and the shape of the scat-terer is then identified using the computed locationof the scatterer and the measured data at a regularwavenumber. Numerical experiments show the pro-posed method is computationally e�cient and workswith full or phaseless backscattering data of smallaperture.

Asynchronous Decentralized Con-sensus with Delayed StochasticGradient

Xiaojing YeGeorgia State University, USA

Decentralized consensus optimization plays a criticalrole in the emerging technology of decentralized net-work computing. In contrast to traditional central-ized computing, the nodes in a decentralized network

hold partial objective/data privately and the goal isto let them collaboratively solve for the global opti-mum. In this talk, we provide several e�cient decen-tralized consensus algorithms. In particular, we focuson the accelerated consensus algorithm and the caseswhere only delayed stochastic gradient informationare available during computing process. Numericalresults including application in seismic tomographyusing wireless sensor networks will be presented.

Constrained and Regularized Non-Negative Deconvolution: an Appli-cation to Speech Signal Dereverber-ation

Meng YuKnowles Corp., USA

A multi-channel spectral decomposition method isproposed based on a constrained and regularized non-negative deconvolution framework. The reverbera-tion of speech signal is modeled as a convolutionoperation in the spectral domain. Using the gener-alized Kullback-Leibler (KL) divergence, we decom-pose the reverberant magnitude spectrum into cleanmagnitude spectrum convolved with a deconvolutionfilter. The multi-channel deconvolutions are jointlyestimated by enforcing the cross-channel cancellationconstraint and iteratively solved by a multiplicativealgorithm to achieve multi-channel speech derever-beration.


Special Session 99: Theory and Applications of Boundary-DomainIntegral and Pseudodi↵erential Operators

Sergey Mikhailov, Brunel University London, EnglandDavid Natroshvili, Georgian Technical University, Rep of Georgia

Integral equation technique is a very e�cient tool in the study of boundary and initial-boundary value prob-lems for linear and nonlinear partial di↵erential equations arising in many static and dynamic mathematicalmodels of physical, biological, and engineering processes. The goal of this special session is to discuss recentprogress in the theory of boundary-domain integral and pseudodi↵erential operators and their applicationsin Mathematical Physics, Solid and Fluid Mechanics, Wave Scattering Problems, Engineering Mathematics,etc.

Asymptotic Analysis of Dynami-cal Interface Crack Problems forMetallic and Electro-Magneto Elas-tic Composite Structures

Otar ChkaduaRazmadze Mathematical Institute, Rep of GeorgiaT.Buchukuri, D. Natroshvili

We investigate the solvability and asymptotic prop-erties of solutions to 3-dimensional dynamical inter-face crack problems for metallic and electro-magneto-elastic composite bodies. We give a mathematicalformulation of the physical problems when the metal-lic and electro-magneto-elastic bodies are bondedalong some proper parts of their boundaries whereinterface cracks occur.Using the Laplace transform, potential theory andtheory of pseudodi↵erential equations on a manifoldwith boundary, the existence and uniqueness the-orems are proved. We analyse the regularity andasymptotic properties of the mechanical and electro-magnetic fields near the crack edges and near thecurves where the di↵erent boundary conditions col-lide. In particular, we characterize the stress singu-larity exponents and show that they can be explic-itly calculated with the help of the principal homoge-neous symbol matrices of the corresponding pseudod-i↵erential operators. For some important classes ofanisotropic media we derive explicit expressions forthe corresponding stress singularity exponents andshow that they essentially depend on the materialparameters. The questions related to the so calledoscillating singularities are treated in detail as well.Acknowledgements:This research was supported byRustaveli Foundation grant No. FR/286/5-101/13:Investigation of dynamical mathematical modelsof elastic multicomponent structures with regardto fully coupled thermo-mechanical and electro-magnetic fields.

A Time-Dependent Boundary-Field Equation Approach to Fluid-Thermoelastic Solid Interaction

George HsiaoUniversity of Delaware, USA

This paper presents a combined field and boundaryintegral equation method for solving time-dependentscattering problem of a thermoelastic body im-mersed in a compressible, inviscid and homogeneousfluid. The approach here is a generation of thecoupling procedure employed in [G.C. Hsiao, J.F.Sayas and R.J. Weinacht. A Time-Dependent Fluid-Structure Interaction, Math. Meth. Appl. Sci., DOI:10.1002/sim.0000] for treating the time-dependentfluid-structure interaction problem. By using inte-gral representation for the solution in the infinite ex-terior domain occupied by the fluid, the problem isreduced to one defined only over the finite region oc-cupied by the solid, with nonlocal boundary condi-tions. We analyze this nonlocal boundary problemas the Lubich approach for time-dependent bound-ary integral equations via the Laplace transform withan essential feature in terms of data in the time do-main directly. Existence and uniqueness results areestablished. Galerkin semi-discretization approxima-tions are derived and error estimates are obtained. Afull discretization based on the convolution quadra-ture method is also outlined. Finally, to demonstratethe validity of the method, a numerical experimentis included for the special case of the time-dependentfluid-structure interaction.

The Haseman Boundary ValueProblem with Slowly OscillatingData

Yuri KarlovichUniversidad Autonoma del Estado de Morelos,Mexico

The talk is devoted to studying the Haseman bound-ary value problem� + � ↵ = G�� + g on a star-likeCarleson curve � composed by logarithmic spirals inthe setting of Lebesgue spaces, where� ± are angu-lar boundary values of an unknown analytic function� on � ,G and g are given functions, and ↵ is anorientation-preserving homeomorphism of � onto it-self. This problem is reduced to the equivalent sin-gular integral operator with a shift T = V↵P+

+GP�


on a Lebesgue space Lp(�), where the operatorsP± = 2�1(I ± S

�

) are related to the Cauchy sin-gular integral operator S

�

, and the shift operator V↵is given by V↵f = f � ↵. Applying the theory ofMellin pseudodi↵erential operators with non-regularsymbols of limited smoothness, we establish a Fred-holm criterion and an index formula for the operatorT provided that the shift derivative ↵‘ and the coef-ficient G are slowly oscillating functions on �.

Boundary Value Problems for Non-linear Brinkman and Navier-StokesEquations with Variable Coe�cientsin Lipschitz Domains

Mirela KohrBabes-Bolyai University, Cluj-Napoca, RomaniaMassimo Lanza de Cristoforis, Sergey E.Mikhailov, Wolfgang L. Wendland

In this talk we present recent existence and unique-ness results in Sobolev and Besov spaces for boundaryvalue problems for nonlinear Brinkman and Navier-Stokes equations with variable coe�cients in Lips-chitz domains in R3 and in compact Riemannianmanifolds. First, we analyze boundary value prob-lems for the linear Stokes and Brinkman systems withvariable coe�cients, and show the well-posedness ofsuch a linear problem. A key role in this analysisis played by the mapping properties of Newtonianand layer potential operators, which describe theseequations in appropriate Sobolev and Besov spaces.Next the well-posedness results in the linear case arecombined with a fixed point theorem to show the ex-istence of a solution in Lp-based Sobolev spaces for aboundary value problem for variable-coe�cient non-linear Brinkman or Navier-Stokes equations.

A Functional Analytic Approach tothe Analysis of a Two-ParameterHomogenization for a NonlinearRobin Problem

Massimo Lanza de CristoforisUniversity of Padua, ItalyPaolo Musolino

We consider a nonlinear Robin problem for the Pois-son equation in an unbounded periodically perforateddomain. The domain has a periodic structure, andthe size of each cell is determined by a positive pa-rameter �. The relative size of each periodic perfora-tion is instead determined by a positive parameter ✏.We prove the existence of a family of solutions whichdepends on ✏ and � and we analyze the behavior ofsuch a family as (✏, �, ) tends to (0, 0) by an approachwhich is alternative to that of asymptotic expansionsand of classical homogenization theory.

Brinkman-Oseen TransmissionProblem

Dagmar MedkovaMathematical Institute, Czech RepM. Kohr, W .L. Wendland

We develop a layer potential analysis in order to showthe well-posedness result of a transmission problemfor the Oseen and Brinkman systems in open sets inEuclidean spaces with compact Lipschitz boundariesand around a lower dimensional solid obstacle, whenthe boundary data are q-integrable.


Analysis of Segregated Boundary-Domain Integral Equations forVariable-Coe�cient Scalar BVPswith General Data

Sergey MikhailovBrunel University London, England

Segregated direct boundary-domain integral equa-tions (BDIEs) based on a parametrix and associ-ated with the Dirichlet and Neumann boundary valueproblems for the linear stationary di↵usion partialdi↵erential equation with a variable coe�cient areformulated. The PDE right hand sides belong tothe Sobolev space H�1(⌦) or H�1(⌦), when nei-ther classical nor canonical co-normal derivatives arewell defined. Equivalence of the BDIEs to the origi-nal BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as Fredholm property andinvertibility of the BDIE operators are analysed inSobolev (Bessel potential) spaces. It is shown thatthe BDIE operators for the Neumann BVP are notinvertible, and appropriate finite-dimensional pertur-bations are constructed leading to invertibility of theperturbed operators. The contribution is based onand develops some results of [1-3].

References

[1] O. Chkadua, S.E. Mikhailov, and D.Natroshvili.Analysis of direct boundary-domain integralequations for a mixed BVP with variable coe�-cient, I: Equivalence and invertibility. J. IntegralEquations Appl., 21(4):499–543, 2009.

[2] S.E. Mikhailov. Finite-dimensional perturba-tions of linear operators and some applica-tions to boundary integral equations. Eng. Anal.Bound. Elem., 23:805–813, 1999.

[3] S.E. Mikhailov. Analysis of SegregatedBoundary-Domain Integral Equations forVariable-Coe�cient Dirichlet and NeumannProblems with General Data, ArXiv:1509.03501,http://arxiv.org/abs/1509.03501, 1-32, 2015.

The BMO-Dirichlet Problem forElliptic Systems in the Upper-HalfSpace and Quantitative Characteri-zations of VMO

Dorina MitreaUniversity of Missouri, USAJ.M. Martell, I. Mitrea, M. Mitrea

Let L be a homogeneous, second order, constantcomplex coe�cient elliptic system L in Rn. In thistalk I will discuss the well-posedness of the Dirich-let problem in Rn

+

for L with boundary data inBMO(Rn�1) in the class of functions u for which theLittlewood-Paley measure associated with u, namelydµu(x‘, t) := |ru(x‘, t)|2 t dx‘dt, is a Carleson mea-sure in Rn

+

. The regularity result for this problemcorresponding to the boundary datum being in Sara-son’s space VMO(Rn�1) will be discussed as well.

Moreover, a new characterizations of the space VMOas the closure in BMO of classes of smooth func-tions contained in BMO within which uniform con-tinuity may be suitably quantified (such as the classof smooth functions satisfying a Holder or Lipschitzcondition) will be presented. This improves on Sara-son’s classical result describing VMO as the closure inBMO of the space of uniformly continuous functionswith bounded mean oscillations.

How to Custom-Tailor BoundaryLayer Potentials for a Given Di↵er-ential Operator on a Manifold

Marius MitreaUniversity of Missouri, USA

Given an elliptic, second-order di↵erential operatorL acting between two vector bundles over a Rieman-nian manifold M , I will discuss a procedure that as-sociates to L a double-layer operator with respectto some uniformly rectifiable subdomain of M . Thisis a natural adaptation of the blue-print accordingto which the familiar harmonic double-layer is asso-ciated with the Laplacian. Indeed, this generalizeddouble-layer turns out to enjoy all trademark proper-ties that the classical harmonic double-layer possessessuch as nontangential maximal function estimates,jump relations, and square-function estimates.

Potential Operators Methodfor Domains with Smooth Un-bounded Boundaries for AnisotropicHelmholtz Operators

Vladimir RabinovichNational Polytechnic Institute of Mexico, Mexico

The paper is devoted to the potential operatorsmethod for the boundary and transmission problemsin domains in Rn with smooth unbounded boundariesfor the anisotropic Helmholtz operators

Hu(x) = r · a(x)ru(x) + b(x)u(x), x2 Rn

with variable coe�cients, where a =�ak,l

�nk,l=1

are

real-valued symmetric matrix on Rn with entries ak,l

2 C1b (Rn), k, l = 1, ..., n. We suppose that the op-

erator H is strongly elliptic, b(x) = !2b0

(x), ! > 0is the frequency of the harmonic vibrations, b

0

(x) isthe refractive index which satisfies the following con-ditions: b

0

2 C1b (Rn),

Rb0

(x) > 0, Ib0

(x) � 0, x 2 Rn, lim infRn3x!1

Ib0

(x) > 0.

We introduce the single and double layer poten-tials associated with the operator H, and reduce bymeans of these potentials the Dirichlet, Neumann,Robin, and transmission problems for domains withunbounded smooth boundary @D to pseudodi↵eren-tial equations on @D. Applying the limit operatorsmethod we study the Fredholm properties and theinvertibility of the boundary pseudodi↵erential oper-ators in the Sobolev spaces Hs(@D), s 2 R.


Special Session 100: Nonstandard Analysis, Quantizations and SingularPerturbations

Kiyoyuki Tchizawa, Tokyo City University, JapanSiavash H. Sohrab, Northwestern University, USA

Nonstandard Analysis is e↵ective in the context of discretizations and changes of scale, for example blow-ups.Notably in the case of singular perturbations and quantum-mechanics phenomena are studied which involvesimultanuously the small and the observable. The di↵erent orders of magnitude of the parameters may thenbe related by general principles of nonstandard analysis, like the Transfer Principle. In some cases blowingup, the Idealization and the Standardization Principle may also be needed.

Extended Wiener Measure by Non-standard Analysis for FinancialTime Series

Shuya KanagawaTokyo City University, JapanKiyoyuki Tchizawa

We propose a new approach to construct an extendedWiener measure using nonstandard analysis by E.Nelson. For the new definition we construct non-standardized convolution of probability measure forindependent random variables. As an application weconsider a simple calculation of financial time series.

No Waste Assumption, Construc-tion of Optimal Solutions, andTransversality Condition for InfiniteHorizon Problems

Takashi NittaMie University, JapanCai Dapeng

By imposing a no waste assumption, which assumesthat the entire stock is depleted before the planninghorizon terminates, we modify the usual optimalitycriterion that is based on the standard overtakingcriterion.

Some Implications of InvariantModel of Statistical Mechanics toSet Theory, Nonstandard Analysis,and Noncommutative Geometry

Siavash SohrabNorthwestern University, USA

A scale-invariant model of statistical mechanics is ap-plied to describe the foundation of multicomponent(mixture) set theory and some of its implicationsto ZFC axioms. Normalized spacing between non-

trivial zeros of Riemann zeta function are shown tofollow normalized Maxwell-Boltzmann distributionfunction. The zeros of zeta function are related tothe zeros of particle velocities hence their stationarystates. An invariant theory of hyperreal ordered setson a line composed of nonstandard infinitesimals, re-als, and nonstandard transfinite is introduced. Phys-ical space or Casimir vacuum is identified as a quan-tum tachyon fluid, Dirac stochastic ether, that is gov-erned by Heisenberg matrix mechanics and hence de-scribed by noncommutative geometry. The implica-tion of modified theory of nonstandard analysis tomicrostructure of normal shock waves is described.

Canards in R3 and R4 with Co-Dimension 2

Kiyoyuki TchizawaTokyo City University, Japan

This paper gives a survey about the existence of thecanards in a slow-fast system in R1+2 and R2+2 withan invariant manifold. It has a 4-dimensional canardhaving a relatively stable region when there existsthe invariant manifold near the pseudo singular nodepoint.

Nonstandard Methods in Combina-torics

Keita YokoyamaJapan Advanced Institute of Science and Technology,Japan

Nonstandard methods are available in various fieldsof mathematics, as well as analysis. In this talk, I willgive a brief introduction of an application of nonstan-dard methods to combinatorics and graph theory.With nonstandard methods, the relation between fi-nite combinatorial statements and infinite combina-torial statements can be understood clearly. More-over, proof-theoretic analysis of nonstandard meth-ods give more information to combinatorics, whichprovides some useful information to computer sci-ence.


Special Session 101: Randomness meets Life

Peter Hinow, University of Wisconsin - Milwaukee, USABlerta Shtylla, Pomona College, USA

We will bring together researchers that employ stochastic methods to model a rich set of biological phenom-ena. Stochastic e↵ects are important when modeling small scale phenomena such as biochemical reactions,intra-cellular interactions, cellular pattern formation and transport due to the small scale of the environment.In these cases stochasticity is not a mere afterthought but the engine that mediates exquisitely complex phe-nomena. Stochastic e↵ects are also important at larger scales, as in animal dispersion, spread of epidemics,mathematical immunology and cancer. Despite the large di↵erence in scales the techniques used for modeldevelopment and analysis are often similar. We intend to bring researchers together that have worked atboth large and small scales in order to stimulate the exchange of ideas. An important aspect of the sessionis that we will encourage speakers to give large picture ideas of their respective fields which is importantfor cross pollination and also for young researchers interested in work on mathematical modeling of newbiological applications.

An Stochastic Model of Cancer Evo-lution and Treatment Resistance inthe Bone

David BasantaH. Lee Mo�tt Cancer Center, USAArturo Araujo, Leah Cook, Conor Lynch

The American Cancer Society predicts that approx-imately 6 Floridian men will die from prostate can-cer each day in 2015. These deaths are due to thecancer spreading to secondary sites. Prostate can-cer frequently metastasizes to the skeleton where itpromotes extensive bone destruction and formationcausing great pain to the patient. The bone metas-tases are incurable and thus we need to identify newtherapies and rapid approaches in regards to howtherapies are applied to our patients. Experimen-tal animal models provide insights but are limited interms of their ability to tease apart the drivers ofcancer growth and evolution. Computational modelson the other hand produce rich and high-resolutiondata, can yield new clinically interesting hypothesesand can accelerate biological research. We proposethat integrating computational modeling with bio-logical experimentation will allow for the rapid opti-mization of existing and new therapies for the treat-ment and cure of bone metastatic prostate cancer.Here, we illustrate how a biology-driven agent-basedcomputational model can dissect the complex e↵ectsof inhibiting new therapeutic targets such as, trans-forming growth factor Beta (TGF-Beta). TGF-Betais a powerful growth factor in the bone microenvi-ronment that can have di↵erential e↵ects on bonebuilding, bone destroying and cancer cells. Becauseof its important and diverse roles, we thought it to bethe ideal candidate with which to begin to test ourintegrated computational and biological approaches.Computational results predict that TGF-Beta inhi-bition, applied in a specific therapeutic window, im-pacts prostate cancer cell growth directly but also,by limiting bone destruction and unexpectedly, pro-moting bone formation; results that in turn werevalidated in biologically relevant mouse models ofthe disease. We also demonstrate how the compu-tational model can be modified on an individual pa-tient basis to address cancer cell heterogeneity andtheir response to applied therapies. Collectively, our

results demonstrate the power of a combined com-putational/experimental approach in optimizing thee�cacy of applied therapies and measuring their im-pact on the tumor-bone microenvironment. We be-lieve that the development, refinement and validationof our computational models will have a profoundimpact on the health of Floridian men su↵ering withbone metastatic prostate cancer.

Periodically Driven Noisy NeuronalModels: a Spectral Approach

Alla BorisyukUniversity of Utah, USAFiras Rassoul-Agha

Neurons are often driven by (noisy) periodic or peri-odically modulated inputs. In many such cases neu-ronal firing can be characterized by a stochastic phaseresponse map (SPRM) that maps phase of the cur-rent spike into the phase of the subsequent spike.More generally, SPRMs represent Markov chains ona circle. In our spectral approach to studying suchmaps, we analyze path-wise dynamic properties ofthe Markov chain, such as stochastic periodicity (orphaselocking) and stochastic quasiperiodicity, andshow how these properties are read o↵ of the geome-try of the spectrum of the transition operator. I willalso discuss how SPRMs can be computed for someneuronal models, their relationship with phase re-sponse curves, and how they are a↵ected by changesin the ionic channels.

Computer Simulations of YeastMating Reveal Robustness Strate-gies for Cell-Cell Interactions

Ching-Shan ChouOhio State University, USAWeitao Chen, Tau-Mu Yi, Qing Nie

Cell-cell communication is important to cell function-ality. Successful cell communication requires coor-dination of intricate intracellular and extracellularpathways with the presence of noises. In this work,


we build a computational framework that accountsfor the molecular dynamics inside and outside thecell, as well as cell morphogenesis. Through com-puter simulations, we found strategies that buddingyeast cells use for e�cient and successful mating.

Ergodicity and Loss of Capacity fora Family of Concave Random Maps

Peter HinowUniversity of Wisconsin - Milwaukee, USAAmi Radunskaya

Random fluctuations of an environment are com-mon in ecological and economical settings. We con-sider a family of concave maps on the unit interval,f�(x) = x(1+��x), that model a self-limiting growthbehavior. The maps are parametrized by an indepen-dent, identically distributed random variable � withvalues in the unit interval. We show the existenceof a unique invariant ergodic measure of the result-ing random dynamical system for arbitrary param-eter distributions supported on certain subintervalsof [0, 1]. Moreover, there is an attenuation of themean of the state variable compared to the constantenvironment with the averaged parameter. We alsoprovide an example of a family of just two maps suchthat the invariant probability measure is supportedon a Cantor set.

This work has been supported by the grant “Collab-orative Research: Predicting the Release Kinetics ofMatrix Tablets“ (DMS 1016214 and DMS 1016136)of the National Science Foundation of the UnitedStates of America.

Cross Scale Dynamics and The Evo-lutionary Emergence of InfectiousDiseases

Ruian KeNorth Carolina State University, USASebastian J. Schreiber, Claude Loverdo,Miran Park, Prianna Ahsan, James O. Lloyd-Smith

Emerging infectious diseases typically involvepathogens that are exposed to novel environments,such as pathogens from animal populations enter-ing human populations or human pathogens exposedto new drug regimens. When pathogens are poorlyadapted to these novel environments, emergence re-quires the pathogen to evolve su�ciently fast to avoidextinction. This evolution involves pathogen traitsat multiple scales, such as within-host viral repli-cation rates and between-host transmissibility. Weintroduce and analyze a stochastic cross-scale model, and determine how the likelihood of pathogen emer-gence is governed by selective pressures at these twoscales. Our cross-scale analysis opens the door for anew generation of integrative risk assessment modelswhich will link growing streams of data collected inlaboratories and field surveillance programs.

Modelling Evolution of Post-Menopausal Human Longevity:the Grandmother Hypothesis

Peter KimUniversity of Sydney, AustraliaJohn McQueen, James Coxworth, KristenHawkes

Human post-menopausal longevity makes us uniqueamong primates, but how did it evolve? One explana-tion, the Grandmother Hypothesis, proposes that asgrasslands spread in ancient Africa displacing foodsancestral youngsters could e↵ectively exploit, olderfemales whose fertility was declining left more de-scendants by subsidizing grandchildren and allowingmothers to have new babies sooner. As more ro-bust elders could help more descendants, selectionfavoured increased longevity while maintaining theancestral end of female fertility.We develop a probabilistic agent-based model thatincorporates two sexes and mating, fertility-longevitytradeo↵s, and the possibility of grandmother help.Using this model, we show how the grandmothere↵ect could have driven the evolution of humanlongevity. Simulations reveal two stable life-histories,one human-like and the other like our nearest cousins,the great apes. The probabilistic formulation showshow stochastic e↵ects can slow down and prevent es-cape from the ancestral condition, and it allows us toinvestigate the e↵ect of mutation rates on the trajec-tory of evolution.

Fluctuation Models for Suspensionsof Swimming Microorganisms

Peter KramerRensselaer Polytechnic Institute, USAYuzhou Qian, Patrick Underhill

The collective dynamics of swimming microorgan-isms (“microswimmers“) such as bacteria and algalcells have been of considerable recent interest, bothas paradigms of collective patterns arising from in-dividual autonomous agents and for their relevanceto technological issues such as biofilm formation andpower sources for microdevices. We will discuss somerecent e↵orts to characterize stochastic fluctuationsin a continuum “mean field“ partial di↵erential equa-tion framework for the e↵ective microswimmer dy-namics in a suspension.

Evolutionary Games on the Lattice:Best-Response Dynamics

Nicolas LanchierArizona State University, USAStephen Evilsizor

The best-response dynamics is an example of evolu-tionary game where players are located on the infi-nite square lattice and update their strategy in or-der to maximize their payo↵. In the presence of twostrategies, and calling a strategy selfish or altruistic


depending on a certain ordering of the coe�cients ofthe underlying payo↵ matrix, a simple analysis of thenon-spatial mean-field approximation of this processshows that a strategy is evolutionary stable if andonly if it is selfish, making in particular the systembistable when both strategies are selfish. The mainobjective of this talk is to show that, in contrast withthe mean-field approximation, only the most selfishstrategy is evolutionary stable for the stochastic pro-cess. The main ingredients of the proof are mono-tonicity results and a coupling between the processproperly rescaled in space with bootstrap percola-tion.

Di↵usion in a Randomly SwitchingEnvironment

Sean LawleyUniversity of Utah, USA

Driven by diverse applications to biochemistry andphysiology, several recent models impose randomlyswitching boundary conditions on either a PDE orSDE. The PDE models arise from considering a den-sity of particles di↵using in a random environment,whereas the SDE models arise from considering onlyfinitely many particles di↵using in a random environ-ment. In this talk, I will describe the mathematicaltools for analyzing these systems and highlight theinteresting behavior that they can exhibit. Specialattention will be given to establishing mathematicalconnections between these classes of stochastic pro-cesses.

Mesoscopic Modeling of DNATransport in an Array of EntropicBarriers

Anastasios MatzavinosBrown University, USA

In this talk, we discuss dissipative particle dynam-ics (DPD) simulations of the dispersion of DNAmolecules conveyed by a pressure-driven fluid flowacross a periodic array of entropic barriers. We com-pare our simulations with nanofluidic experiments,which show the DNA to transition between varioustypes of behaviors as the pressure is increased, anddiscuss physical insights a↵orded by the ability of theDPD method to explicitly model flows in the sys-tem. Finally, we present anomalous di↵usion phe-nomena that emerge in both experiment and simula-tion, and we illustrate similarities between this sys-tem and Brownian motion in a tilted periodic poten-tial. This is a joint work with Clark Bowman, DanielKim, and Derek Stein.

Anomalous Di↵usion and RandomEncounters in Living Systems

Scott MckinleyTulane University, USARebecca Borchering, Jason Flynn, StevenBellan, Juliet Pulliam

Due to the rapid growth of animal movement dataobtained by GPS, radio tracking collars and othermeans, there is a growing recognition that classi-cal models of encounter rates among animal popula-tions should be revisited. Recent theoretical investi-gations have demonstrated that biologically relevantmodifications to classical assumptions about individ-ual behavior can bring about non-trivial changes inthe formulation of population-scale dynamical sys-tems. In particular, the combination of tracking datawith habitat information has revealed the substantialimpact that environmental factors have on animalmovement and sociality. In this talk, I will reviewsome of the existing conventional wisdom that sup-ports the use of so-called “Levy flight“ models thatseek to describe animal movement in the absence ofenvironmental cues. However, through a few exam-ples, I will make the case that animal movement pat-terns should not be separated from the spatial en-vironmental features that shape them. In fact, ani-mal sensing and decision-making are “leading-order“e↵ects, and their study gives rise to new ecologicalobservations and novel mathematical challenges.

Sensory Feedback in a Bump At-tractor Model of Path Integration

Daniel PollUniversity of Houston, USAKhanh Nguyen, Zachary Kilpatrick

Mammalian spatial navigation systems utilize severaldi↵erent sensory information channels.This information is converted into a neural codethat represents the animal’s current position in spaceby engaging place cell, grid cell, and head direc-tion cell networks. In particular, sensory landmark(allothetic) cues can be utilized in concert with ananimal’s knowledge of its own velocity (idiothetic)cues to generate a more accurate representation ofposition than path integration provides on its own(Battaglia et al, 2004). We develop a computationalmodel that merges path integration with feedbackfrom external sensory cues that provide a reliablerepresentation of spatial position along an annulartrack. Starting with a continuous bump attractormodel, we explore the impact of synaptic heterogene-ity and noise fluctuations, which disrupt the positioncode of the path integration process. We use asymp-totic analysis to reduce the bump attractor model toa single scalar equation. Such imperfections causeerrors to build up when the network performs path


integration, but these errors can be corrected by anexternal control signal representing the e↵ects of sen-sory cues. We demonstrate that there is an optimalstrength and decay rate of the control signal whencues appear either periodically or randomly.

Dimension Reduction for StochasticConductance Based Neural Modelswith Time Scale Separation

Deena SchmidtUniversity of Nevada, Reno, USAPeter Thomas, Roberto Galan

The Stochastic Shielding Approximation (SSA) is afast, accurate simplification of randomly gated ionchannel models (Schmandt and Galan 2012, Schmidtand Thomas 2014). Viewing the channel as a dis-crete process on a directed graph, driven by an in-dependent noise source for each edge, the SSA accu-rately represents the process using independent noisesources for only a small subset of the edges. This ap-proximation preserves the mean field behavior whileselectively incorporating only the underlying noisesources that contribute the most significantly to ob-servable system behavior. Thus the stochastic shield-ing heuristic provides an analytically tractable exam-

ple of incorporating noise in a manner relevant to thenetwork’s physiological function. Here we investigatethe limits of the SSA by studying its accuracy for sys-tems exhibiting large separation of time scales, whichis often the case in neural systems.

Stochastic Models of Force Genera-tion and Measurement in DividingCells

Blerta ShtyllaPomona College, USA

In this talk, we discuss mathematical models thattrack bias generation by nano-machines operatingduring cell division. Several nano-machines are in-volved in cell division and their e�cient operationrequires specific interactions with dynamic bio poly-mers. We provide two examples of such nano-machines: one working in eukaryotic cells and onein prokaryotic cells. We use first passage techniquesto derive mesoscale properties for these constructsusing microscale rates and reactions. In each case,we show how these nano-machines could be employedby cells to make the precise measurements needed forequipartition of DNA.


Special Session 102: Recent Developments of High-Order NumericalMethods

Ying Wang, University of Oklahoma, USAYang Yang, Michigan Technological University, USA

This mini-symposium is to bring researchers together to discuss the recent advances and exchange ideas in thealgorithm design of high-order numerical methods for di↵usion-dominated and other high-order partial di↵er-ent equations, including the implementation, numerical analysis, and applications. In the mini-symposium,the speakers will apply those high-order numerical methods to computational fluid, biology and physics, etc.This mini-symposium provides a good opportunity for discussions among researchers from di↵erent areas,and explore more applications and future research collaborations.

Maximum-Principle-SatisfyingThird Order Direct Discontinu-ous Galerkin Methods for TimeDependent Convection Di↵usionEquations on Unstructured Trian-gular Mesh

Zheng ChenOak Ridge National Laboratory, USAHongying Huang, Jue Yan

In this talk, we show Direct discontinuous Galerkinmethod (DDG) and its variations satisfy the strictmaximum principle with at least third order accu-racy for convection di↵usion equations on unstruc-tured triangular meshes. The key contribution ofDDG is the introduction of numerical flux to ap-proximate the solution derivative at the discontin-uous element boundaries. We carefully calculate thenormal derivative numerical flux across the elementedges and prove that with proper choice of parame-ters in the numerical flux and suitable CFL condition,the piecewise quadratic polynomial solution satisfiesstrict maximum principle and maintains the third or-der accuracy at the same time. The implementationis very e�cient, with a simple Maximum-Principle-Satisfying (M-P-S) limiter applied after each timestepping. These methods have potential applicationsin physical models which require the solutions sat-isfying maximum principle or preserving positivityduring time evolution. Numerical examples includ-ing incompressible flows are carried out to show theoptimal 3rd order of accuracy is maintained with theM-P-S limiter applied.

Reduced Basis Methods for Linearand Nonlinear Equations and TheirApplications in Data Science

Yanlai ChenUniversity of Massachusetts Dartmouth, USA

Models of reduced computational complexity is indis-pensable in scenarios where a large number of numer-ical solutions to a parametrized problem are desiredin a fast/real-time fashion. These include simulation-based design, parameter optimization, optimal con-trol, multi-model/scale analysis, uncertainty quan-tification. Thanks to an o✏ine-online procedureand the recognition that the parameter-induced so-lution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM)and reduced collocation method (RCM) can improvee�ciency by several orders of magnitudes. The accu-racy of these spectrally convergent methods is main-tained through a rigorous a posteriori error estimatorwhose e�cient development is critical.In this talk, I will give a brief introduction of theRBM, discuss recent and ongoing e↵orts to developRCM for linear and nonlinear equations, explainhow the newly-designed Reduced Basis Decomposi-tion can be used for data compression and face recog-nition.

Semi-Implicit Integration FactorMethods on Sparse Grids for High-Dimensional Systems

Weitao ChenUniversity of California, Irvine, USADongyong Wang, Qing Nie

Numerical methods for partial di↵erential equationsin high-dimensional spaces are often limited by thecurse of dimensionality. Though the sparse grid tech-nique is popular for handling challenges such as thoseassociated with spatial discretization, the stabilityconditions on time step size due to temporal dis-cretization, such as those associated with high-orderderivatives in space and sti↵ reactions, remain. Here,we incorporate the sparse grids with the implicit in-tegration factor method (IIF) that is advantageousin terms of stability conditions for systems contain-ing sti↵ reactions and di↵usions. We combine IIF,in which the reaction is treated implicitly and thedi↵usion is treated explicitly and exactly, with vari-


ous sparse grid techniques based on the finite elementand finite di↵erence methods and a multi-level com-bination approach. The overall method is found tobe e�cient in terms of both storage and computa-tional time for solving a wide range of PDEs in highdimensions. In particular, the IIF with the sparsegrid combination technique is flexible and e↵ectivein solving systems that may include cross-derivativesand non-constant di↵usion coe�cients.

A New Discontinuous GalerkinMethod, Conserving the DiscreteH2-Norm, for Third-Order LinearEquations in One Space Dimension

Bo DongUniversity of Massachusetts Dartmouth, USAYanlai Chen, Bernardo Cockburn

We introduce a Bassi-Rebay type discontinuousGalerkin (DG) method for both stationary and time-dependent third-order linear equations. This methodis the first DG method which conserves the massand the L2-norm of the approximations of the so-lution and that of its first and second derivatives.For the stationary case, L2-projections of the er-rors (in the approximation of the solution, its firstand second derivatives) are proven to have optimalconvergence rates when the polynomial degree k iseven and the mesh is uniform, and to converge sub-optimally, but sharply, with order k when k is oddor the mesh is non-uniform. We show that suitablydefined projections of the errors superconverge withorder k+ 1+min{k, 1

2

} on uniform meshes and con-verge optimally on non-uniform meshes. The numer-ical traces are proven to superconverge with order2k if k is odd or the mesh is non-uniform. For evenk and uniform meshes, we show that the numericaltraces superconverge with order 2k + 3

2

. If in addi-tion, the number of intervals is odd, the convergenceorder is improved to 2k + 3

2

+ min{k, 1

2

}. This al-lows us to use an element-by-element postprocessingto construct new approximations that superconvergewith the same orders as the numerical traces. Forthe time-dependent case, the errors are proven to beof order k + 1 for even k on uniform meshes, andof order k when k is odd or the mesh is nonuniform.Numerical results are displayed which verify all of theabove-mentioned theoretical orders of convergence aswell as the conservation properties of the method.We also show that the orders of convergence of thethe stationary case also hold for the time-dependentcase.

Computational Methods for Ex-tremal Steklov Problems

Chiu-Yen KaoClaremont McKenna College, USAEldar Akhmetgaliyev, Braxton Osting

We develop a computational method for extremalSteklov eigenvalue problems and apply it to study theproblem of maximizing the p-th Steklov eigenvalue asa function of the domain with a volume constraint. Incontrast to the optimal domains for several other ex-tremal Dirichlet- and Neumann-Laplacian eigenvalueproblems, computational results suggest that the op-timal domains for this problem are very structured.We reach the conjecture that the domain maximizingthe p-th Steklov eigenvalue is unique (up to dilationsand rigid transformations), has p-fold symmetry, andan axis of symmetry. The p-th Steklov eigenvalue hasmultiplicity 2 if p is even and multiplicity 3 if p � 3is odd.

Ultraconvergence of Finite ElementMethod by Richardson Extrapola-tion

Runchang LinTexas A&M International University, USAWen-ming He, Zhimin Zhang

In this talk, two novel Richardson extrapolation op-erators are proposed to investigate local 2kth or-der ultraconvergence properties of the kth order La-grange finite element method for the second orderelliptic problems. We show that, for both tensorproduct Qk element and simplicial Pk element, thepost-processed displacement and gradient both have2kth order ultraconvergence at interior mesh nodesaway from the boundary. Numerical results are pro-vided to demonstrate the theoretical findings.

A Partitioned Approach for Com-puting Fluid-Structure Interaction

Jin WangUniversity of Tennessee at Chattanooga, USA

The interactions between fluid flows and immersedsolid structures are nonlinear multi-physics phenom-ena that have applications to a wide range of sci-entific and engineering disciplines. In this talk, wepresent a new partitioned approach to compute fluid-structure interaction (FSI) by extending the originaldirect-forcing technique and integrating it with theimmersed boundary method. The fluid and struc-tural equations are calculated separately via their re-spective disciplinary algorithms, and their solutionsonly communicate at the fluid-structure interface.The computational framework is capable of handlingFSI problems with sophisticated structures describedby detailed constitutive laws.


Two-Competing-Species Chemo-taxis Models

Qi WangSouthwestern University of Finance and Economics,Peoples Rep of ChinaJ. Hu, J. Yang, L. Zhang

In this talk, we consider a two-species chemotaxismodel with Lotka-Volterra dynamics. We show theexistence of both stationary and time-periodic pat-terns to this system. The e↵ects of chemotaxis rateand cellular growth are examed. It is shown that thissystem has very rich spatial-temporal dynamics.

The Modified Buckley-LeverettEquation

Ying WangUniversity of Oklahoma, USAYulong Xing

In this talk, I will present the modified Buckley-Leverett (MBL) equation, which models the under-ground oil recovery process. I will show some prelim-inary results for solving this equation using the DGmethods. This is a joint work with Yulong Xing.

A Posteriori Error Estimates of Lo-cal Discontinuous Galerkin Methodsfor the Generalized Korteweg-DeVries Equations

Yulong XingUniversity of California Riverside, USA

The Korteweg-de Vries (KdV) equation is a nonlinearmathematical model for the unidirectional propaga-tion of waves in a variety of nonlinear, dispersive me-dia. Recently it has attracted increasing attention astest-bed for the competition between nonlinear anddispersive e↵ects leading to a host of analytical issuessuch global existence and finite time blowup, etc.In this presentation, we construct, analyze, and nu-merically validate a class of discontinuous Galerkinschemes for the generalized KdV equation. We willprovide a posteriori error estimate through the con-cept of dispersive reconstruction, i.e. a piecewisepolynomial function which satisfies the GKdV equa-tion in the strong sense but with a computable forc-ing term enabling the use of a priori error estimationtechniques to obtain computable upper bounds forthe error. Both semi-discrete and fully discrete ap-proximations are studied.

Direct Discontinuous GalerkinMethod and Its Variations forSecond Order Elliptic Equations

Jue YanIowa State University, USAHongying Huang, Zheng Chen

In this paper, we study direct discontinuous Galerkinmethod and its variations for 2nd order ellipticproblems. A priori error estimate under energynorm is established for all four methods. Opti-mal error estimate under L2 norm is obtained forDDG method with interface correction and symmet-ric DDG method. A series of numerical examplesare carried out to illustrate the accuracy and capa-bility of the schemes. Numerically we obtain optimal(k + 1)th order convergence for DDG method withinterface correction and symmetric DDG method onnone uniform and unstructured triangular meshes.An interface problem with discontinuous di↵usion co-e�cients is investigated and optimal (k + 1)th orderaccuracy is obtained. Peak solutions with sharp tran-sitions are captured well. Highly oscillatory wave so-lutions of Helmholz equation are well resolved.

Discontinuous Galerkin Methodsfor Chemotaxis Models

Yang YangMichigan Technological University, USAXingjie Li, Chi-Wang Shu

In this talk, we will focus on local discontinu-ous Galerkin methods for Chemotaxis model, whichmight yield blow-up solutions. We first give the er-ror estimates based on two di↵erent finite elementspaces, and then proceed to the positivity-preservingtechnique to obtain positive numerical approxima-tions. Finally, we will numerically demonstrate howto find the blow-up time.

Recent Results on the OptimalError Estimates of the Local Dis-continuous Galerkin Method WhenSolving the Convection Di↵usionEquations

Qiang ZhangNanjing University, Peoples Rep of China

In this talk we will present two results about the op-timal error estimates of local discontinuous Galerkinmethods when solving the time-dependent convectiondi↵usion equations. The first one is the optimal er-ror estimates in the L2-norm when the generalizedalternating numerical flux is used, where the gener-alized Gauss-Radau projection plays the importantrole. The second one is the optimal error estimates inthe L2-norm on the fully-discrete local discontinuousGalerkin methods with the implicit-explicit Runge-Kutta time-marching. Some numerical experimentsare also given.


Special Session 103: Mixing in Dynamical Systems: Theory, Modelling,and Applications, from Micro- to Geophysical Scales

Sanjeeva Balasuriya, University of Adelaide, AustraliaGary Froyland, Univesity of New South Wales, Australia

The notion of mixing is central to the study of dynamical systems. This special session is devoted to variousaspects of mixing, including the optimisation or control of mixing, the quantification of mixing by statisticallaws, and the mathematical modelling of mixing processes in physical and geophysical systems. The sessionbrings together theoreticians, modellers, and practitioners from small scale microfluidic devices to planetaryscale geophysical processes.

Eigenmode Analysis of Advective-Di↵usive Transport by the CompactMapping Method

Patrick AndersonEindhoven University of Technology, NetherlandsMFM Speetjens, M. Giona

The present study concerns an e�cient spectral anal-ysis of advective-di↵usive transport in periodic flowsby way of a compact version of the di↵usive map-ping method. Key to the compact approach is therepresentation of the scalar evolution by only a smallsubset of the eigenmodes of the mapping matrix, andcapturing the relevant features of the transient to-wards the homogeneous state. This has been demon-strated for purely advective transport in an earlierstudy by Gorodetskyi et al. Phys. Fluids 24, (2012).Here this ansatz is extended to advective-di↵usivetransport and more complex 3D flow fields, moti-vated primarily by the importance of molecular dif-fusion in many mixing processes. The study exposedan even greater potential for such transport problemsdue to the progressive widening of the spectral gapsin the eigenvalue spectrum of the mapping matrixwith increasing di↵usion. This facilitates substan-tially larger reductions of the eigenmode basis com-pared to the purely advective limit for a given ap-proximation tolerance. The compact di↵usive map-ping method is demonstrated for a representativethree-dimensional prototype micro-mixer. This re-vealed a reliable prediction of (transient) scalar evo-lutions and mixing patterns with reductions of theeigenmode basis by up to a factor 2000. The accurateestimation of the truncation error from the eigen-value spectrum enables systematic determination ofthe spectral cut-o↵ for a desired degree of approxima-tion. The validity and universality of the presumedcorrelation between spectral cut-o↵ and truncationerror has been established. This has the importantpractical consequence that the cut-o↵ can a priori bechosen such that the truncation error remains withina preset tolerance. This o↵ers a way to systematically(and reliably) employ the compact mapping methodfor in-depth analysis of advective-di↵usive transport.

Detecting Coherent Sets withSpacetime Di↵usion Maps

Ralf BanischUniversity of Edinburgh, ScotlandPeter Koltai

Intuitively, coherent sets are subsets of the configu-ration space that stay together under the (possiblychaotic) dynamics. Many di↵erent approaches formaking this notion precise exist in the literature. Forexample, one approach defines coherent sets via spec-tral properties of the transfer operator, and anotherdefines coherent sets as tight bundles of trajectoriesby specifying a euclidean distance metric in space-time. We show that these two approaches can bereconciled: By replacing the Euclidean distance inspacetime with an augmented version of the distanceused in di↵usion maps, one can make contact withthe transfer operator notion of coherence in the in-finite data limit. The resulting numerical method,which can be used to extract coherent sets directlyfrom trajectory data, is related to similar methodsthat have been discussed in the past. We demon-strate its performance on several examples.

Mixing Asymptotics and Limit Lawsfor Random Intermittent Maps

Chris BoseUniversity of Victoria, CanadaWael Bahsoun

Non-uniformly expanding maps of the interval havebeen intensively studied, starting at least 20 yearsago, primarily since they are amongst the simplestsystems that model intermittency: for example poly-nomial mixing rates and non-CLT-type limit laws.Here we study random maps constructed from a pa-rameterized family of intermittent maps, drawing twoconclusions: 1. The speed of correlation decay (forHolder data) is completely determined by the mapwith the fastest mixing rate, independent of the ran-domizing process and 2. in cases where correlationdecay fails to be square summable, establishing aCLT or stable law (as appropriate) is dependent onboth the maps and the randomizing process. Specificexamples will be discussed.


Optimal Mixing Enhancement ofFlows

Gary FroylandUniversity of New South Wales, AustraliaNaratip Santitissadeekorn

We introduce a general purpose method for optimis-ing the mixing rate of steady or periodically-drivenflows. A nearby vector field is selected from a pre-specified neighborhood of the original vector field soas to maximise the mixing rate for flows generated byvector fields in that neighborhood. A linear optimiza-tion problem is solved to identify the optimal vectorfield. The perturbed flow may be easily constrainedto preserve the same invariant density (e.g. preservevolume) as the original flow, and various other natu-ral geometric constraints can be simply applied.

Transport and Mixing in DynamicalSystems Via Transfer Operators

Cecilia Gonzalez TokmanUniversity of Queensland, AustraliaGary Froyland, Anthony Quas, Thomas Wat-son

Transport and mixing properties of dynamical sys-tems are of interest to geophysical (among other)scientists, and they pose interesting challenges formathematicians. For example, large scale structuressuch as oceanic eddies and atmospheric vortices areconnected with important features of the global cli-mate, and their detection and tracking in complexmodels of the real world has been an active topicof mathematical research in recent years. In thistalk we will discuss recent advances and challenges,arising from the use of transfer operators to inves-tigate analytical and computational problems in thearea, ranging from mixing optimization to the de-tection and approximation of coherent structures innon-autonomous dynamical systems.

On Fast Computation of Finite-Time Coherent Sets Using RadialBasis Functions

Oliver JungeTechnical University Munich, GermanyGary Froyland

Finite-time coherent sets inhibit mixing over finitetimes. The most expensive part of the transfer oper-ator approach to detecting coherent sets is the con-struction of the operator itself. We present a numeri-cal method based on radial basis function collocationand apply it to a recent transfer operator construc-tion that has been designed specifically for purely ad-vective dynamics. The construction is based on a dy-namic Laplacian operator and minimises the bound-ary size of the coherent sets relative to their volume.

The main advantage of our new approach is a sub-stantial reduction in the number of Lagrangian tra-jectories that need to be computed, leading to largespeedups in the transfer operator analysis when thiscomputation is costly.

Hyperbolic Mixing Sets in Control-A�ne Systems

Christoph KawanUniversity of Passau, GermanyAdriano Da Silva

A control-a�ne system is a control system of the form

⌃ : x(t) = f0

(x(t)) +mX

i=1

ui(t)fi(x(t)), u 2 U ,

where f0

, f1

, . . . , fm are vector fields on a smoothmanifold M and U denotes the set of admissiblecontrol functions u : R ! Rm. Assuming U =L1(R, U) for a compact and convex set U ⇢ Rm,the set U can be equipped with the weak⇤-topologyof L1(R,Rm) = L1(R,Rm)⇤ and becomes a compactmetrizable space. A natural dynamical system on Uis the shift flow ✓tu = u(·+ t), t 2 R. Assuming thatfor every u 2 U and every initial value x (at time0) the solution '(t, x, u) of the above ODE is uniqueand defined on R, the map ' : R⇥M⇥U! M turnsout to be a cocycle over ✓ and

� : R⇥(U⇥M) ! U ⇥M, �t(u, x) = (✓tu,' (t, x, u)),

is a continuous skew-product flow, the so-called con-trol flow of ⌃. Mixing properties of � are stronglyrelated to controllability properties of ⌃. Underthe assumption of local accessibility, there is one-to-one correspondence between the maximal subsetsof M of approximate controllability (called controlsets) and the maximal topologically mixing subsetsof U ⇥ M . In this talk, we present several resultsabout the structure and the properties of control setswhich admit a uniformly hyperbolic splitting, reveal-ing remarkable analogies to the classical theory ofuniformly hyperbolic dynamical systems.

Coherent Families: Spectral Theoryfor Transfer Operators in Continu-ous Time

Peter KoltaiFreie Universitat Berlin, GermanyGary Froyland

The decomposition of the state space of a dynami-cal system into metastable or almost-invariant sets isimportant for understanding macroscopic behavior.This concept is well understood for autonomous dy-namical systems, and has recently been generalizedto non-autonomous systems via the notion of coher-ent sets. We elaborate here on the theory of coherentsets in continuous time for periodically-driven flowsand describe a numerical method to find families ofcoherent sets without trajectory integration.


Response Operators for MarkovProcesses in a Finite State Space:Radius of Convergence and Link tothe Response Theory for Axiom aSystems

Valerio LucariniUniversity of Hamburg, Germany

Using linear algebra we derive response operators de-scribing the impact of small perturbations to finitestate Markov processes. The results can be used forstudying empirically constructed finite state approx-imation of statistical mechanical systems. Recent re-sults on the convergence of the statistical propertiesof finite state Markov approximation of the dynamicsof a system in the limit of finer and finer partitionsof its phase space suggest that our results are robustin the Axiom A case. Our findings give closed for-mulas for the response theory at all orders of pertur-bation and provide matrix expressions that can bedirectly implemented in any coding language, plusproviding bounds on the radius of convergence of theperturbative theory. We relate the convergence tothe rate of mixing of the unperturbed system. Ourformulas can be used to recover previous findings ob-tained on the response of continuous time Axiom Asystems, by considering the generator of time evolu-tion for the measure and for the observables. A verybasic, low-tech, and computationally cheap analysisof the response of the Lorenz ‘63 model provides en-couraging results regarding the possibility of usingthe approximate representation given by finite stateMarkov processes to compute a system’s response.

Vortices in a Street Canyon

Rua MurrayUniversity of Canterbury, New ZealandGary Froyland, Jamie de Jong, MiguelMoyers-Gonzalez, Benjamin Roberts, PhilWilson

Among the many factors a↵ecting urban air qualityis the pattern of air circulation around clusters oftall buildings. Under steady wind conditions, each“street canyon” can support vortices that recirculatecontaminated air and delay the dispersal of pollu-tants. The number of vortices can depend on the rel-ative heights and spacing of the structures. This talkwill report on the use of transfer operator methodsfor open systems to identify vortices and study theirdependence on the geometry of the street canyon.

Memory Loss for NonstationaryOpen Dynamical Systems

William OttUniversity of Houston, USABrett Geiger, Andrew Torok

This talk is about an analog of decay of correlationsfor nonstationary open dynamical systems. By non-stationary, we mean that the dynamical model itselfvaries in time. Examples include dynamical processesevolving in slowly varying environments and dynam-ical systems with time-dependent parameters. Byopen, we mean that the phase space contains (possi-bly moving) holes through which trajectories escape.We formulate a notion of memory loss appropriate fornonstationary open systems and we provide a theo-retical framework that allows one to prove that mem-ory loss occurs at an exponential rate. We then ap-ply this theory to a sample setting - a class of mapsstudied by Cowieson and Saussol. We emphasize thatunlike the random dynamical systems setup, the sta-tionarity of the process is entirely irrelevant for ourpurposes.

Cluster-Based Extraction of Finite-Time Coherent Sets from TrajectoryData

Kathrin Padberg-GehleTU Dresden, GermanyGary Froyland

Coherent features in time-dependent dynamical sys-tems are di�cult to identify. Most identification al-gorithms require knowledge of the dynamical systemor high-resolution trajectory information, which inapplications may not be available. We present a fastand simple method that is based on spatio-temporalclustering of trajectory data. It provides a rough andrapid coherent structure analysis and is particularlyaimed at situations where the available information ispoor: there are few trajectories, the available trajec-tories do not span the full time duration under con-sideration, and there are missing observations withintrajectories.

Multiobjective Optimal Control ofCoherent Structures Using ReducedOrder Modeling

Sebastian PeitzPaderborn University, GermanySina Ober-Bløbaum, Michael Dellnitz

In a wide range of applications it is desirable tooptimally control a dynamical system with respectto concurrent, potentially competing goals. Thisgives rise to a multiobjective optimal control prob-lem where, instead of computing a single optimalsolution, the set of optimal compromises has to beapproximated. When the problem under consider-ation is described by a partial di↵erential equation(PDE), as is the case for fluid flow, the computa-


tional cost rapidly increases and renders a directtreatment infeasible. Reduced order modeling is apopular method to reduce the computational cost, inparticular in a multiquery context such as optimiza-tion. In this presentation, we show how to combinereduced order modeling and multiobjective optimalcontrol techniques in order to e�ciently solve mul-tiobjective optimal control problems constrained byPDEs. We consider a global, derivative free opti-mization method as well as a local, gradient basedapproach for which we derive the optimality system.The methods are compared with regard to the solu-tion quality as well as the computational e↵ort, andthey are illustrated using the example of the two-dimensional incompressible flow around a cylinderwhere we want to minimize the occurence of coher-ent structures (i.e. vortex shedding) and the controlcost.

Global B-Pullback Attractors forCocycles Generated by Discrete-Time Cardiac Conduction Models

Volker ReitmannSt. Petersburg State University, RussiaAnastasia Maltseva

We investigate the existence and the structure ofglobal B-pullback attractors for cocycles generatedby nonautonomous di↵erence equations. As an ex-ample of such equations some discrete-time modelsof cardiac conduction systems are considered. Weuse the transfer function of the linear part of thesystem and some properties of the nonlinear part inorder to prove the dissipativity of the given systemwith the help of the Yakubovich-Kalman frequencytheorem. Employing the dissipativity the existenceof the global B-pullback attractor for the discretesystem under a perturbation, which is considered asa discrete-time cocycle, is shown. We state condi-tions under which the following is true: if the class ofperturbations is almost periodic then the global B-pullback attractor is also almost periodic, in the caseof perturbations given by a stationary Gaussian pro-cess, which is mixing, the resulting global B-pullbackattractor of the cocycle is also represented by a mix-ing stationary Gaussian process.

Lagrangian Coherent Structuresand The Non-Advective Transportof Potential Vorticity in TropicalCyclones

Blake RutherfordNorthwest Research Associates, USA

During tropical cyclogenesis, a flow topology emergesin a wave-relative reference frame that indicates aquasi-closed circulation in the lower troposphere.Considering time-dependent flows introduces topo-logical changes in the flow field. While a layer-wise2D approximation is reasonable under tropical ap-proximations, various 3D flow components are im-portant to the evolution of the vorticity field. These

3D mechanisms may arise from small-scale convec-tion or from large-scale environmental flow, but theyhave important consequences for development of thecyclone. Specifically, they can further alter the topol-ogy of the flow field and allow for the combination ofregions with very di↵erent thermodynamic proper-ties. By viewing Lagrangian coherent structures thatarise purely from the 2D approximation, we can seeall of the layer-wise topological changes in the flow.For any monotonic vertical coordinate, the imperme-ability principle states that there is no net verticaltransport of the potential vorticity (pv). Variationsin pv follow from a pv tendency equation and repre-sent changes due to vortex tilting, diabatic heatingand friction. An evaluation of the tendency equa-tions along time evolving Lagrangian coherent struc-tures gives a complete picture of the mixing of pv asa layer-wise 2D component tied to topological rear-rangement plus a 3D flux of pv that can be viewed asacting only at the boundary due to Stokes‘ theorem.

Linear Response in the IntermittentFamily

Benoit SaussolUniversity of Brest, FranceWael Bahsoun

A standing question in dynamical systems is to un-derstand how statistical properties of a perturbedsystem are related to the original system. In partic-ular the physical measure (or SRB measure) of thesystem may depend in a di↵erentiable way on the pa-rameters of the system. In the physics literature thisis called linear response. We prove a linear responseformula for the intermittent family x 7! x+ 2↵x1+↵,that is the di↵erentiability of the density h↵ with re-spect to the parameter ↵.

Bounds for Generalised LyapunovExponents for Random Products ofShears

Rob SturmanUniversity of Leeds, EnglandJean-Luc Thi↵eault

We give lower and upper bounds on both the Lya-punov exponent and generalised Lyapunov exponentsfor the random product of positive and negative shearmatrices. These types of random products arise inapplications involving randomized stirring devices.The bounds, obtained by considering invariant conesin tangent space, give excellent accuracy comparedto standard and general bounds, and are increas-ingly accurate with increasing shear. Bounds on gen-eralised exponents are useful for testing numericalmethods, since these exponents are di�cult to com-pute.


Improving Mixing in Micro FluidicSystems by Introducing Partial Slipto Break Symmetry

Pushpavanam SubramaniamI.I.T. Madras, IndiaGowtham Sankaran, Piyush Garg, JasonPicardo

Mixing in flow systems in micro-fluidic systems is achallenge in view of the small length scales which pre-vail in these devices. In the literature several mecha-nisms for mixing have been proposed which also endup increasing the pressure drop. In this work wediscuss how a serpentine geometry can be used toimprove mixing. Here the naturally occurring Deansvortices cause mixing in the direction normal to theflow. The symmetry in these vortices is broken byintroducing a a periodic slip on the walls. The peri-odicity in the two walls is phase shifted. This resultsin creating a situation which is similar to the classicalblinking vortex. The blinking vortex now evolves spa-tially. as the liquid moves through the channel. Theflow is analysed using Poincare maps and other scalarindicators. In addition to single phase flows we dis-cuss how the analysis can be extended to a slug flowregime in two phase flows.. The partial slip resultsin a decrease in the pressure drop across the channel.The flow field is computed using a semi- analyticalapproach which makes the algorithms computation-ally e�cient.

Scalar Density Evolution with La-grangian Measures

Wenbo TangArizona State University, USAPhillip Walker

In integrable shear flows, the second order momentsof a passively advected and di↵usive scalar can becomputed analytically to obtain the spatial- andtemporal-dependent e↵ective di↵usivity. In non-integrable flows, such analytical results are not ac-cessible. In this talk, we discuss a semi-analyticframework, which reconstructs scalar density evolu-tion from finite-time Lagrangian measures. We payclose attention to the stretching and shearing com-ponents of the deformation tensor, both playing im-portant roles in shaping scalar patches. With properchoice of time-scale for mapping the scalar densityforward, we show that the scalar density field can beresolved without having to solve the transport equa-tion.

Reduced Transfer Operator Ap-proach to Mixing and Stability inChaotic and Stochastic Systems

Alexis TantetIMAU, Utrecht University, NetherlandsMickael D. Chekroun, Valerio Lucarini,J. David Neelin, Henk A. Dijkstra, FrankLunkeit

Much can be learned about systems exhibiting com-plex dynamics by studying the evolution of proba-bility densities rather than single trajectories. Thisevolution is governed by the semigroup of transfer op-erators which allows to connect the correlation func-tion to the Liouville or the Fokker-Planck equation.Yet, their approximation quickly becomes intractablewhen the dimension of the phase space is large.We propose to approximate the transfer operatorsby Markov operators on a reduced space. Whilethese Markov operators do not in general constitutea semigroup, rigorous results can be obtained regard-ing their spectral properties, in particular allowing toreconstruct correlation functions and quantify mixingin the reduced space.The approach is applied to the study of the variabil-ity and the stability of chaotic and stochastic systemsrelevant for climate. New analytical and numericalresults are shown for the Hopf bifurcation with addi-tive noise, bringing new insights on the phenomenaof noise-induced oscillation and phase di↵usion. Fi-nally, it is found for a chaotic attractor crisis thatthe slowing down of the decay of correlations is as-sociated with the shrinkage of the reduced spectralgap, improving our understanding of early-warningsignals for high-dimensional systems.

Can We Define Finite-Time Gener-alized Lyapunov Exponents?

Jean-Luc Thi↵eaultUniversity of Wisconsin – Madison, USAMarko Budisic

One way to define Lyapunov exponents is in terms ofthe stretching of small vectors. If the relative growthof vectors is a random variable `(t), then the expo-nent is � = t�1hlog `i, where the brackets are anensemble average over trajectories. Under suitableassumptions, this quantity converges as t ! 1 or asthe number of samples increases. This double con-vergence allows the definition of finite-time Lyapunovexponents, which have proved useful in analyzing thedetailed structure of mixing flows. Generalized Lya-punov exponents are defined in terms of the expec-tation h`pi, for some constant p. These have verydi↵erent convergence properties, which we discuss.We ask if anything can be done to define these ina finite-time sense, and what could be learned fromsuch a definition.


Central Limit Theorems for Se-quential and Random IntermittentDynamical Systems

Andrew TorokUniversity of Houston, USAM. Nicol, S. Vaienti

We establish self-norming central limit theorems fornon-stationary time series arising as observations onsequential maps possessing an indi↵erent fixed point.These transformations are obtained by perturbingthe slope in the Pomeau-Manneville map. We alsoobtain quenched central limit theorems for randomcompositions of these maps.


Special Session 104: Nonlinear Elliptic Equations and FractionalLaplacian

Qianqiao Guo, Northwestern Polytechnical University, Peoples Rep of ChinaWenxiong Chen, Yeshiva University, USA

We will exchange new developments, ideas, and methods in the area of nonlinear elliptic partial di↵erentialequations and systems including those involving fractional Laplacians and other nonlocal operators. We willstudy qualitative properties, such as existence, symmetry, classification, monotonicity, asymptotic behavior,and nonexistence of solutions for the above mentioned equations.

Global Dynamics of CompetitionModels with Fractional Laplacian/Nonlocal Dispersal

Xueli BaiNorthwestern Polytechnical University, Peoples Repof ChinaFang Li

In this talk, we consider the global dynamics of sevralcompetition models with fractional Laplacian or non-local dispersal. Since all the systems we consid-ered here are monotone, thus our aim is deriving theuniqueness of the positive steady state..

Uniform Approach to Sharp Hardy-Littlewood-Sobolev Type Inequali-ties

Qianqiao GuoNorthwestern Polytechnical University, Peoples Repof ChinaJingbo Dou, Meijun Zhu

In this talk we try to give a (new) uniform approachto prove the sharp Hardy-Littlewood-Sobolev typeinequalities, including reversed Hardy-Littlewood-Sobolev type inequalities, which should be useful tosome integral equations.

Ground State of NonlinearSchrodinger Equation with Frac-tional Laplacian

Zhiqing HanDalian University of Technology, Peoples Rep ofChinaZupei Shen, Chuanfang Zhang

In this talk, I will present some results for the equa-tion involving a fractional Laplacian: (��)↵ + u =f(u), x 2 RN . By using Mountain Pass Theoremwith a rearrangement argument, we prove the exis-tence of symmetry mountain pass solutions withoutthe Ambrosetti-Rabinowitz condition. We aslo get anonnegative symmetry ground state in the fractionalSobolev space H↵(RN ). rearrangement

The Di↵usive Competition Problemwith a Free Boundary in Heteroge-neous Time-Periodic Environment

Fengquan LiDalian University of Techology, Peoples Rep ofChinaQiaoling Chen, Feng Wang

In this talk, we will discuss the di↵usive competitionproblem with a free boundary and sign-changing in-trinsic growth rate in heterogeneous time-periodic en-vironment, consisting of an invasive species with den-sity u and a native species with density v. We assumethat v undergoes di↵usion and growth in RN , and uexists initially in a ball Bh0(0), but invades into theenvironment with spreading front {r = h(t)}. Thee↵ect of the dispersal rate d

1

, the initial occupyinghabitat h

0

, the initial density u0

of invasive species u,and the parameter µ (see (1.3)) on the dynamics ofthis free boundary problem are studied. A spreading-vanishing dichotomy is obtained and some su�cientconditions for the invasive species spreading and van-ishing are provided. Moreover, when spreading ofu happens, some rough estimates of the spreadingspeed are also given.

Symmetry Results for a SystemInvolving the Fractional Laplacians

Yan LiYeshiva University, USAPei Ma

In this paper, we study non-negative solutions for asystem involving the fractional Laplacians:

8<

:

(�4)↵/2u(x) = f(v(x)),(�4)�/2v(x) = g(u(x)),u, v � 0, x 2 Rn,

(1)

where ↵,� 2 (0, 2). Using a direct method of themoving planes, we obtain symmetry results of solu-tions.


Nonexistence of Positive Solutionsfor a System of Semilinear IndefiniteFractional Laplacian Problem

Ye LiCentral Michigan University, USAJingbo Dou

We consider a system of semilinear indefinite equa-tions involving the fractional Laplacian in the Eu-clidean space Rn:

⇢(��)↵/2u(x) = f(xn)v

p(x)(��)↵/2v(x) = g(xn)u

q(x)

in the subcritical case 1

Controllability for a Class of Semi-linear Fractional Evolution SystemsVia Resolvent Operators

Yansheng LiuShandong Normal University, Peoples Rep of ChinaDaliang Zhao

This talk is concerned with the exact control fora class of fractional evolution systems in a Banachspace. First, we introduce a new concept of exactcontrol and give the mild solutions of considered evo-lution system via re-solvent operators. Second, byutilizing the semi-group theory, fixed point strategyand measure of non-compactness, the exact control ofthe evolution system has been investigated withoutLipschitz continuity and growth conditions imposedon nonlinear functions. The results are establishedunder the hypothesis that re-solvent operator is dif-ferentiable and analytic respectively instead of sup-posing that the semi-group is compact. An exampleis provided to illustrate the abstract results.

The Pohozaev Identity of the Frac-tional Laplacian System

Pei MaYeshiva University, USAFengquan Li, Yan Li

In this paper, we study the Pohozaev identity asso-ciated with a Henon-Lane-Emden system involvingthe Fractional Laplacian:

8><

>:

(��)su = |x|avp, x 2 ⌦,

(��)sv = |x|buq, x 2 ⌦,

u = v = 0, x 2 Rn\⌦,

in a star-shaped and bounded domain for s 2 (0, 1).And then using the Pohozaev Identity result, we de-rive the nonexistence of positive solutions.

Time-Harmonic Solutions to Non-linear Maxwell Equations on aBounded Domain

Jaroslaw MederskiNicolaus Copernicus University, PolandThomas Bartsch

We find solutions E : ⌦ ! R3 of the problem(r⇥ (µ(x)�1r⇥ E)� !2"(x)E = @EF (x,E) in ⌦

⌫ ⇥ E = 0 on @⌦

on a bounded Lipschitz domain⌦ ⇢ R3 with exte-rior normal ⌫ : @⌦ ! R3. Here r⇥ denotes the curloperator in R3. The equation describes the propaga-tion of the time-harmonic electric field <{E(x)ei!t}in an anisotropic material with a magnetic perme-ability tensor µ(x) 2 R3⇥3 and a permittivity ten-sor ✏(x) 2 R3⇥3. The boundary conditions are thosefor ⌦ surrounded by a perfect conductor. It is re-quired that µ(x) and ✏(x) are symmetric and posi-tive definite uniformly for x 2 ⌦, and that µ,✏ 2L1(⌦,R3⇥3). The nonlinearity F : ⌦ ⇥ R3 ! R issuperquadratic and subcritical in E, the model non-linearity being of Kerr-type: F (x,E) = |�(x)E|p.

On Nonnegative Solutions to El-liptic Di↵erential Inequalities onRiemannian Manifolds

Yuhua SunNankai University, Peoples Rep of ChinaA.Grigor‘yan

We provide optimal condition in terms of the volumegrowth of a Riemannian manifold that ensures thatany non-negative solution to the inequality ellipticdi↵erential inequalities on this manifold is identicallyequal to 0.

Properties of Solutions of Sub-Elliptic Equations with SingularNonlinearities on Heisenberg Group

Xinjing WangNorthwestern Polytechnical University, Peoples Repof ChinaPengcheng Niu

In this work, we consider positive solutions for thezero Dirichlet boundary condition to the singularsemilinear sub-elliptic equation on the Heisenberggroup

��Hu =1u�

+ f(u),

in a bounded smooth domain in the Heisenberggroup. We provide the monotonicity and the sym-metry of cylindrical solutions to the problem. Themain technique is the generalization of moving planemethod to the Heisenberg group.


Unified Weighted Poincare Inequal-ities in Metric Measure Space andApplications

Huiju WangNorthwestern Polytechnical University, Peoples Repof ChinaPengcheng Niu

In this work we establish unified weighted Poincareinequalities in metric measure spaces. A new class ofhigher order Poincare inequalities in the Euclideanspace is given. We obtain weighted higher orderPoincare inequalities in the Euclidean space andstratified Lie groups, respectively.

Some Stability Results of the Large-Amplitude Traveling Fronts forTra�c Flow Models

Lina WangBeijing Technology and Business University, PeoplesRep of ChinaJingyu Li, Tong Li, Yaping Wu

In this talk, we will focus on the stability of the trav-eling fronts for tra�c flow models with uniform roadwidth and non-uniform road width. By applying ge-ometric singular perturbation method, special Evansfunction estimates, detailed spectral analysis and C

0

semigroup theories, the linear exponential stabilityof the non-degenerate waves with large wave strengthin some exponentially weighted spaces will be shown.At the same time, the convergence rate of solutionsto traveling fronts with small wave strength will alsobe given.

Green Functions for WeightedSubelliptic p-Laplacian OperatorsConstructed by Hormander’s Vec-tor Fields

Leyun WuNorthwestern Polytechnical University, Peoples Repof ChinaPengcheng Niu

This work deals with the following weighted subellip-tic p-Laplacian constructed by Hormander’s vectorfields:

Lpu = divX⇣hA(x)Xu(x), Xu(x)i

p�22 A(x)Xu(x)

⌘,

where u 2 W 1,p(U,w), 1 < p < Q,A(x) is a boundedmeasurable and m⇥m symmetric matrix satisfying

��1w(x)2/p|⇠|2 6 hA(x)⇠,⇠ i 6 �w(x)2/p|⇠|2,⇠ 2 Rm, w(x) 2 Ap.

We first prove existence of the modified Green func-tion of Lp by virtue of Minty-Browder theorem andthen existence of the Green function of Lp by provingthe convergence of sequence of modified Green func-tions. Next, we derived upper bounds of the modified

Green function of Lp by establishing the interpolationinequality in the weighted weak Lp spaces. Finally,the bounds of the Green function of Lp are also ob-tained by virtue of results for the modified Greenfunction and the weighted compact embedding theo-rem.

On the Principal Eigenvalue of aPerturbed Fractional Laplace Oper-ator

Guangyu ZhaoCentral Michigan University, USA

This talk presents our recent study on the principaleigenvalue problem of a perturbed fractional Laplaceoperator:

⇢(��)su+ c(x)u = �u, in⌦ ,u = 0 in Rn \ ⌦.

Here ⌦ is a bounded domain of Rn (n � 1) withsmooth boundary, c 2 L1(⌦), and s 2 (0, 1). Ourwork extends a number of well-known propertiesof the principal eigenvalue of the Laplace operatorto the aforementioned fractional Laplace operator.More specifically, the established results, among theother things, reveal the equivalence between the va-lidity of a strong maximum principle and the positiv-ity of the principal eigenvalue. The similar charac-terizations were also obtained for a fractional Laplaceoperator associated with weakly coupled cooperativesystems. As an application, these results are utilizedto investigate the spatio-temporal dynamics of a fewmathematical models that arise from population bi-ology and mathematical epidemiology.

Extension of Hardy-Littlewood-Sobolev Inequality on CompactManifolds

Meijun ZhuUniversity of Oklahoma, USA

In this talk, I will survey our recent work onthe extension of Hardy-Littlewood-Sobolev inequal-ity on compact Riemannian manifolds with or with-out boundary. The study of the sharp forms andrelated integral equations will be addressed.


Special Session 105: Recent Advances in Computational PDEs and theirApplications

Xinfeng Liu, University of South Carolina, USAHong Wang, University of South Carolina, USA

Huanzhen Chen, Shandong Normal University, Peoples Rep of China

This minisymposium focuses on recent advances on e�cient and accurate numerical methods for PDEs andtheir various applications in physical or biological systems. The invited researchers from diverse backgroundwill discuss a wide range of computational methods ranging from e�cient finite element and finite di↵erencemethods, adaptive methods, multiscale methods, to spectral methods and kinetic Monte Carlo simulations.Computational challenges will be discussed, and new computational techniques will be introduced for variousapplications.

Adjoint-Free Calculation Methodfor Conditional Nonlinear OptimalPerturbations

Ming CuiBeijing University of Technology, Peoples Rep ofChina

Adjoint-free calculation method is proposed to com-pute conditional nonlinear optimal perturbations(CNOP) combined with initial perturbations andmodel parameter perturbations. The new approachavoids the use of adjoint technique in the opti-mization process. CNOPs respectively generated byensemble-based and adjoint-based methods are com-pared based on a simple theoretical model.

Continuous Galerkin Method forDelay Di↵erential Equations of Pan-tograph

Qiumei HuangBeijing University of Technology, Peoples Rep ofChinaHermann Brunner, Xiuxiu Xu

We analyze the optimal global convergence and localsuperconvergence properties of continuous Galerkin(CG) solutions on uniform meshes and quasi-geomteric meshes for delay di↵erential equations withproportional delay. It is shown that the attainableorder of nodal superconvergence of CG solutions un-der quasi-geomteric meshes is higher than of the oneunder uniform meshes. The theoretical results areillustrated by a broad range of numerical examples.

Numerical Simulation forFractional-Order Di↵usion Equa-tions

Chen HuanzhenShandong Normal University, Peoples Rep of China

In this talk we adopt the saddle-point theoreti-cal framework to analyze the conservative space-fractional di↵usion equations. By introducing an in-termediate Hilbert space and a fractional-order fluxas auxiliary variable, we establish the well-posednessof the saddle-point variational formulation and bet-

ter regularity of the solution. A locally-conservativemixed finite element procedure based on the formu-lation is proposed to approximate the unknown, itsderivative and the fractional flux directly. Existenceand uniqueness results are proven and the error es-timates are derived. Numerical experiments are in-cluded that confirm our theoretical findings.

Petrov-Galerkin Methods for Frac-tional Convection Di↵usion Problem

Bangti JinUniversity College London, EnglandRaytcho Lazarov, Zhi Zhou

In this work, we develop variational formulationsof Petrov-Galerkin type for boundary value prob-lems involving either a Riemann-Liouville or Ca-puto derivative of order ↵ 2 (3/2, 2) in the leadingterm and a convection term. The well-posedness ofthe formulations and sharp regularity pickup of theweak solutions are established. A novel finite ele-ment method is developed, which employs contin-uous piecewise linear finite elements and “shifted”fractional powers for the trial and test space, respec-tively. It admits optimal error estimates in both L2-and H1-norms. Extensive numerical results are pre-sented to verify the theoretical analysis and robust-ness of the numerical scheme.

E�cient and Tunably AccurateSpectral Methods for FractionalDi↵erential Equations on the Half-Line

Anna LischkeBrown University, USAMohsen Zayernouri, George Karniadakis

In this talk, we introduce new Laguerre Petrov-Galerkin spectral methods for fractional di↵erentialequations on the half-line. We demonstrate the tun-able accuracy of these methods and the sensitivityof the accuracy due to the tuning parameter usingnumerical experiments. We also show that these ap-proaches result in computationally e�cient methodsfor solving multi-term FDEs on the half-line.


Integration Factor Method for aClass of Di↵erential Equations

Xinfeng LiuUniversity of South Carolina, USA

In this talk, we will present an e�cient high-order in-tegration factor method for solving a family of highorder di↵erential equations, in which the linear highorder derivatives are explicitly handled and the com-putational cost and storage remain the same as to theclassic integration factor methods for second-orderproblems. In particular, the proposed method candeal with not only sti↵ nonlinear reaction terms butalso various types of homogeneous or inhomogeneousboundary conditions. Also such method has recentlybeen extended to solve a hydrodynamic phase fieldmodel for a binary fluid mixture of two immiscibleviscous fluids.

New Central Schemes on Over-lapping Cells for Solving IdealMagnetohydrodynamic Equations

Yingjie LiuGeorgia Institute of Technology, USAZhiliang Xu

We develop a new central DG-type method on over-lapping cells for solving MHD equations on triangu-lar meshes. This method is fully conservative for themagnetic field. New features are introduced to re-duce the complexity.

Polynomial Approximate Solutionsfor a Non-Darcy Groundwater FlowEquation

Aleksey TelyakovskiyUniversity of Nevada, Reno, USAJe↵rey Olsen, Je↵ Mortensen

Certain kinds of flows in groundwater aquifers aremodeled by a nonlinear Forchheimer equation. Weconsider semi-infinite aquifer that is initially dry. Atthe inlet boundary conditions are specified. For cer-tain types of the boundary conditions problem canbe reduced using similarity variables to a boundary-value problem for a nonlinear ordinary di↵erentialequation. We derive polynomial approximate solu-tions to that nonlinear ordinary di↵erential equation.Our approach shows good results when it is comparedwith the numerical solution obtained with a rescalingalgorithm.

Local High Order Absorbing Bound-ary Conditions in Terms of FarfieldExpansions

Vianey VillamizarBrigham Young University, USASebastian Acosta, Blake Dstrup

A new local high order absorbing boundary condi-tion (ABC) for scattering of time-harmonic wavesfrom obstacles of arbitrary shape is devised. Firstthe infinite domain ⌦ is truncated by means of anartificial boundary B. This results in a division ofthe original infinite domain into a finite computa-tional domain⌦ � and an exterior infinite domain⌦+. Then, we define interface conditions at the arti-ficial boundary B from truncated versions of Wilcox’sfarfield expansion in 3D and Karp’s farfield expansionin 2D. As a result, we obtain a new local ABC for abounded problem on⌦ �, which e↵ectively accountsfor the outgoing behavior of the scattered field. Con-trary to what happens to other ABCs previously de-fined, the order of approximation of the farfield pat-tern can be increased to any order. We accomplishthis by adding as many terms as needed to the trun-cated farfield expansions. We include numerical re-sults which demonstrate the improved accuracy whencompared to other absorbing boundary conditions.

A Fast Collocation Method for aBond-Based Peridynamic Model

Hong WangUniversity of South Carolina, USAXuhao Zhang

We develop a fast collocation method for a two-dimensional linear steady-state bond-based peridy-namic model, which provides an appropriate descrip-tion of the planar deformation of a continuous elasticbody involving discontinuities or other singularities.The method reduces the computational cost of evalu-ating and assembling the sti↵ness matrix from O(N2)to O(N), where N is the number of unknowns in thediscrete system. The method also reduces the compu-tational work from O(N2) to O(N log N) per Krylovsubspace iteration and the memory requirement fromO(N2) to O(N). All of this is achieved by carefullyexploring the structure of the sti↵ness matrix of thecollocation scheme, without any lossy compressioninvolved. Numerical results are presented to showthe utility of the method.


Split-Step Orthogonal Spline Col-location Methods for NonlinearSchrodinger Equations in One,Two, and Three Dimensions

Shanshan WangNanjing University of Aeronautics and Astronautics,Peoples Rep of ChinaLuming Zhang

Split-step orthogonal spline collocation (OSC) meth-ods are proposed for one-, two-, and three-dimensional nonlinear Schrodiner (NLS) equationswith time-dependent potentials. The original NLSequations are separated into two nonlinear equations,and one or more one-dimensional equations by thesplit-step method. Discrete-time OSC Schemes areapplied to solve the linear subproblems. Commonly,the nonlinear subproblems could be integrated di-rectly and accurately, but it fails when the time-dependent potential cannot be integrated exactly. Inthis case , three approximations based on the quadra-ture formulae is used, and split order in not reduced.Extensively numerical tests are carried out to verifythe reliability and e�ciency of the present method.The three approximations are applied to the split-step finite di↵erence methods and the time-splittingspectral methods, and they also work well.

A Phase Field Model and EnergyStable Scheme for Simulating Multi-Phase Viscoelastic IncompressibleFlow

Zhiliang XuUniversity of Notre Dame, USAShixin Xu, Mark Alber

Di↵usion interface (phase field) method is one of themost important approaches for studying multi-phasefluids due to its treatment of the interface as a physi-cally di↵use thin layer. In this talk, we will present athermodynamically consistent model for the mixtureof Newtonian and viscoelastic fluids derived by usingthe Energetic Variational Approach. Elastic propertyof the viscoelastic fluid is described by the deforma-tion gradient tensor in an Eulerian framework. Dif-ferent densities and viscosities of distinct fluid phasesare taken into account. This new model is shown tosatisfy the law of energy dissipation automatically.An energy stable numerical scheme is proposed tosolve the coupled system of model equations. Nu-merical experiments using problem with large den-sity ratio and problem with mixture of Newtonianand viscoelastic fluids are carried out to validate themodel and the scheme. Specifically, the model is usedto simulate deformation of a blood clot under shearflow.


Special Session 106: Nonlinear Waves: Coherent Structures andComplex Dynamics

S. Roy Choudhury, University of Central Florida, USAAndrei Ludu, Embry-Riddle Aeronautical University, USA

This session will consider recent advances in the field of nonlinear wave propagation, including coherentstructures, Bose-Einstein condensates, and a variety of dynamical behaviors. Application will primarily bedrawn from, but will not limited to, water waves and nonlinear optics.

Quantitative Dynamics and Routesto Chaos in a Chemical System

Sudipto ChoudhuryUniversity of Central Florida, USA

A global study of an generic family of chemical re-actions is carried out to analyze their primary com-plex dynamical behaviors. The study proceeds from aconstruction of periodic solutions of the system con-structed via a harmonic balance method. Bifurca-tions of such periodic orbits are then systematicallyconsidered, with secondary Hopf (Neimark-Sacker)bifurcations leading to quasiperiodic solutions andone route to chaos, a first period doubling initiat-ing an infinite sequence in another well-known pathto chaos, and so on. The analytical results give afairly comprehensive general picture of possible sys-tem behaviors in the parameter space, and comparewell when tested against numerical simulations. Thepredictions also provide a good guide to numericallysearching for di↵erent dynamical behaviors in variousparts of the parameter space.

Fractional Partial Di↵erential Equa-tion for Rapid Growing Systems

Andrei LuduEmbry-Riddle Aeronautical University, USA

A short review on fractional ordinary/partial di↵er-ential equations (FODE, FPDE) and some of theirnovel applications developed in the last decade willbe presented, followed by a presentation of the im-proved fractional sub-equation method and the spec-tral method to solve such equations, including non-linear PDE inspired by solitons in fluid dynamics. Anew type of super-dynamical FPDE will be presentedtogether with its potential applications in very fasttime growing systems like those related to Kryder’s,Reed’s, Nielsen’s, and Carlson’s laws, etc.

JONSWAP Rogue Waves OverNon-Constant Backgrounds

Constance SchoberUniversity Central Florida, USAAnna Calini

JONSWAP rogue waves are large amplitude waveson a non-constant background that are distinct fromprior analytical models, including spatially peri-odic breathers and rational solutions of the Nonlin-ear Schrodinger (NLS) equation, which are all con-

structed over a constant background. In this talkwe classify JONSWAP rogue waves using the inversespectral theory of the NLS equation and constructNLS solutions that model JONSWAP rogue waves.The associated spectral configurations are more com-plex that those of prior models, but also in thesecases proximity to instabilities is the main indica-tor of rogue wave occurrence. To support this claim,we correlate the maximum wave strength as well asthe higher statistical moments with elements of thenonlinear spectrum. The result is a diagnostic toolwidely applicable to both model and field data forpredicting the likelihood of rogue waves.

Solitary Waves for Long Wave-ShortWave Interaction System

Sharad SilwalJe↵erson College of Health Sciences, USASantosh Bhattarai

In this talk, we prove the existence of solitary wavesolutions of a system of long wave-short wave inter-action equations. The system here describes the in-teraction between multiple short waves and a longwave. The existence of solitary waves is establishedvia a variational method and using the concentrationcompactness argument. We also prove the stabilityof the set of minimizers.

Numerical Investigation of the Dy-namics of High-Intensity UltrashortLight Pulses

Michail TodorovTechnical University of Sofia, Bulgaria

The pulse propagation in nonlinear bulk medium gov-erned by the (3+1)D nonlinear Schrodinger equation(NLSE) as well as by the (3+1)D nonlinear enve-lope equation (NEE) is investigated. Implicit finite-di↵erence schemes for the both dynamical systemsare developed. An operator splitting by physical fac-tors of these types of equations is applied and itssuitability is grounded. For the non-linear terms ofthe equations is used the so-called internal iteration.Numerical treatment in complex arithmetic is carriedand the generalized Thomas algorithm for multidiag-onal complex banded matrices with pivoting is used.


Significant and consistent physical results are ob-tained. The minimal number of processes providinga self-compression of the pulse is established. A newregime of propagation of the pulse is found and acontrollable guidance concept as alternative to thesoliton concept based on (3+1)D NLSE is proposed.

For (3+1)D NEE realistic propagation regimes areconsidered and a self-compression of the pulse as wellas a stable propagation of the compressed pulse arefound out. The strong influence of ionization on thegroup-velocity dispersion can cause even its inversionfrom positive to negative one.


Special Session 107: Analysis of Nonlinear Dispersive Wave Equationsand Integrable Systems

Stephen Anco, Brock University, CanadaRobin Ming Chen, University of Pittsburg, USAYue Liu, University of Texas at Arlington, USA

Changzheng Qu, Ningbo University, Peoples Rep of China

This session will bring together researchers at all career stages to share their recent results on various aspectsin the analysis of nonlinear dispersive wave equations and integrable structures. It will focus on (but notbe restricted to) developments connected with derivation of nonlinear equations from physical models andintegrable structures, including well-posedness, stability analysis, blow-up, wave breaking, and geometricaspects. Of special interest are equations related to the theory of water waves, such as the Camassa-Holmequation, Whitham equation, Hunter-Saxton equation, Degasperis-Procesi equation, NLS equation, andtheir various recent generalizations.

On the Symmetry of Traveling WaveSolutions to the Whitham Equation

Gabriele BruellNTNU Trondheim, NorwayMats Ehrnstrom, Anna Geyer, Long Pei

The Whitham equation is a nonlocal, nonlinear dis-persive wave equation introduced by G. B. Whithamas an alternative wave model equation for theKorteweg-de Vries equation, describing the wave mo-tion at the surface on shallow water. Knowing thattraveling wave solutions to the Whitham equationexist, we prove that any solitary wave solution issymmetric and has exactly one crest. Moreover, thestructure of the Whitham equation allows to con-clude that conversely any classical symmetric solu-tion constitutes a traveling wave. In fact, the latterresult holds true for a large class of partial di↵erentialequations sharing a certain structure.

Asymptotic Behavior of BoundStates for a Class of Schrodinger-Poisson System

Jianqing ChenFujian Normal University, Peoples Rep of ChinaYongqing Li, Zhengping Wang, Xiaoju Zhang

For the following Schrodinger-Poisson equation(Pµ)⇢

��u+ V (x)u+ ��(x)u = µu+ |u|p�1u, x 2 R3,�� = u2, lim|x|!+1 �(x) = 0,

we study the existence of ground state and boundstate as well as their asymptotic behavior depend-ing on µ. Here p 2 (3, 5), � > 0, V 2 C(R3,R+)and lim|x|!+1 V (x) = 1. We prove that for any� > 0, there exists �

1

(�) > 0 such that for µ1

< µ <µ1

+ �1

(�), problem (P) has a nonnegative groundstate, which bifurcates from zero solution; problem(P) has a nonnegative bound state, which bifurcatesfrom a solution of (Pµ1). Here µ

1

is the first eigen-value of �� + V . We also analysis the stability ofbound states. Similar results are also proven for amore general version of SP system.

Asymptotic Analysis of the ModelsArising from the Shallow Waterwith the Coriolis E↵ect

Guilong GuiNorthwest University, Peoples Rep of ChinaYue Liu, Ting Luo, Junwei Sun

This talk is concerned with asymptotic analysis of themodels arising from the shallow water with the Cori-olis e↵ect. It is shown that the rotation-Camassa-Holm (R-CH) equation captures stronger nonlin-ear e↵ects than the classical nonlinear dispersiverotation-Korteweg-de Vries (R-KdV) equation. It isalso demonstrated that the classical KdV equation isthe limit approximation of the R-KdV model as theCoriolis e↵ect vanishes. Moreover, it is establishedthat the rotation-free limit of the R-CH equation isthe classical CH equation.

On a 4-Parameter Family of Equa-tions with Peakon Traveling Waves

Alex HimonasUniversity of Notre Dame, USADionyssios Mantzavinos

The Cauchy problem for a novel 4-parameter familyof evolution equations, which are nonlinear and non-local and possess peakon traveling wave solutions, isstudied on both the line and the circle. It is provedthat this family of equations is well-posed in the senseof Hadamard when the data belong to the Sobolevspaces Hs, s > 5/2. Also, it is shown that the data-to-solution map is not uniformly continuous. How-ever, it is Holder continuous in a weaker Sobolevnorm.


Liouville Correspondence Betweenthe Short-Pulse Hierarchy and TheSine-Gordon Hierarchy

Jing KangNorthwest University, Peoples Rep of China

This talk considers the whole hierarchy of bi-Hamiltonian integrable equations associated to eachof the Short-Pulse (SP) equation and the Sine-Gordon (SG) equation. We prove that the trans-formation that relates the SP equation with the SGequation also serves to establish the correspondencebetween their flows and Hamiltonian conservationlaws in respective hierarchy.

Stability of the Camassa-HolmPeakons and Train of Peakons inthe Dynamics of a Shallow-Water-Type System

Xiaochuan LiuNorthwest University, Peoples Rep of China

The stability of the Camassa-Holm (periodic) peaksand train of peakons in the dynamics of an integrableshallow-water-type system is investigated. A varia-tional approach with the use of the Lyapunov methodis presented to prove the variational characterizationand the orbital stability of the peaked wave patterns.Furthermore, the energy method is used to show thestability issue of the train of peakons.

Orbital Stability of Solitary Wavesin the Two-Component Camassa-Holm Systems

Ting LuoThe University of Texas at Arlington, USA

In this talk, we consider the orbital stability of soli-tary waves in the two-component Camassa-Holm sys-tems. Using the property of almost monotonicity andthe local coercivity of the solitary-wave solution, it isshown that the train of N-smooth solitary waves ofthe generalized two-component Camassa-Holm sys-tem is dynamically stable to perturbations in energyspace with a range of parameters. In addition, sta-bility results on multi-peakons and multi-antipeakon-peakons of the two-component Camassa-Holm sys-tem on the ground state is obtained by taking ad-vantage of the conservation laws.

Multiple Solutions for Kirchho↵Type Problems Involving Super-Linear and Sub-Linear Terms

Cao XiaofeiSoutheast University, Peoples Rep of ChinaJunxiang Xu

The paper concerns the multiplicity of solutions fora class of Kirchho↵ type problems with concave andconvex nonlinearities on an unbounded domain. Un-der suitable hypotheses, it is proved that the Kirch-ho↵ problem has at least two positive solutions, oneof which has negative energy and the other positiveenergy. The proof is based on Ekeland’s variationalprinciple, Jeanjean’s monotone method and the Po-hozaev identity.

Blow-Up Phenomena and Persis-tence Property for the Modifiedb-Family of Equation

Ying WangUniversity of Electronic Science and Technology ofchina, Peoples Rep of China

We investigate the blow-up mechanism and persis-tence property of solutions to the modified b-familyof Equation. The dynamics of the blow-up quantityalong the characteristics is established by the Riccati-type di↵erential inequality. Furthermore, the persis-tence results for the solution in weighted spaces areestablished in the case of data decaying slower thansolitons.

The Existence and Mass Concentra-tion of L2-Normalized Solutions forNonlinear Fractional SchrodingerEquations

Fubao ZhangSoutheast University, Peoples Rep of ChinaMiao Du, Lixin Tian, Jun Wang

In this talk we will prove the existence and mass con-centration of L2-normalized solutions for the follow-ing Schrodinger equations with fractional Laplacian:(��)su+ V (x)u = µu+ af(u), x 2 RN .

Blow-Up of Solutions to the PeriodicModified Camassa-Holm Equationwith Varying Linear Dispersion

Min ZhuNanjing forestry University, Peoples Rep of ChinaShuanghu Zhang

Considered herein is the blow-up mechanism to theperiodic modified Camassa-Holm equation with vary-ing linear dispersion. We first consider the case whenlinear dispersion is absent and derive a finite-timeblow-up result. The key feature is the ratio betweensolution and its gradient. Using the continuity of


the solutions and the right transformation, we thenobtain this blow-up criterion to the case with nega-tive linear dispersion and determine that the finitetime blow-up can still occur if the initial momentumdensity is bounded below by the magnitude of the

linear dispersion and the initial datum has a localmild-oscillation region. Finally, we demonstrate thatwhen the linear dispersion is non-negative, formationof singularity can be induced by an initial datum witha su�ciently steep profile.


Special Session 108: New Developments in Porous Media

Akif Ibragimov, Texas Tech University, USAYuliya Gorb, University of Houston, USALuan Hoang, Texas Tech University, USA

Porous media problem attract attention of scientists and mathematician since classical work by HenryDarcy. Fundamental developments in the area of porous media led to big spectrum of application spanningproblems from engineering, and Geo-sciences to biochemistry and environmental sciences. Many problemsin the porous media are highly non-linear involving modeling of coupling physical and chemical processes atdi↵erent scales. The goal of special session is to present a new development of the modeling of the processes inthe porous media from mathematical analysis point of view and discuss most challenging up-today problems.

Multiscale Numerical Methods forSolving Nonlinear ForchheimerEquation in Highly HeterogeneousPorous Media

Manal AlotibiTexas A&M University, USAYalchin Efendiev, Eric Chung

Abstract. In this talk, I will present a local mul-tiscale model reduction for nonlinear flows in het-erogeneous porous media. I will consider generalizedForchheimer equations. The generalized Forchheimerequation describes flows at Darcy scales and ariseswhen the pore-scale velocity is large. We consider thetwo term law form of Forchheimer equation and writethe resulting system in terms of a degenerate nonlin-ear flow equation for the pressure. Our multiscalemodel reduction can be considered a generalizationof recently introduced upscaling and numerical ho-mogenization techniques, where the authors considerproblems with scale separation. In the proposed ap-proach, we construct local reduced-order model byconstructing appropriate snapshot spaces and localspectral problems within the framework of General-ized Multiscale Finite Element Method (GMsFEM).To save the computational time, we use empiricalinterpolation techniques in estimating the nonlinearterms. I will discuss the use of adaptive proceduresboth in o✏ine and online stages of the computation.We present numerical and theoretical results for theproposed method.

Some New Approaches to Simu-lating Two-Phase Flow in PorousMedia on Hexahedral Meshes

Todd ArbogastUniversity of Texas at Austin, USAMaicon Correa, Chieh-Sen Huang, Zhen Tao,and Xikai Zhao

Subsurface geology often dictates that relatively gen-eral hexahedral computational meshes be used whensimulating flow in porous media. The equationsof two-phase flow divides into a parabolic pressureequation for the flow and a degenerate parabolic(convection-di↵usion) saturation equation for thetransport. We present three new approaches. (1)The elliptic part of both equations is often approx-imated using mixed finite elements, which are de-

fined by mapping from a reference cube using the Pi-ola transformation. This destroys the approximationproperties of the method. We describe a new familyof finite elements (AC elements) that overcomes theproblem. (2) The saturation equation exhibits degen-eracy in its elliptic di↵usion term due to loss of capil-larity when a phase is lost. We present a new mixedformulation that is stable when approximating de-generacies. It is suitable for approximation with thenew AC elements. (3) The convection part of the sat-uration equation is often approximated, for example,using discontinuous Galerkin (DG) methods, sinceDG can handle general meshes. Traditional WENOmethods are very accurate but restricted to rectan-gular meshes. We present a new approach using highorder WENO reconstructions on logically rectangularmeshes.

Productivity Index for Darcy andPre-/post-Darcy Flow

Lidia BloshanskayaSUNY New Paltz, USAAkif Ibragimov, Fahd Siddiqui, Mohamed Y.Soliman

We investigate the impact of nonlinearity of high andlow velocity flows on the well productivity index (PI).Experimental data shows the departure from the lin-ear Darcy relation for high and low velocities. High-velocity (post-Darcy) flow occurring near wells andfractures is described by Forchheimer equations andis relatively well-studied. While low velocity flow re-ceives much less attention, there is multiple evidencesuggesting the existence of pre-Darcy e↵ects for slowflows far away from the well. This flow is modeledvia pre-Darcy equation. We combine all three flowregimes, pre-Darcy, Darcy and post-Darcy, under onemathematical formulation subjected to certain criti-cal transitional velocities. This allows to use our pre-viously developed framework to obtain the analyticalformulas for the PI for the cylindrical reservoir. Westudy the impact of non-Darcy e↵ect on the PI de-pending on the well-flux and the parameters of theequations.


Studying Generalized ForchheimerFlows in Heterogeneous Porous Me-dia

Emine CelikTexas Tech University, USALuan Hoang

We study the generalized Forchheimer flows ofslightly compressible fluids in heterogeneous porousmedia where the derived nonlinear partial di↵eren-tial equation for the pressure can be singular anddegenerate in the spatial variables, in addition to be-ing degenerate for large pressure gradient. Suitableweighted Lebesgue norms for the pressure, its gradi-ent and time derivative are estimated. The contin-uous dependence on the initial and boundary datais established for the pressure and its gradient withrespect to those corresponding norms. Asymptoticestimates are derived even for unbounded boundarydata as time tends to infinity. We also obtain theestimates for the L1-norms of the pressure and itstime derivative by implementing De Giorgi’s iterationin the context of the above weighted norms. This isa joint work with Luan Hoang.

Self-Similar Viscous Gravity Cur-rents in Heterogeneous PorousMedia: Second-Kind Solutions

Ivan ChristovPurdue University, USAZ. Zheng, H. A. Stone

We summarize our recent combined experimental-theoretical-computational study of the e↵ects of hor-izontal heterogeneities on the propagation of viscousgravity currents with applications to porous mediaflows. Our model geometry is a horizontal chan-nel (specifically, a Hele-Shaw cell) with variable gapthickness in the streamwise direction in the form ofa power law. We demonstrate that two types of self-similar behaviors emerge as a result of such horizon-tal heterogeneity: (a) a “first-kind” solution is foundusing dimensional analysis for currents that propa-gate away from the origin (a point of zero perme-ability); (b) a “second-kind” solution is found usinga phase-plane analysis for viscous gravity currentsthat propagate toward the origin. Using the phase-plane formalism, we are able to construct the univer-sal second-kind self-similar current shape. Addition-ally, still employing self-similar intermediate asymp-totics and the phase-plane formalism, we identify self-similar behaviors in the post-closure regime, i.e., oncethe current reaches the geometric origin and beginsto fill the model porous medium. The theoreticalpredictions show good agreement with lab-scale ex-periments using Hele-Shaw cells and also numericalsolutions of the governing partial di↵erential equa-tion developed under the lubrication approximation.Z. Zheng, I.C. Christov, H.A. Stone, J. Fluid Mech.747 (2014) 218-246, doi:10.1017/jfm.2014.148.

Analysis of a Turbulence K-EpsilonModel with Applications in PorousMedia

Hermenegildo de OliveiraUniversidade do Algarve, PortugalAna Paiva

In this talk we will analyse a one-equation turbu-lence model of the k-epsilon type that is being usedto describe turbulent flows through porous media.The considered equations are in the steady-state andwe supplement them with homogeneous Dirichletboundary conditions. The novelty of the problemrelies on the consideration of the classical Navier-Stokes equations with feedback’s forces field, whosepresence in the momentum equation will a↵ect theequation for the turbulent kinetic energy (TKE) witha new term that is known as the production and rep-resents the rate at which TKE is transferred fromthe mean flow to the turbulence. For the consideredproblem, we prove the existence and uniqueness ofweak solutions by assuming suitable growth condi-tions together with monotone conditions on the feed-back terms. In this talk we will also address the issueof existence by considering strongly nonlinear feed-back terms and the question of partial regularity ofthe solutions will be analyzed as well.

An Expanded Mixed FEM forGeneralized Forchheimer Flows ofSlightly Incompressible Fluids inPorous Media

Thinh KieuUniversity of North Georgia, USAAkif Ibragimov

We study the expanded mixed finite element methodapplied to the generalized Forchheimer equation withthe Dirichlet boundary condition. The bounds forthe solutions are established. In both continuous anddiscrete time procedures, utilizing the monotonicityproperties of Forchheimer equation and boundednessof solutions, we establish the error estimates for thesolutions in several Lebesgue norms. A numerical ex-ample using the lowest order Raviart-Thomas (RT

0

)mixed element agrees our theoretical result regardingconvergence rate.

Interior W 1,q Estimates for Solutionsof Nonlinear Degenerate ParabolicSystems

Truyen NguyenUniversity of Akron, USA

We consider nonlinear parabolic systems of the formut = divA(x, t, u,ru)+B(x, t, u,ru) which includethose of p-Laplacian type. In this talk, we will discusssome results concerning local integrability of gradi-ents of weak solutions to the system. In particular,we derive interior Lq estimates for the gradient when


A is possibly dicontinuous in the x variable. The de-pendence of the principal part on the u variable madeit di�cult to perform any scaling analysis and wehandle it by using the intrinsic geometry method ofDiBenedetto together with our two-parameter scal-ing technique.

Reconstruction of Dynamic Tortu-osity from Data at Distinct Fre-quencies

Yvonne OuUniversity of Delaware, USA

Dynamic tortuosity plays an important role in theenergy dissipation of wave propagation in poroelasticmaterials by being the kernel function of the mem-ory terms in the time-domain Biot-Johnson-Koplik-Dashen (JKD) wave equations. In this talk, the inte-gral representation formula (IRF) of dynamic tortu-osity will be presented and the mathematical strat-egy for constructing the dynamic tortuosity functionfrom dynamic permeability at distinct frequencies,utilizing the IRF, will be explained. Numerical re-sults for JKD tortuosity will be demonstrated in thistalk.

Reduced Order Hybrid Modelingfrom Pore-Scale to Core-Scale

Malgorzata PeszynskaOregon State University, USATim Costa, Anna Trykozko

We propose a new paradigm for modeling flow andtransport when the pore-scale geometry is changingdue to, e.g., reactive transport, phase transitions,bioclogging, proppant, and/or matrix swelling; (de-noted by the proxy u). Such changes are some-times accounted for with ad-hoc algebraic relation-ships such as Carman-Kozeny for Darcy conductivi-ties K(�). Based on our experience with real pore-scale imaging data, we propose a new reduced or-der hybrid dynamic methodology for K(u). We donot require transient simulations at pore-scale, butrather we rely on a set of values computed o✏inebased on a) a stochastic parametrization of the mod-ified pore-geometries, b) pore-scale flow solver withan Immersed Boundary, c) and a reduced order modelwhich approximates K(u).We (i) use a probability distribution K(�,! ) insteadof K(�) to accurately account for the evolving pore-scale. Next, (ii) given the character of u (e.g., pore-filling, or pore-coating), we sample e�ciently fromthe corresponding distribution of K(�,! ). In ad-dition, (iii) we account for the local in time andspace changes in K(u) by introducing the intermedi-ate third scale of pore-network. The latter step pre-vents the prohibitive complexity of local pore-scaletransport computations.

Phase-Field Modeling of Proppant-Filled Fractures in a PoroelasticMedium

Mary WheelerThe University of Texas At Austin, USAS. Lee, A. Mikelic, T. Wick

This work presents proppant and fluid-filled fracturewith quasi-Newtonain fluid in a poroelastic medium.Lower-dimensional fracture surface is approximatedby using the phase field function. The two-fielddisplacement phase-field system solves fully-coupledconstrained minimization problem due to the crackirreversibility. This constrained optimization prob-lem is handled by using active set strategy. Thepressure is obtained by using a di↵raction equationwhere the phase-field variable serves as an indicatorfunction that distinguishes between the fracture andthe reservoir. Then the above system is coupled viaa fixed-stress iteration. The transport of the prop-pant in the fracture is modeled by using a power-lawfluid system. The numerical discretization in space isbased on Galerkin finite elements for displacementsand phase-field, and an Enriched Galerkin method isapplied for the pressure equation in order to obtainlocal mass conservation. The concentration is solvedwith cell-centered finite elements. Nonlinear equa-tions are treated with Newton’s method. Predictor-corrector dynamic mesh refinement allows to capturemore accurate interface of the fractures with reason-able number for degree of freedoms.

References

[1] Lee, S., Wheeler, M., and Wick, T.; Pressureand fluid-driven fracture propagation in porousmedia using an adaptive finite element phasefield model

[2] Lee, S., Mikelic, A., Wheeler, M., and Wick, T.;Phase-field modeling of proppant-filled fracturesin a poroelastic medium

E↵ect of Pore Size Distribution onthe State of Hydrocarbon PhasesDuring Pressure Depletion

Xiaolong YinColorado School of Mines, USA

In shale gas and shale oil reservoirs, hydrocarbonsare generally stored in pores that are nanometers totens of nanometers in size. In these pores, capillarypressure a↵ects the phase behavior of hydrocarbonmixtures. In this talk, I will present modeling ofthe e↵ect of pore size distribution on vapor-liquidequilibrium in porous media with narrow pores andstrong capillary pressures. Such models are impor-tant for predicting the saturations and properties ofgas and oil phases during primary production. Takea porous medium that is initially saturated with oilas an example. Gas saturation will appear first inlarge pores; the initially formed gas changes the com-


position of the remaining oil, and the compositionalchange in turn alters the equilibrium condition atwhich oil is vaporized in smaller pores. To properlypredict the state of phases confined in this porousmedium, one therefore must trace the entire pressure-saturation history and cannot just rely on a singlephase diagram. Our calculations show that the cap-illary force increases with increasing gas saturation,and it is likely that the smallest pores are always filledwith liquid during production, no matter whether thereservoir is initially filled with oil or gas.

Di↵usion in Random Networks

Duan ZhangLos Alamos National Laboratory, USAJuan C. Padrino

We study mass di↵usion in an ensemble of randomnetworks consisting of junction pockets connectedby tortuous microchannels. Inside the channels, themass di↵usion is governed by the one-dimensional dif-fusion equation. Using the ensemble averaging tech-nique to derive an averaged equation for these pro-cesses, we find that the average concentration evolu-tion is governed by an integro-di↵erential equation.In the case of di↵usion in a semi-infinite domain, thisequation predicts that for an early time compared tothe characteristic time of channel di↵usion, there isa similarity variable xt�1/4 for the average concen-tration in these inhomogeneous media, instead of thetraditional xt�1/2 in a homogeneous medium, wherex is the distance from the boundary, and t is thetime. This early time similarity is a result of thetime required to establish the linear concentrationprofile inside the channels and can be explained bythe random walk theory.Work sponsored by LANL LDRD project20140002DR.

Flow Regimes for Fluid InjectionInto a Confined Porous Medium

Zhong ZhengPrinceton University, USABo Guo, Ivan C. Christov, Michael A. Celia,Howard A. Stone

We report theoretical and numerical studies of theflow behaviour when a fluid is injected into a con-fined porous medium saturated with another fluid ofdi↵erent density and viscosity. For a two-dimensionalconfiguration with point source injection, a nonlinearconvection-di↵usion equation is derived to describethe time evolution of the fluid-fluid interface. In theearly time period, the fluid motion is mainly drivenby the buoyancy force and the governing equation isreduced to a nonlinear di↵usion equation with a well-known self-similar solution. In the late time period,the fluid flow is mainly driven by the injection, andthe governing equation is approximated by a nonlin-ear hyperbolic equation that determines the globalspreading rate; a shock solution is obtained when theinjected fluid is more viscous than the displaced fluid,whereas a rarefaction wave solution is found whenthe injected fluid is less viscous. In the late time pe-riod, we also obtain analytical solutions including thedi↵usive term associated with the buoyancy e↵ects(for an injected fluid with a viscosity higher than orequal to that of the displaced fluid), which providethe structure of the moving front. Numerical simu-lations of the convection-di↵usion equation are per-formed; the various analytical solutions are verifiedas appropriate asymptotic limits, and the transitionprocesses between the individual limits are demon-strated. The flow behaviour is summarized in a dia-gram with five distinct dynamical regimes: a nonlin-ear di↵usion regime, a transition regime, a travelingwave regime, an equal-viscosity regime, and a rar-efaction regime.


Special Session 109: Applied and Integrable Nonlinear PDEs

Stephen Anco, Brock University, Canada

This session will focus on a number of recent developments in the very active area of nonlinear PDEs:integrable systems; Hamiltonian structures; symmetry reductions and constraints;Hamiltonian splitting;special solutions e.g. solitons on non-zero backgrounds, rogue waves, peakons, compactons, lump solutionsand complexiton solutions; exact solution methods. Directions of work related to new methods and theirapplications to integrable systems and nonlinear PDEs will be emphasized, with the aim of bringing togethera number of leading researchers and young scientists working in on these topics.

Equivalence Group of a GeneralizedKuramoto-Sivashinski Equation andConservation Laws

Maria BruzonUniversity of Cadiz, SpainRafael de la Rosa

The aim of this paper is to carry out an exhaus-tive analysis of the symmetries of a GeneralizedKuramoto-Sivashinsky equation with dispersive ef-fects. To achieve this objective, we obtain the contin-uous equivalence transformations of this class. Thegenerators of the equivalence group allow us to de-termine for which types of arbitrary functions theequation admits additional symmetries. In addition,we get the corresponding group invariant solutions.Furthermore, we derive some conservation laws byapplying the direct construction method of Anco andBluman.

Symmetry Analysis and Conserva-tions Laws for Some Equations withCompacton Solutions

Maria Luz GandariasUniversity of Cadiz, SpainMaria Rosa

In this talk we consider some equations that admitcompacton solutions induced by a non-convex con-vection. We derive symmetry reductions and we findsome of the reduced ordinary di↵erential equationssuitable for a qualitative analysis. By using the mul-tipliers method we find a classification of the low-order conservation laws for these equations.

An Integrable Hamiltonian Hierar-chy in sl(2,R) and Its Counterpartin so(3,R) with Three Potentials

Xiang GuUniversity of South Florida, USAWen-Xiu Ma, Wen-Ying Zhang

Starting from two specific matrix spectral problemsassociated respectively with sl(2,R) and so(2,R) ma-trix Lie algebras, we engender two integrable Hamil-tonian hierarchies with three potentials. The compu-

tation and analysis on their Hamiltonian structuresby means of the trace identity reveal the Liouvilleintegrability of both hierarchies; namely, that theyboth consist of infinitely many independent commut-ing conserved functionals and symmetries.

Residual Symmetries of the MKdVEquation and The AKNS Equations

Ping LiuUniversity of Electronic Science and Technology ofChina Zhongshan Institute, Peoples Rep of China

The residual symmetries of the mKdV equation andthe AKNS equations are obtained by the truncatedPainleve analysis. The residual symmetries for themKdV equation are proved to be nonlocal and thenonlocal residual symmetries are extended to the lo-cal Lie point symmetries by means of enlarging themKdV equations. It is noted that we researchedthe twofold residual symmetries by means of tak-ing the mKdV equation as an example. Similaritysolutions and the reduction equations are demon-strated for the extended mKdV equations related tothe twofold residual symmetries. The residual sym-metries for the AKNS equations are proved to benonlocal and the nonlocal residual symmetries areextended to the local Lie point symmetries of a pro-longed AKNS system. The local Lie point symme-tries of the prolonged AKNS equations are composedof the residual symmetries and the standard Lie pointsymmetries, which suggests that the residual symme-try method is a useful complement to the classical Liegroup theory. Several types of exact solutions for theAKNS equations are obtained with the help of thesymmetry method and the Backlund transformationsbetween the AKNS equations and the SchwarzianAKNS equation.


The Darboux Transformation fora Generalized Associated CamassaHolm Equation

Lin LuoShanghai Second Polytechnic University, PeoplesRep of China

In this talk, we discuss an integrable generalizationof the associated Camassa Holm equation. The gen-eralized system is shown to be integrable in the senseof Lax pair. Meanwhile, the Darboux transformationfor the system is derived with the help of the gaugetransformation between two Lax pairs. As an appli-cation, soliton and periodic wave solutions are giventhrough the Darboux transformation.

Quasideterminant Solutions of NCPainleve II Equation with the TodaSolution at n = 1 As a Seed Solutionin Its Darboux Transformation

Irfan MahmoodUniversity of the Punjab, Pakistan

In this paper, the Darboux-quasideterminant solu-tions for the noncommuting elements � and ofnoncommutative (NC) Toda system at n = 1 arepresented. Their Darboux transformations have con-structed with the zero curvature representations ofthe associated systems of non-linear di↵erential equa-tions. I have also derived the quasideterminant solu-tions to the NC Painleve II equation by taking theToda solutions at n = 1 as a seed solution in its Dar-boux transformations. Further by iteration, I gener-alize the Darboux transformations of these solutionsto the N -th form.

A Two-Component Short Pulse Sys-tem Produced Through a NegativeIntegrable Flow

Zhijun (George) QiaoUniversity of Texas - Rio Grande Valley, USAQilao Zha, Qiaoyi Hu

In this paper, we study a two-component short pulsesystem, which was produced through a negative inte-grable flow associated with the WKI hierarchy. TheCauchy problem and the multi-soliton solutions forthe two short pulse system investigated, in particu-lar, one-, two-, three-loop soliton, and breather soli-ton solutions are discussed in details with interestingdynamical interactions and shown through figures.

A Nonlinear Generalization ofCamassa-Holm and ModifiedCamassa-Holm Equations withMulti-Peakon Solutions

Elena RecioBrock University, CanadaStephen Anco

In this work, a 2-parameters family of equations gen-eralizing the Camassa-Holm equation and the mod-ified Camassa-Holm equation is considered. Thisequation reduces to the Camassa-Holm equationwhen p = 2 and the modified Camassa-Holm equa-tion when p = 3 and shares one of the Hamiltonianstructures of both equations. In addition, this equa-tion admits multi-peakon solutions.

Applications of Wu Method forPDE Symmetry Calculation, Clas-sification, Decision and Extension

Chaolu TermuerShanghai Maritime University, Peoples Rep of China

In this talk, we give a review on characteristic setalgorithm for PDE symmetry calculation, classifica-tion, decision and extension done in recent years byauthor. A key step of solving these problems is toanalysis and solves so called determining system. Theessential idea of our research for the purposes is in thefollowing. First we turn the problems of PDE sym-metry calculation, classification, decision and exten-sion equivalently into ones of dealing with the zeropoint set of di↵erential polynomial system (d.p.s).The target of the idea is that we focus our atten-tion on the analysis of the zero point set and algebraproperties of the d.p.s corresponding to determiningequations of the symmetry. Then we use the charac-teristic set theory and algorithm fundamental tool indi↵erential algebra on the transferred problems andsolve original problems. Doing so, we can get an al-ternative ways to discuss these problems from di↵er-ential algebra point and get automatic algorithm toobtain concrete symmetry and decision of (non clas-sical) symmetry existence for a PDE. Meantime, thealgorithm can give symmetry classification and sym-metry extension of a PDE with arbitrary parameters.


Complexiton Solutions and LinearSuperposition Principle to BilinearPDEs

Yuan ZhouUniversity of South Florida, USAWen-Xiu Ma

Based on previous work of Ma and his collabora-tors, we analyzed the linear superposition principleof complexitons (exponential and trigonometric trav-eling waves), we proposed a method to construct a

sub-class of exact solutions to some bilinear PDEs bylinear combinations of complexitons and solitons un-der weak conditions. We obtained multi-complexitonsolutions for some integrable systems such as KdV,KP equations via applying Hirota’s direct method.We also considered the counterpart of generalized bi-linear derivative case. Finally, We presented someexamples in di↵erent situations.


Special Session 110: Computational and Mathematical Methods forComplex Biological Systems

Andrew Nevai, University of Central Florida, USALibin Rong, Oakland University, USA

Zhisheng Shuai, University of Central Florida, USA

The characterization in space and time scales often leads to extraordinary complexity in biology systems.Mathematics and computation now play a much greater role in the study of complex biological systems. Thisspecial session will gather researchers who are actively in development and application of computational andmathematical methods in mathematical biology. The purpose is to present the most cutting-edge studies inthe field and to promote further method development and practical application.

The Importation, Establishmentand Transmission Dynamics of theMosquito-Borne Disease

Jing ChenUniversity of Miami, USAShigui Ruan

From 2006-2013, all the cases of chikungunya virusinfection in the United States were travelers visitingor returning to the United States from a↵ected areas.From the beginning of 2014, 12 locally-transmissioncases were reported from Florida in 2014. As thedevelopment of transportation systems, it is moreconvenient and frequent to have long-distance traveltoday. Therefore, they are not the mosquito, butthe humans who carry the viruses, move to anothercountry and spread the disease. Our hypothesis isthe imported cases first spread the mosquito-bornedisease to local mosquitoes, which later cause localinfections on humans. Based on this, we propose amathematical model to study how these movementsof humans a↵ect the establishment and transmissiondynamics of the mosquito-borne disease.

On the Turnover Rate of the HIVLatent Reservoir in Untreated Pa-tients

Ruian KeNorth Carolina State University, USAKai Deng

The stability of the HIV latent reservoir, a popula-tion of cells latently infected by replication compe-tent HIV but do not actively produce viruses, rep-resents a major barrier to cure HIV infection. Themechanism underlying this stability is not clear. Itmay be due to a low death rate of latently infectedcells; alternatively, it may be due to a dynamical bal-ance between high production and high death rate.Recent data shows that the frequency of cytotoxic Tlymphocyte (CTL) escape mutant viruses in the la-tent reservoir rises to 98% in a majority of untreatedpatients between 6 months to 2 years after HIV infec-tion. Here, we construct mathematical model to de-scribe the dynamics of transmitted founder virus andCTL escape mutants in the plasma and the reservoir,and use the model to estimate the rate of turnover ofthe HIV latent reservoir from the recent data. Theresults suggest that the half-life of the reservoir is

at least 10 times shorter in untreated patients thanthe half-life estimated from treated patients. We fur-ther explore possible mechanisms that drive a fastturnover rate in untreated patients. This work shedslight on the dynamic nature of the reservoir and hasimplications for HIV eradication strategies.

Systematic Measures of ODE-Modeled Complex Networks

Yao LiUniversity of Massachusetts Amherst, USAYingfei Yi

Proposed in systems biology, systematic measuresof complex biological systems, including degeneracy,complexity, and robustness, have been frequentlyused by biologists. Measuring the ability of struc-turally di↵erent components to perform the samefunction, degeneracy is known to have close ties withboth structural complexity and robustness of com-plex systems. In this talk I will report our e↵ortsof quantifying these systematic measures. In addi-tion, we will also discuss our results about connec-tions among degeneracy,complexity, and robustness.

On the Principle for Host Evolutionin Host-Pathogen Interactions

Maia MartchevaUniversity of Florida, USANecibe Tuncer, Yena Kim

We use a two-host one pathogen immuno-epidemiological model to argue that the principlefor host evolution, when the host is subjected toa fatal disease, is minimization of the case fatalityproportion F . This principle is valid whether thedisease is chronic or leads to recovery. In the caseof continuum of hosts, stratified by their immuneresponse stimulation rate a, we suggest that F(a)has a minimum because a trade-o↵ exists betweenvirulence to the host induced by the pathogen andvirulence induced by the immune response. We findthat the minimization of the case fatality proportionis an evolutionary stable strategy (ESS) for the host.


E↵ects of Macroalgal Toxicity andOverfishing on the Resilience ofCoral Reef

Samares PalUniversity of Kalyani, IndiaJoydeb Bhattacharyya

Competition between macroalgae and corals for oc-cupying the available space in sea bed is an impor-tant ecological process underlying coral-reef dynam-ics. While herbivorous reef-fish play a beneficial rolein decreasing the growth of macroalgae, macroalgaltoxicity and overfishing of herbivores leads to theproliferation of macroalgae in coral reef ecosystem,which eventually changes the community structuretowards macroalgae-dominated reef ecosystem. Weanalyze a mathematical model of interactions be-tween coral, macroalgae and herbivores to investigatecoral-macroalgal phase shifts by assuming that thegrowth of herbivorous Parrotfish is limited by coralcover. It is observed that in presence of macroal-gal toxicity and overfishing of Parrotfish the systemexhibits hysteresis through saddle-node bifurcationand transcritical bifurcation. We examine the e↵ectsof macroalgal toxicity and herbivore-harvesting inthe resilience of coral-macroalgae coexistence steadystate. Further, we study the non-autonomous versionof the model by incorporating synchronous or asyn-chronous seasonal variations in di↵erent parameters.By using of Mawhin’s continuous theorem of coinci-dence degree theory, a su�cient condition is obtainedfor the existence of a positive periodic solution. Com-puter simulations have been carried out to illustratedi↵erent analytical results.

Stochastic Models in a Heteroge-neous Host Population

Donald PorchiaUniversity of Central Florida, USAR. N. Mohapatra, Zhisheng Shuai

Heterogeneity and stochasticity are two importantcharacteristics in modeling infectious diseases. Wepropose and investigate basic stochastic models in aheterogeneous host population. Simulations are car-ried out to demonstrate the impact of heterogeneityand stochasticity.

Virus and T Cell Dynamics in HIV-Infected Individuals

Libin RongOakland University, USA

In this talk, I will present some recent work on model-ing HIV infection and treatment, such as HIV latencyand persistence, viral blips, virus dynamics under dif-ferent classes of drugs, treatment intensification withadditional drugs, and the slow time scale of CD4+

T cell depletion in untreated patients. Model formu-lation, mathematical analysis, numerical simulationand comparison with data will be presented. Impli-cations of modeling results for viral control strategieswill also be discussed.

Modeling Zika As a Mosquito-Borneand Sexually Transmitted Disease

Shigui RuanUniversity of Miami, USADaozhou Gao, Yijun Lou, Daihai He, TravisC. Porco, Yang Kuang, Gerardo Chowell

The ongoing Zika virus (ZIKV) epidemic poses amajor global public health emergency. It is well-known that ZIKV is spread by Aedes mosquitoes,recent studies show that ZIKV can also be trans-mitted via sexual contact and several cases of sex-ually transmitted ZIKV of the current outbreak havebeen confirmed in the U.S. and France. We presenteda mathematical model to investigate the impact ofmosquito-borne transmission and sexual transmis-sion on prevention and control of ZIKV and used themodel to fit the ZIKV data up to February 2016 inBrazil, Colombia, and El Salvador. Our study indi-cates that sexual transmission increases the risk of in-fection and epidemic size and prolongs the outbreak.In order to prevent and control the transmission ofZIKV, it must be treated as not only a mosquito-borne disease but also a sexually transmitted disease.

A Mathematical Model for FelineLeukemia

Je↵ SharpeUniversity of Central Florida, USAAndrew Nevai

A model of feline leukemia in a population of feralcats is studied. Building upon a multi-compartmentmodel involving kittens, adult females, and adultmales, the model accounts for multiple methods ofvirus transmission. A basic reproduction R

0

can bedefined which distinguishes between disease persis-tence and disease eradication. Vaccination strategiesare also considered.

Mathematical Models for the Tro-jan Y-Chromosome EradicationStrategy of an Invasive Species

Xueying WangWashington State University, USAJay R Walton, Rana D Parshad

The Trojan Y-Chromosome (TYC) strategy, a ge-netic biocontrol method, has been proposed to elimi-nate invasive species by introducing sex-reversed tro-jan females. Because constant introduction of thetrojans for all time is not possible in practice, therearises the question: What happens if this injection isstopped after some time? Can the invasive speciesrecover? To answer that question, we study this


strategy through deterministic and stochastic mod-els. Our results show that: (1) with the inclusion ofan Allee e↵ect, the number of the invasive femalesis not required to be very low when this injection isstopped, and the remaining trojan population is suf-ficient to induce extinction of the invasive females;(2) incorporating di↵usive spatial spread does notproduce a Turing instability, which would have sug-gested that the TYC eradication strategy might beonly partially e↵ective; (3) the probability distribu-tion and expectation of the extinction time of invasivefemales are heavily shaped by the initial conditionsand the model parameters; (4) elevating the constantnumber of the trojan females being introduced intothe population will lead to a decrease in the expectedextinction time for wild-type females, as opposed toan increase in the extinction probability within anapplication time.

Evaluation on Outcomes of VaccineStrategies for Areas with Di↵erentDemographic Structures

Yanyu XiaoUniversity of Cincinnati, USAM. Laskowski, N. Charland S. M. Moghadas

Ongoing research and technology developments holdthe promise of rapid production and large-scale de-ployment of strain-specific or cross-protective vac-cines for novel influenza viruses. We sought to inves-tigate the impact of early vaccination on age-specificattack rates and evaluate the outcomes of di↵erentvaccination strategies that are influenced by the levelof single or two-dose vaccine-induced protections.We developed and parameterized an individual-basedmodel for two population demographics of urban andremote areas in Canada. Our results demonstratethat there is a time period before and after the onsetof epidemic, during which the outcomes of vaccina-tion strategies may di↵er significantly and are highlyinfluenced by demographic characteristics.


Special Session 111: Geometric Methods in Mechanics and Di↵erentialEquations

Dmitry Zenkov, North Carolina State University, USAIrina Kogan, North Carolina State University, USA

This session will concentrate on the contemporary geometric techniques in ordinary and partial di↵erentialequations, finite and infinite dimensional mechanics, and control. The geometric viewpoint provides a unifiedand systematic framework for addressing a broad range of questions arising in variational calculus, variationalgeometry, fluid dynamics, mechanics of coupled systems, and mathematical physics. Geometric methodinclude structure-preserving integration, symmetry reduction, moving frame calculus, and integrability. Thissession will become a venue for mathematicians, applied mathematicians, and engineers to explore theircommon interest in the geometric approach to mechanics, and more generally to non-linear di↵erentialequations.

Conservation Laws with CoincidingShock and Rarefaction Curves

Michael BenfieldNC State University, USAKris Jenssen, Irina Kogan

Hyperbolic conservation laws are used to modelwaves propagating at finite speed. Solving conserva-tion laws involves analyzing wave interactions: whathappens when two waves collide? A particularly sim-ple case is when the shock and rarefaction curves ofthe conservation law coincide. Here we prove sev-eral results related to such systems and introduce theclass of quadratically interacting systems.

Control and Sensitivity Analysis inFluid-Elasticity Interactions

Lorena BociuNC State University, USA

Free and moving boundary fluid-structure interac-tions are ubiquitous in nature, with most knownexamples coming from industrial processes, aero-elasticity, and biomechanics. We consider optimalcontrol problems subject to interactions between vis-cous, incompressible fluid and nonlinear elasticity.We discuss existence of optimal controls, sensitiv-ity equations, and optimality conditions. One ofthe main challenges of applying optimization tools tofree and moving boundary interactions is the properderivation of the adjoint sensitivity information withcorrect adjoint balancing conditions on the commoninterface. As the coupled fluid-structure state isthe solution of a system of PDEs that are coupledthrough continuity relations defined on the free in-terface, sensitivity of the fluid state, which is an Eu-lerian quantity, with respect to the motion of thesolid, which is a Lagrangian quantity, falls into mov-ing shape analysis framework.

Lipschitz Metric for VariationalWave Equation

Geng ChenGeorgia Institute of Technology, USAAlberto Bressan

The nonlinear wave equation: utt�c(u)[c(u)ux]x = 0is a natural generalization of the linear wave equa-tion. In this talk, we will discuss a recent break-through addressing the Lipschitz continuous depen-dence of solutions on initial data for this quasi-linearwave equation. Our earlier results showed that thisequation determines a unique flow of conservative so-lution within the natural energy space H1(R). How-ever, this flow is not Lipschitz continuous with re-spect to the H1 distance, due to the formation of sin-gularity. To prove the desired Lipschitz continuousproperty, we constructed a new Finsler type metric,where the norm of tangent vectors is defined in termsof an optimal transportation problem. For paths ofpiecewise smooth solutions, we carefully estimatedhow the distance grows in time. To complete theconstruction, we proved that the family of piecewisesmooth solutions is dense, following by an applicationof Thom’s transversality theorem. This is a collabo-ration work with Alberto Bressan.

Nonholonomic Dynamics As Limitof Infinite Friction

Jaap ElderingUniversidade de Sao Paulo - ICMC, Brazil

Nonholonomic dynamics is Lagrangian mechanics ex-tended with d‘Alembert’s a principle to incorporateconstraints on velocities. Such no slipping constraintsoften show up in mechanical systems and it is nat-ural to think of them as enforced by strong contactfriction forces.I will show how this idea can be made rigorous byconsidering the limit dynamics with infinite (linear)friction forces added, using singular perturbation the-ory. We indeed recover nonholonomic dynamics (thusproviding a justification for d‘Alembert’s principle),but moreover, obtain a simplified method to studysystems with non-ideal, i.e. large but finite friction.This improves previous results by Karapetian, Bren-delev and others.


I will illustrate the results applied to the Chaplyginsleigh toy model and discuss its possible use in study-ing the tippe top.

Geometric Integration of SurfaceIsometric Embeddings

Thomas IveyCollege of Charleston, USAJeanne Clelland, Ben McKay, Peter Vassiliou

We formulate the system for isometrically embeddinga surface into Euclidean 3-space as an exterior di↵er-ential system on the product of frame bundles. Weidentify surface metrics for which this system is inte-grable as those realized by certain surfaces of revolu-tion. We develop two complementary methods for in-tegrating these equations, the first using Weierstrass-type representation, and the second using superposi-tion formulas and the action of the Vessiot group.

Multiscale Dynamics and Synchro-nization in Vibrationally DrivenNonholonomic Systems

Scott KellyUNC Charlotte, USA

A mechanical system subject to periodic forcing inthe presence of nonholonomic constraints may ex-hibit aperiodic behavior. This principle underpinsthe self-propulsion of a class of planar robotic vehi-cles that includes the snakeboard and the Chaplyginsleigh surmounted by a balanced rotor. Internal ac-tuation enables each of these vehicles to propel itselfatop a stationary platform, but endowing the plat-form with dynamics of its own can induce a dynami-cally rich response from a similar vehicle without in-ternal actuation. This talk will explore the dynamicsof simple wheeled vehicles atop non-stationary plat-forms that are subject to exogenous forcing or thatpassively permit energy transfer between vehicles.

Hyperbolic Conservation Laws withPrescribed Eigenfields.

Irina KoganNorth Carolina State University, USAMichael Benfield, Kris Jenssen

Eigenvectors of the Jacobian of a flux play an impor-tant role in determining wave curves of a hyperbolicsystem of conservation laws, and hence in construct-ing solutions of the system. Since eigenvectors de-pend on a point in the state space, they are calledeigenfields. In this talk, we concentrate on systemsof three equations in one spatial dimension and dis-cuss the following questions. Given three indepen-dent vector fields on an open subset of R3 (a localframe), does there exist a strictly hyperbolic system,such that the given three vector fields are eigenfieldsof its flux? If yes, what is the degree of freedom forfinding such systems? What happens if we are only

given two independent vector fields (an incompletelocal frame) on an open subset of R3? Is there astrictly hyperbolic flux, such that the given two vec-tor fields are eigenfields of the flux? If yes, how manyare there? These questions are part of a larger projectof determining the e↵ects of geometric properties ofthe wave curves on the behavior of the solutions ofthe conservative systems.

Global Asymptotic Controllabilityfor Control Systems with Un-bounded Inputs

Anna Chiara LaiSapienza University of Roma, ItalyMonica Motta, Franco Rampazzo

We present some results on control systems charac-terized by a Lagrangean with non-negative values andby the unboundedness of controls. In the first partof the talk we show su�cient conditions for globalasymptotic controllability and a state-dependent up-per bound for the infima. The result holds undera quite mild assumption concerning the dependenceof the data on inputs: such condition is met, forinstance, by control vector fields that are compo-sitions of Lipschitz maps with polynomials and ex-ponentials of the control variable. We then discussthe particular case of systems with a polynomialdependence on controls. This class of systems in-cludes control-quadratic systems, that are suitableto model mechanical systems controlled via time-varying,frictionless, holonomic constrains. We showthat algebraic and convexity properties of control-polynomial systems provide simplified versions of themain result.

Local Study of Causal Geometriesand Related Structures

Omid MakhmaliMcGill University, Canada

A causal structure on Mn+1 is given by a codimen-sion one sub-bundle C ⇢ PTM with tangentially non-degenerate fibers. There structures are special classesof cone structures whose study is motivated partlyby the Hwang-Mok program of di↵erential geomet-ric characterization of uniruled varieties and, by thegeometrization program of di↵erential equations inthe sense of Cartan. In this work, the equivalenceproblem for causal structures is solved using Cartan’smethod of equivalence, and the local invariants areobtained. The result shows that the relation betweenthese structures and Finsler geometry are analogousto that of conformal structures and Riemannian ge-ometry. Moreover, causal geometries are the struc-tures associated to certain distributions with growthvector (n, 2n�1, 2n) on an 2n-dimensional manifold.Several properties and related structures obtained viatwistor correspondences will be discussed.


On the Construction and Proper-ties of Weak Solutions DescribingDynamic Cavitation

Alexey MiroshnikovUniversity of Massachusetts Amherst, USAAthanasios Tzavaras

In this work we study the problem of dynamic cavityformation in isotropic compressible nonlinear elas-tic media. Cavitating solutions were introducedby J.M. Ball (1982) in elastostatics and by K.A.Pericak-Spector and S. Spector (1988) in elastody-namics. They turn out to decrease the total mechan-ical energy and provide a striking example of non-uniqueness of entropy weak solutions (in the senseof hyperbolic conservation laws) for polyconvex en-ergies. In our work we established various furtherproperties of cavitating solutions. For the equationsof radial elasticity we construct self-similar weak so-lutions that describe a cavity emanating from a stateof uniform deformation. For dimensions d = 2,3 weshow that cavity formation is necessarily associatedwith a unique precursor shock. We also study thebifurcation diagram and do a detailed analysis of thesingular asymptotics associated to cavity initiation asa function of the cavity speed of the self-similar pro-files. We show that for stress-free cavities the criti-cal stretching associated with dynamically cavitatingsolutions coincides with the critical stretching in thebifurcation diagram of equilibrium elasticity.

Controlled Lagrangians and Stabi-lization of Discrete Spacecraft withRotor

Yuanyuan PengClaflin University, USASyrena Huynh, Dmitry V. Zenkov, AnthonyM. Bloch

The method of controlled Lagrangians for discretemechanical systems is extended to the problem ofstabilization of the rotations of a spacecraft with asymmetric rotor. In particular, stabilization aboutits intermediate axis of inertia is considered. TheMoser-Veselov discretization is used to obtain thediscrete dynamics of the system. Stabilization condi-tions for the continuous model and its discretizationare compared. It is shown that stability of the dis-crete system is su�cient for stability of its continuouscounterpart but not vice versa.

Optimal Control of NonholonomicMechanical Systems

Stuart RogersUniversity of Alberta, CanadaVakhtang Putkaradze, Stuart Rogers

We investigate the optimal control of mechanicalsystems with nonholonomic constraints. For alge-braic simplicity, we focus mainly on Suslov’s prob-lem, which is a classical example of a nonholonomicmechanical system. This system considers the mo-tion of a rigid body rotating about a fixed point sub-ject to the constraint⌦ · ⇠ = 0, where ⌦ denotesthe angular velocity and ⇠ is a prescribed vector,both expressed in the body frame. Permitting ⇠ tovary with time, we derive the optimal control equa-tions given a cost function C(⌦, ⇠ ,⇠). In additionto numerical solutions, we show that in several par-ticular cases, we can either find analytical solutionsor asymptotic solutions using singular perturbationmethods. We shall also touch briefly on the opti-mal control of a substantially more complicated non-holonomic system, namely, a ball with moving inter-nal masses rolling without slipping on a horizontalsurface. Time permitting, derivations of the uncon-trolled and controlled equations of motion, as wellas numerical simulations, for this system will be pre-sented.

Lie-Dirac Reduction on SemidirectProducts and Nonholonomic Me-chanics

Hiroaki YoshimuraWaseda University, JapanFrancois Gay-Balmaz

For the sake of modeling nonholonomic systems, thetheory of Lagrange-Dirac systems is quite a usefultool since Dirac structures systematically show howthe system components are interconnected throughenergy flow, even for the case of degenerate La-grangians. In particular, the case in which a config-uration space is given by a Lie group can be under-stood in the context of the so-called Lie-Dirac reduc-tion theory. In this talk, we consider Lagrange-Diracsystems for nonholonomic mechanical systems on Liegroups with broken symmetry. We show reductionof Lagrange-Dirac systems as well as Hamilton-Diracsystems in the context of the Lie-Dirac reductionwith advected parameters. We illustrate our the-ory with some examples of Chaplygin’s ball and thesecond-order Rivlin-Ericksen Navier-Stokes fluids.


Weak* Solutions of ConservationLaws

Robin YoungUniversity of Massachusetts, USAAlexey Miroshnikov

We introduce a new concept of solution for systemsof conservation laws, which allows us to rigorouslyhandle certain measure-valued solutions such as thevacuum in a Lagrangian frame. In particular, thePDEs can be regarded as (weak*) solutions of anODE in Banach space, and inherit certain regular-ity. We show that BV weak solutions are weak* so-lutions, and describe solutions with vacuums as anexample. We further discuss approximations to suchsolutions.

Hamel’s Formalism for Infinite-Dimensional Mechanics

Dmitry ZenkovNorth Carolina State University, USA

Separation of the position and velocity measurementsin mechanics originated in Euler’s work on the dy-namics of rigid body and fluid. Hamel extendedthis formalism from the rigid body setting to arbi-trary finite-dimensional Lagrangian mechanical sys-tems. This talk will introduce Hamel’s formalism forinfinite-dimensional mechanics, with an emphasis onthe dynamics of constrained continuum-mechanicalsystems.


Special Session 113: Inverse Problems, Variational Inequalities, andApplications

Akhtar A. Khan, Rochester Institute of Technology, USAPatrizia Daniele, University of Catania, Italy

Baasansuren Jadamba, Rochester Institute of Technology, USA

The special session “Inverse Problems, Variational Inequalities, and Applications” will focus on recent de-velopments in variational methods for inverse problems, variational inequalities, the cross fertilization ofideas in these two disciplines, and their applications focusing on projected dynamical systems, equilibriumproblems, imaging and signal processing, elasticity imaging, regularization, among others.

A New Topological Degree The-ory and Applications to NonlinearHyperbolic Problems with Non-monotone Nonlinearities

Te↵era AsfawVirginia Tech, USA

Let H be a real Hilbert space. Let T : H ◆D(H) ! 2H be maximal monotone, C : H ! Hbe demicontinuous such that there exist nonegativeconstants ⌧ and ⌫ satisfying kCxk ⌧kxk + ⌫ forall x 2 H, S : H ! 2H be bounded of type (S

+

)and L : H ◆ D(L) ! H be linear, closed, rangeclosed and L�1 : R(L) ! H be compact. A newdegree theory is developed for operators of the typeL(T + S) +C. The generalization of this theory alsoholds if S is bounded pseudomonotone and T + S isof type (S). The theory developed herein is appliedto prove existence of weak solution (s) for nonlinearhyperbolic equations of the type

8>>>><

>>>>:

utt � �u� @@t

�u+ ut + ↵(�u� �2u) + g(x, t, u)

= f(x, t) (x, t) 2 QT

u(x, t) = 0 (x, t) 2 @QT

u(x, 0) = u(x, T ) x 2 ⌦,

where ↵ � 0, QT = ⌦ ⇥ (0, T ), @QT = @⌦ ⇥ (0, T )and g : ⌦ ⇥ [0, T ] ⇥ R ! R satisfies suitable sublin-earity condition. Analogous existence theorems areapplied to show solvability of hyperbolic variationalinequality problems in appropriate Hilbert spaces.

Inverse Problems of IdentifyingNonlinear Parameters in Varia-tional Problems

Manki ChoRochester Institute of Technology, USABaasansuren Jadamba, Akhtar A. Khan

This work is on a reliable computational frame-work for an inverse problem of identifying nonlin-ear parameters using an output least-squares(OLS)approach. The proposed framework is based on asecond-order adjoint method for the computation of

the second-order derivative of the regularized OLSfunctional. The feasibility of the proposed frame-work is supported by numerical experiments. Thisis joint work with Baasansuren Jadamba and AkhtarA. Khan.

Generalized Nash Games, Vacci-nation Policies and EnvironmentalAccords

Monica CojocaruUniversity of Guelph, Guelph ON, CanadaE. Wild, A. Small

We present two numerical methods to compute theentire solution sets of Nash points in a generalizedNash game. We then show how two very di↵erentmodels of policy games can be analyzed by beingformulated as generalized Nash games with sharedconstraints. This is joint work with Erin Wild andAllison Small.

Nonlinear Parameter Identificationin Variational Problems

Baasansuren JadambaRochester Institute of Technology, USA

Inverse problems of identifying parameters in par-tial di↵erential equations (PDEs) constitute an im-portant class of problems emerging from diverse ap-plications. These inverse problems are most con-veniently studied using an optimization frameworkand there are various optimization formulations. Al-though a non-convex output least-squares (OLS) ob-jective is commonly used, a convex modified outputleast-squares (MOLS) gained considerable interest inrecent years. However, the convexity of the MOLShas only been established for parameters appearinglinearly in the PDEs. The primary objective of thiswork is to introduce and analyze a variant of theMOLS for the inverse problem of identifying param-eters that appears nonlinearly in general variationalproblems. We are interested in understanding whatgeometric properties of the original MOLS can beretained for the nonlinear case. Besides giving an ex-istence result for the optimization formulation of theinverse problem, we give a thorough derivation of thefirst-order and second-order derivative formulas for


the new objective functional. The derivative formu-lae suggest that the convexity of the MOLS cannotbe retained for the parameters appearing nonlinearlywithout imposing additional assumptions on the datainvolved.

Parameter Identification in Vari-ational and Quasi-Variational In-equalities

Akhtar KhanRochester Institute of Technology, USAJoachim Gwinner, Baasansuren Jadamba,Miguel Sama

In this talk, we will focus on the inverse problem ofidentifying variable parameters in certain variationaland quasi variational inequalities. We develop a tri-linear form based optimization framework that hasbeen used quite e↵ectively for parameter identifica-tion in variational equations emerging from partialdi↵erential equations. An abstract nonsmooth reg-ularization approach is developed that encompassesthe total variation regularization and permits theidentification of discontinuous parameters. We inves-tigate the inverse problem in an optimization settingusing the output-least squares formulation. We giveexistence and convergence results for the optimiza-tion problem. We also penalize the variational in-equality and arrive at optimization problem for whichthe constraint variational inequality is replaced bythe penalized equation. For this case, the smooth-ness of the parameter-to-solution map is discussedand convergence analysis and optimality conditionswill be discussed.

Coupling of Regularization andH-(hp-Adaptive) BEM for Hemi-variational Inequalities Modelling aDelamination Problem

Nina OvcharovaUniversity of the Army Munich, GermanyLothar Banz

We couple the regularization techniques of nondi↵er-entiable optimization with the h resp. hp-adaptiveversions of the boundary element method (BEM) tosolve nonsmooth variational problems arising in con-

tact mechanics. As a model example we consider thedelamination problem. The variational formulationof this problem leads to hemivariational inequalitywith a nonsmooth functional de↵ned on the contactboundary. This problem is first regularized and then,discretized by a h- resp. hp-BEM. We give conditionsfor the uniqueness of the solution, prove convergenceof the BEM Galerkin solution of the regularized prob-lem in the energy norm, and obtain an a-priori er-ror estimate for the regularized problem based on anovel Cea-Falk approximation lemma. Furthermore,we derive a reliable a-posteriori error estimate basedon an equivalent regularized mixed formulation, thusenabling hp- adaptivity. Numerical experiments il-lustrate the behavior, strengths and weaknesses ofthe proposed approximation scheme.

Some Properties of A�ne Varia-tional Inequalities

Stephen RobinsonUniversity of Wisconsin-Madison, USA

A�ne variational inequalities (AVI) are basic to mod-eling in economics and in several areas of engineering.For some problems, the AVI provide suitable mod-els; for other problems the models require variationalinequalities with nonlinear functions but AVI canprovide suitable approximations to support Newton-type methods for solution.Although AVI look quite simple, proofs of someof their useful properties require considerably morework than proofs of analogous properties of ordi-nary linear equations (which are the simplest cases ofAVI, in which the underlying set is the whole space).In particular, the property of injectivity—which inlinear equations is equivalent to nonsingularity of asquare matrix—has been characterized by a varietyof di↵erent arguments, some accessible to students inthe fields noted above and some not. Even now, afteryears of work, the characterization can be expressedin either of two di↵erent ways, which must be equiv-alent but of which the equivalence is not evident.In this lecture we will look at some basic propertiesof AVI, including injectivity, and will ask how theproofs supporting those properties might be made astransparent as possible so that students with a basicmathematical background could follow the proofs.


Special Session 114: Uncertainty Quantification

Kody Law, Oak Ridge National Laboratory, USAClayton Webster, Oak Ridge National Laboratory, USA

Uncertainties in the parameters which define di↵erential equations give rise to distributions of solutions,hence distributions of functionals of the solutions. Quantities of interest can be represented as expectationsof such functionals. The resulting high-dimensional integrals, involving expensive function evaluations, haslead to a wealth of new Mathematics. Further complexity is introduced if data is available. The Bayesianframework gives rise to a probabilistic interpretation of inverse problems. This special session aims to bringtogether researchers in high-dimensional approximation theory, numerical and computational methods forstochastic partial di↵erential equations, and Bayesian inverse problems to share ideas and discuss theseinteresting problems and the interplay between them.

Sparse Sampling Methods for Neu-tron Data

Rick ArchibaldORNL, USA

This talk will focus on mathematics developed tohelp with the mathematical challenges face by theDOE at the experimental facilities. This talk willdiscuss sparse sampling methods and fast optimiza-tion developed specifically for neutron tomography.Sparse sampling has the ability to provide accuratereconstructions of data and images when only par-tial information is available for measurement. Sparsesampling methods have demonstrated to be robustto measurement error. These methods have the po-tential to scale to large computational machines andanalysis large volumes of data.

A Partial Domain Inversion Ap-proach for Large-Scale BaysianInverse Problems in High Dimen-sional Parameter Spaces

Tan Bui-ThanhThe University of Texas at Austin, USAVishwas Hebbur Venkata Subba Rao

While Bayesian inference is a systematic approachto account for uncertainties, it is often prohibitivelyexpensive for inverse problems with large-scale for-ward equation in high dimensional parameter space.We shall develop a partial domain inversion strat-egy to only invert for distributed parameters that arewell-informed by the data. This is an e�cient data-driven reduction method for both forward equationand parameter space. This approach induces severaladvantages over existing methods. First, dependingon the size of truncated domain, solving the trun-cated forward equation could be much less computa-tionally demanding. Consequently, the adjoint (andpossibly incremental forward and adjoint equationsif Newton-like method is used is much less compu-tationally intensive. Second, since the parameter tobe inverted for is now restricted, the curse of dimen-sionality encountered when exploring the parameterspaces (to compute statistics of the posterior density,for example) is mitigated. Third, this approach iswell-suited for current and future extreme-scale com-

puting systems (with decreasing memory per node)since it is naturally a low memory demanding algo-rithm. Fourth, it supports methods that are e�cientfor small-to-medium scale problems, such as (paral-lel) direct solvers.

Hierarchical Bayesian Level SetInversion

Matthew DunlopUniversity of Warwick, EnglandMarco Iglesias, Andrew Stuart

The level set approach has proven widely successfulin the study of inverse problems, since its system-atic development in the 1990s. Recently it has beenemployed in the context of Bayesian inversion, al-lowing for the quantification of uncertainty withinreconstruction methods. However the Bayesian ap-proach is very sensitive to the length and amplitudescales encoded in the prior probabilistic model. Wedemonstrate how the scale-sensitivity can be circum-vented by means of a hierarchical approach, using asingle scalar parameter. Together with careful con-sideration of the development of algorithms whichencode probability measure equivalences as the hier-archical parameter is varied, this leads to well-definedGibbs based MCMC methods found by alternatingMetropolis-Hastings updates of the level set func-tion and the hierarchical parameter. These methodsdemonstrably outperform non-hierarchical Bayesianlevel set methods.

Multilevel Markov Chain MonteCarlo Method for Bayesian InverseProblems

Viet Ha HoangNanyang Technological University, SingaporeChristoph Schwab, Andrew M. Stuart

For Bayesian inverse problems of partial di↵eren-tial equations with unknown random coe�cients, theplain Markov Chain Monte Carlo (MCMC) methodthat straightforwardly combines Finite Element ap-proximation with MCMC sampling is prohibitivelycostly. We present the Multilevel MCMC methodthat achieves a prescribed level of accuracy for ap-proximating the posterior expectation of a quantityof interest, with essentially optimal complexity. Nu-merical results confirm our rigorous theory.


Multilevel Sequential Monte CarloSamplers

Kody LawORNL, USAAjay Jasra, Yan Zhou, Alex Beskos, RaulTempone

This talk will review the probabilistic formulationof the inverse problem, the sequential Monte Carlo(SMC) sampling framework, and the standard multi-level Monte Carlo (MLMC) framework. These ideas

will coalesce into the MLSMC sampling algorithm forBayesian inverse problems. A numerical example ofpermeability inversion through an elliptic PDE givenobservations of pressure will illustrate the theoreticalresults.


Special Session 117: Partial Di↵erential Equations from Fluid Dynamics

Hongjie Dong, Brown University, USARobin Ming Chen, University of Pittsburg, USADong Li, University of British Columbia, Canada

This session will serve to promote and disseminate recent developments on evolutionary PDEs governingthe motion of fluids. The fundamental prototypes are the Navier-Stokes (NS) and Euler equations. Theseequations appear, with fixed or moving interfaces, either alone or coupled with other equations, in the studyof many phenomena in aerodynamics, geophysics, meteorology, plasma physics, etc. This session will focuson (but not restricted to) issues regarding modeling, local well/ill-posedness, gobal well-poseness/finite-timeblowup, and stability.

Review of Some Properties of Leray-Hopf Weak Solutions to the NavierStokes Equations

Hao JiaUniversity of Chicago, USA

In the talk, we will review the properties of LerayHopf weak solutions to the Navier Stokes equationsand some related open problems. In particular, wewill discuss a recent application in the construction ofscale invariant solutions, and explain the connectionof linear spectral property of scale invariant solutionswith uniqueness problem of the weak solutions.

Lp

-Estimates for Stationary StokesSystems with Coe�cients Measur-able in One Direction

Doyoon KimKorea University, KoreaHongjie Dong

We discuss Lp-estimates of solutions to stationaryStokes systems when the coe�cients are measurable(i.e., no regularity assumptions) in one direction. Weprove a priori Lp-estimates of solutions when the sys-tems are defined on the whole Euclidean space or ona half space. We also prove the unique solvability inLp-spaces of stationary Stokes systems on a boundeddomain.

Spatial Asymptotics in the EulerEquation

Robert McOwenNortheastern University, USAPeter Topalov

We prove that the Euler equation describing the mo-tion of an incompressible fluid in Rn is well-posed ina class of functions allowing partial asymptotic ex-pansions of any order as |x| ! 1.

Global Well-Posedness of 2D Non-linear Schrodinger Equations ofIndefinite Signature

Nathan D. TotzUniversity of Massachusetts Amherst, USA

We describe a new method to obtain global a prioribounds in time for solutions to nonlinear Schrodingerequations (NLS) on R2 having power nonlinearitiesof arbitrary odd degree, and with large initial datain Sobolev space. The method presented here ap-plies to both the usual NLS equations associated tothe Laplacian and with a nonlinearity of defocusingsign, as well as to the more di�cult so-called “hy-perbolic” NLS which is associated to an indefinitesignature. The latter is particularly interesting sinceits long time behavior is to date unknown for large(and even very smooth) initial data. We show, byrigorously justifying that these equations govern themodulation limit of an artificially constructed equa-tion with an advantageous structure, that every sub-critical Sobolev norm of the solution increases a prioriat most polynomially in time. Global existence in allsubcritical Sobolev spaces then follows by standardlocal well-posedness results for NLS.

Existence and Qualitative Theoryof Stratified Solitary Water Waves

Samuel WalshUniversity of Missouri, USARobin Ming Chen, Miles H. Wheeler

In this talk, we will report some recent results con-cerning two-dimensional gravity solitary water waveswith hereogeneous density. The fluid domain is as-sumed be bounded below by an impenetrable flatocean bed, while the interface between the water andvacuum above is a free boundary. Our main existenceresult states that, for any smooth choice of upstreamvelocity and streamline density function, there existsa path connected set of such solutions that includeslarge-amplitude surface waves. Indeed, this solutionset can be continued up to (but does not include) an“extreme wave“ that possess a stagnation point.We will also discuss a number of results character-izing the qualitative features of solitary stratifiedwaves. In part, these include bounds on the Froudenumber from above and below that are new even forconstant density flow; an a priori bound on the ve-


locity field and lower bound on the pressure; a proofof the nonexistence of monotone bores for stratifiedsurface waves; and a theorem ensuring that all su-percritical solitary waves of elevation have an axis ofeven symmetry.

Blow-Up of Critical Norms for the3-D Navier-Stokes Equations

Wendong WangDalian University of Technology, Peoples Rep ofChinaZhifei Zhang

In this talk, we‘ll talk about the regularity of weaksolutions for the 3-D Navier-Stokes equationsin the critical norm.This is a joint work with Z.-F.Zhang etc.

Asymptotic and Integral Propertiesof Solitary Waves in Deep Water

Miles WheelerNYU Courant Institute, USA

We consider solitary waves on the surface of an in-finitely deep two- or three-dimensional fluid, bothwith and without surface tension. Under a mild alge-braic decay assumption, we prove precise asymptoticsat infinity, and relate these asymptotics to several in-tegral properties of the wave. This complements pre-vious nonexistence results for waves in deep waterwithout surface tension.

KdV Dynamics and Traveling Wavesin Polyatomic FPU

J. Douglas WrightDrexel University, USA

Using homogenization theory, we can derive andjustify a Korteweg-deVries limit for a polyatomicFermi-Pasta-Ulam lattice problem under the assump-tion that the material parameters vary periodically.While the KdV approximation predicts the existenceof solutions which look like solitary waves for longtimes, it does not guarantee that such solutions re-main coherent forever. We discuss recent results onthe global in time existence of generalized solitarywaves in diatomic FPU lattices.

Global Martingale Solution for theStochastic Boussinesq System withZero Dissipation

Kazuo YamazakiWashington State University, USA

In this talk, we discuss some new results on thestochastic analysis of PDE in fluid. In particu-lar, we discuss the global existence of a martin-gale solution to the 2D stochastic Boussinesq sys-tem with zero dissipation. This result is new incontrast to previous work due to the lack of dissi-pation. If time permits, we also discuss the globalexistence of a martingale solution to the 3D stochas-tic nonhomogeneous (variable-density) magnetohy-drodynamics system, ergodicity and large deviationprinciple results for the 2D stochastic micropolar andmagneto-micropolar fluid systems.

The Weak Solutions to 3D Degen-erate Compressible Navier-StokesEquations

Cheng YuThe University of Texas at Austin, USAAlexis Vasseur

In this talk, we discuss the existence of global weaksolutions for 3D compressible Navier-Stokes equa-tions with degenerate viscosity. The method is basedon the Bresch and Desjardins entropy conservation.The main contribution of this paper is to derive theMellet-Vasseur type inequality for the weak solutions,even if it is not verified by the first level of approxi-mation. This provides existence of global solutions intime, for the compressible Navier-Stokes equations,in three dimensional space, with large initial datapossibly vanishing on the vacuum. This solves anopen problem proposed by Lions.

Regularity Criteria of 3D Navier-Stokes Equations Involving thePressure Term

Xinwei YuUniversity of Alberta, CanadaChuong V. Tran

In this talk we will present some new regularity crite-ria for the 3D Navier-Stokes equations. This new cri-teria involve combinations of the pressure and the ve-locity. They improve the classical Prodi-Serrin con-ditions. This is joint work with Prof. Chuong V.Tran of University of St. Andrews, Scotland.


Special Session 118: Mean Field Games and Applications

Daniela Tonon, CEREMADE Universite Paris Dauphine, FranceAdriano Festa, RICAM, Austria

Mean Field Games theory is a new and challenging mathematical topic which analyses the dynamics ofa very large number of interacting rational agents. Introduced ten years ago, the MFG models are usedin many areas such as economics, finance, social sciences and engineering. In this section we will presentsome recent developments of the topic with elements coming from mean field theories, optimal control andstochastic analysis, calculus of variations and partial di↵erential equations. Modeling and numerical aspectsof the matter will also be presented.

An E�cient Numerical Method forStationary Mean Field Games

Simone CacaceSapienza University of Rome, ItalyFabio Camilli

We propose a new approach to the numerical solutionof Stationary Mean Field Games. It is based on aNewton-like method for solving inconsistent systemsof nonlinear equations, arising in the discretizationof the corresponding Ergodic Hamilton-Jacobi andFokker-Planck equations. We show that our methodis able to solve e�ciently Mean Field Games on Eu-clidean spaces and Networks, also in the case of morecompeting populations. We present several numeri-cal experiments in dimension one and two, showingthe performance of the proposed method in terms ofaccuracy, convergence and computational time.

Bifurcation and Segregation inQuadratic Two-Populations MeanField Games Systems

Marco CirantUniversita di Milano, ItalyGianmaria Verzini

In this talk we will consider stationary (ergodic)Mean Field Games systems of two competing pop-ulations. We will focus on the corresponding ellip-tic system of coupled semilinear equations obtainedvia the Hopf-Cole transformation, and in particularstudy the behavior of the two populations as the vis-cosity parameter goes to zero. We will discuss the ex-istence of non-trivial solutions using variational andbifurcation methods; then, for selected families ofnon-trivial solutions, we will address the appearing ofsegregation in the vanishing viscosity limit by meansof blow-up techniques.

A Semi-Lagrangian Scheme for theHughes Model for Pedestrian Flow

Adriano FestaRICAM - Austrian Academy of Science, AustriaCarlini, Silva, Wolfram

In this talk we present a Semi-Lagrangian scheme fora regularized version of the Hughes model for pedes-trian flow. Hughes originally proposed a coupled non-linear PDE system describing the evolution of a large

pedestrian group trying to exit a domain as fast aspossible. The original model corresponds to a systemof a conservation law for the pedestrian density andan Eikonal equation to determine the weighted dis-tance to the exit. We consider this model in presenceof small di↵usion and discuss the numerical analysisof the proposed Semi-Lagrangian scheme. Further-more we illustrate the e↵ect of small di↵usion on theexit time with various numerical experiments.

Existence of Weak Solutions ofMean-Field Games by MonotonicityMethods

Diogo GomesKing Abdullah University of Science and Technology,Saudi ArabiaRita Ferreira

Here, we consider monotone mean-field games(MFGs) and study the existence of weak solutions.First, we introduce a regularized problem that pre-serves the monotonicity. Next, using variational in-equality techniques, we prove the existence of so-lutions to the regularized problem. Then, usingMinty’s method, we establish the existence of solu-tions for the original MFG. Finally, we examine theproperties of these weak solutions in several exam-ples. Our methods provide a general framework toconstruct weak solutions to MFGs with local, non-local, or congestion terms.

Well-Posedness of Cournot Compe-tition MFG Model

Philip GraberUniversity of Dexas at Dallas, USAAlain Bensoussan

In their chapter on Mean Field Games and Applica-tions in the Paris-Princeton Lectures on Mathemati-cal Finance, Gueant, Lasry, and Lions introduced anMFG model of production of an exhaustible resource.Although such models have been studied numerically,there seem to be very few results in the literature giv-ing well-posedness. Recently, Chan and Sircar intro-duced a similar model with a linear demand sched-ule, which corresponds to Cournot (or Bertrand) typeoligopolistic competition with a large number of pro-


ducers. In this talk we present a theorem on theexistence and uniqueness of smooth, classical solu-tions to the model of Chan and Sircar. The proofdepends on new a priori estimates, particularly forthe “nonlocal coupling term.

Aggregative Control of Large-ScaleMulti-Agent Systems

Sergio GrammaticoEindhoven University of Technology, Netherlands

Many large-scale systems involve the interaction ofa number of agents with loosely coupled dynamicsand decisions, for instance in demand response man-agement in electricity grids, where the agents locallyoptimize their decisions, but their eventual well be-ing also depends on the aggregate of the decisionsof all other agents. For such systems, it is typicallyimpractical to impose a centralized control structurefor a number of reasons (e.g. privacy concerns, com-putational and communication limitations). Insteadone can provide suitable information to the agentsand impose an appropriate penalty/reward feedbackscheme to control the overall population using macro-scopic commands only, so that the population ex-hibits a desirable macroscopic behaviour. In thistalk, we discuss an aggregative control structure viaa game theoretical approach, and present technicalresults based on fixed point operator theory. Finally,some open research opportunities are presented.

Mean Field Type Control with Con-gestion

Mathieu LauriereUniversity Paris 7, FranceYves Achdou

The theory of mean field type control, developed byBensoussan, Frehse and Yam, aims at describing thebehaviour of a large number of interacting agents us-ing a common feedback. A type of problems thathave raised a lot of interest recently concerns con-gestion e↵ects. They model situations in which thecost of displacement of the agents increases in the re-gions where the density is large (as, for instance, incrowd motion). I will present a system of partial dif-ferential equations arising in this setting. The mainresult is the existence and uniqueness of suitably de-fined weak solutions, which are characterized as theoptima of two optimal control problems in duality. Iftime permits, I will also discuss a numerical methodto solve this problem and present some numerical re-sults.

A Dynamic Game Model of Collec-tive Choice in Multi-Agent Systems

Jerome Le NyPolytechnique Montreal, CanadaRabih Salhab, Roland Malhame

We consider a mean field games-like scenario where alarge number of agents have to make a choice amonga set of di↵erent potential target destinations. Eachindividual both influences and is influenced by thegroup’s decision, as well as the mean trajectory of allthe agents. The model can be interpreted as a styl-ized version of opinion crystallization in an electionor of collective decision making in animal societies forexample. For our formulation, we show how to reducethe computation of the approximate Nash equilibriato a fixed point computation in a finite dimensionalspace, where we essentially look for consistent pro-portions of players choosing each destination. If timepermits, we will also discuss a cooperative version ofthe model as well as the role that advertisers can playto influence the outcome.

Some Aspects of Finite-State Mean-Field Games

Roberto Machado VelhoKAUST, Saudi ArabiaDiogo A. Gomes, Marie-Therese Wolfram

In this talk, we consider finite-state mean-field gameproblems. First, we introduce the finite-state MFGproblem and its analogy to the continuous case.Next, we examine for two-state problems and ad-dress their dual and potential formulation. Then, wepropose a numerical method for the limit hyperbolicequation that is based upon the N-agent problem.Finally, we illustrate the behavior of two-state gamesvia some examples and their simulations.

First Order Mean Field Gameswith Density Constraints: PressureEquals Price

Alpar Richard MeszarosUCLA, USAPierre Cardaliaguet, Filippo Santambrogio

We study first order Mean Field Game systems un-der density constraints as optimality conditions oftwo optimization problems in duality. A weak solu-tion of the system contains an extra term, an addi-tional price imposed on the saturated zones. We showthat this price corresponds to the pressure field fromthe models of incompressible Euler’s equations a laBrenier. By this observation we manage to obtain aminimal regularity, which allows to write optimalityconditions at the level of single agent trajectories andto define a weak notion of Nash equilibrium for ourmodel.


Numerical Method for Mean-FieldType Control Problems

Laurent Pfei↵erGraz University, Austria

In this talk, we present two gradient-type numericalmethods to solve mean-field type control problems.These problems consist in optimal control problemof stochastic di↵erential equations, where the costfunction is a function of the probability distributionof the state variable.The first described method is based on a convexityproperty of the reachable set of probability distribu-tions. We provide a convergence result. The seconddescribed method generates controls which are feed-back controls.

Stationary Mean-Field Games inthe Presence of Mild Singularities

Edgard PimentelPPGM-UFSCar, Brazil

In this talk, we address stationary mean-field gamesystems in the presence of mild singularities andprove well-posedness in the class of smooth solutions.Mild singularities have played a major role in reg-ularity theory for (nonlinear) elliptic equations andfree boundary problems, as it accounts - among otherthings - for cavitation and obstacle problems. In therealm of mean-field games, this class of couplings en-codes a particular setting of preferences. More im-portantly, it introduces a new set of di�culties, asthe integrability of mild singularities is weaker thanthe borderline case, i.e., logarithmic nonlinearities.

A Mean-Field Game of Evacuationin a Multi-Level Building

Hamidou TembineNew York University, USABoualem Djehiche, Alain Tcheukam

This work puts forward a simple mean-field networkgame that captures some of the key dynamic fea-tures of crowd and pedestrian flows in multi-levelbuilding evacuations. It considers both microscopicand macroscopic route choice by strategic agents. Toachieve this, we use mean-field di↵erential game withlocal congestion measure based on the location ofthe agent in the building. Including the local mean-field term and its evolution along the path causesa sort of dispersion of the flow: the agents will tryto avoid high density areas in order to reduce theiroverall walking costs and queuing cost at the exits.Each agent state is represented by a simple dynam-ical system. Each agent will move to one the clos-est exits that is safer and with less congested path.We first formulate the problem and derive optimal-

ity equations using maximum principle and dynamicprogramming with boundary conditions. Then, well-posedness and existence results are provided. Nu-merics and simulations are carried out to illustratemean-field equilibria of a safer evacuation process.

Sobolev Regularity for Weak Solu-tions of First Order MFG

Daniela TononCEREMADE Universite Paris Dauphine, FrancePierre Cardaliaguet, Alessio Porretta

The theory of mean field games in the deterministiccase with a local coupling motivates the analysis forHamilton-Jacobi equation with possibly unboundedright-hand side. We show Sobolev estimates and al-most everywhere di↵erentiability for solutions of firstorder Hamilton-Jacobi equations with Hamiltonianswhich are superlinear in the gradient variable andright-hand sides which are only bounded in Lebesguespaces. The proof relies on an inverse Holder inequal-ity. Applications to mean field games are discussed.

Vaccination and Markov Mean FieldGames

Gabriel TuriniciUniversite Paris Dauphine, Paris, France

The vaccination is a public health policy often in-voked to control an epidemic. However it is not ex-empt of debates which can, in severe cases, lead tovaccination campaign failures.Several approaches have been proposed in the liter-ature to model this collective behaviors as equilib-rium between the individual (well-being optimizing)vision and the global, aggregate, dynamics; in thiscontext we present in this talk our works which usethe Mean Field Games framework (a la Lasry - Li-ons and Caines - Huang - Malhame). We show thatthis perspective is very versatile and allows to treatrigorously a diversity of situations. From the techni-cal point of view the individual undergoes a Markovchain dynamics and the society the resulting, deter-ministic, average master equation.We discuss then the numerical approaches to find theequilibriums and apply to some practical situations.

Mean Field Stackelberg Games:Aggregation of Delayed Instructions

Phillip YamChinese University of Hong Kong, Hong KongAlain Bensoussan, Michael Chau

In this talk, we propose to consider an N-player in-teracting strategic game in the presence of a (endoge-nous) dominating player, who gives direct influenceon individual agents, through its impact on their con-trol in the sense of Stackelberg game, and then on thewhole community. Each individual agent is subjectto a delay e↵ect on collecting information, specifically


at a delay time, from the dominating player. The sizeof his delay is completely known by the agent; whileto others, including the dominating player, his de-lay plays as a hidden random variable coming froma common fixed distribution. By invoking a non-canonical fixed point property, we show that, for ageneral class of finite N-player games, each of themconverges to the mean field counterpart which maypossess an optimal solution that can serve as anepsilon-Nash equilibrium for the corresponding finiteN-player game. Secondly, we provide, with explicit

solutions, a comprehensive study on the correspond-ing linear quadratic mean field games of small agentswith delay from a dominating player. A simple su�-cient condition for the unique existence of mean fieldequilibrium is provided by tackling a class of non-symmetric Riccati equations. Finally, via a study ofa class of forward backward stochastic functional dif-ferential equations, the optimal control of the domi-nating player is granted given the unique existence ofthe mentioned mean field equilibrium for small play-ers.


Special Session 119: Geometric Functional Inequalities and Applicationto PDEs

Hidemitsu Wadade, Kanazawa University, JapanYohei Tsutsui, Shinshu University, Japan

Futoshi Takahashi, Osaka City University, JapanMichinori Ishiwata, Osaka University, Japan

This special session mainly focuses on the functional inequalities of Sobolev type embeddings and relatedpartial di↵erential equations. Participants of this session will collect recent their studies of functional in-equalities established by using the methods of the real analysis, Fourier analysis and variational analysisas basic mathematical tools. Our main purpose of this session is to requesting the possibility of combiningthose methods together for applying evolutionary partial di↵erential equations and so on.

Di↵eomorphisms with PrescribedJacobian and Boundary Data onSobolev-Class Domains

Ching-Hsiao ChengNational Central University, TaiwanSteve Shkoller

In this talk we consider the problem of finding di↵eo-morphisms whose Jacobian and boundary data areof Sobolev class and the domain under considerationis also of Sobolev class; that is, finding a di↵eomor-phism between two Sobolev class domains⌦

1

and⌦

2

satisfying

det(r ) = f in⌦1

,

= g on @⌦1

,

where f and g are given functions possessing Sobolevclass regularity. The result for the case that f , g and⌦

1

possess classical regularity is well-known; how-ever, such kind of results appear to undergo loss ofregularity in the sense that when⌦

1

is of class Ck+3,↵

and the forcing are of class Ck,↵, the solution is ofclass Ck+1,↵. Our result show that when studyingthis problem on Sobolev class domain with Sobolevclass forcing, there exists a solution to the equationabove satisfying

kr kHk+1(⌦1)

ChkfkHk

(⌦1)+ kgkHk+0.5

(@⌦1)

i

for some generic constant C depending on the Hk+1-regularity of⌦

1

. Therefore, no loss of regularity isencountered.

On the Elliptic Equations of Hardy-Sobolev Type with Multiple Bound-ary Singularities

Jann-Long ChernNational Central University, TaiwanXiang Fang, Chun-Hsiung Hsia

In this talk, we are interested in how the geometryof boundary singularities can a↵ect the attainabilityof the respective best Ca↵arelli-Kohn-Nirenberg andHardy-Sobolev constant.

Averaged Decay Estimates forFourier Transforms of MeasuresOver Curves with NonvanishingTorsion

Seheon HamKorea Institute for Advanced Study, KoreaYutae Choi, Sanghyuk Lee

For a positive Borel measure with compact sup-port, we consider L2-averaged decay estimates of itsFourier transform. When the average is taken overthe unit sphere, the decay estimates were studied ex-tensively, in connection with the Falconer distanceset problem, by Mattila, Sjolin, Bourgain, Wol↵, Er-dougan. In this talk, we study the case of spacecurves with non-vanishing torsion. We extend theprevious known results for the unit circle to higherdimensions. Also we discuss sharpness of the esti-mates. This is a joint work with Yutae Choi (PohangUniversity of Science and Technology) and SanghyukLee (Seoul National University).

Minimization Problem on theHardy-Sobolev Inequality

Masato HashizumeOsaka City University, Japan

We consider the attainability of the best constantfor the Hardy-Sobolev inequality on bounded do-mains. It is well known that the attainability ofthis constant is a↵ected by the position of the ori-gin. Moreover, in the boundary singularity case,the mean curvature of the boundary at the singu-larity plays a crucial role in investigating the exis-tence of minimizers. In this talk, we consider theattainability of the best constant on the continuousembedding H1(⌦) ⇢ L2

⇤(s)(⌦, |x|�sdx) in the inte-

rior singularity case. Di↵erent from the case of theSobolev inequality, the attainability of the best con-stant changes according to the scale of the domain.


On the Long Time Stability of aTemporal Discretization Scheme forthe Three Dimensional PrimitiveEquations.

Chun-Hsiung HsiaNational Taiwan University, TaiwanMing-Cheng Shiue

Abstract In this joint work with Ming-Cheng Shiue,a semi-discretized Euler scheme to solve three dimen-sional primitive equations is studied. With suitableassumptions on the initial data, the long time stabil-ity of the proposed scheme is shown by proving thatthe H1 norm (in space variables) is bounded.

Existence and Nonexistence of So-lutions for a Heat Equation with aSuperlinear Source Term

Norisuke IokuEhime University, JapanYohei Fujishima

We consider a heat equation with an unboundedinitial data and investigate local in time existence,nonexistence of solutions for the Cauchy problemwith a superlinear source term. In particular, wereveal the threshold integrability of the initial datato classify existence and nonexistence of solutions forthe Cauchy problem without any assumption on thegrowth rate of the nonlinear term.

Gradient Stability of the SobolevInequality: the Case p > 2

Robin NeumayerUniversity of Texas at Austin, USAAlessio Figalli

The sharp Sobolev inequality in Rn gives control ofthe Lp⇤ norm of a function, p⇤ = np/(n�p), in termsof the Lp norm of the gradient. Equality is achievedby the (n+2)-dimensional family of Talenti functions.In this talk, we show that, in the case p � 2, func-tions which almost attain equality in the Sobolev in-equality are quantitatively close to Talenti functionsat the level of gradients. To the furthest degree pos-sible, we extend the Hilbert space methods employedin Bianchi and Egnell’s proof of the analogous resultfor p = 2 (despite the fact that Lp is not a Hilbertspace for p > 2), and then use an interpolation ar-gument to reduce to a weaker stability result alreadyshown by Cianchi, Fusco, Maggi, and Pratelli. Thistalk is based on joint work with Alessio Figalli.

Some Endpoint Estimates for Bilin-ear Paraproducts and Applications

Salvador Rodriguez-LopezStockholm University, SwedenW. Staubach

In this talk we will present some endpoint estimatesfor bilinear paraproducts of Bony’s type. That is,operators of the form

⇧(f, g)(x) =

Z 1

0

Qtf(x)Ptg(x)m(t)dtt,

where Pt and Qt represent frequency localisation op-erators near the ball |⇠| C/t and the annulus|⇠| ⇡ 1/t, respectively, and m is a bounded function.More precisely, we will present some new bounded-ness estimates for bilinear paraproducts operators onlocal bmo spaces.We will motivate this study by giving some appli-cations to the investigations on the boundednessof bilinear Fourier integral operators and bilinearCoifman-Meyer multipliers.

Generalized Critical Hardy Inequal-ities with the Optimal Constant

Megumi SanoOsaka City University, Japan

Let ⌦ be a bounded domain in RN with 0 2 ⌦,R = supx2⌦

|x|, N � 2, a = 1, e and q,�> 1. Inthis talk, we consider the minimization problem as-sociated with the optimal constant of the generalizedcritical Hardy inequality as follows:

F (⌦; a) := inf06=u2W

1,N0 (⌦)

E(u)

:= inf06=u2W

1,N0 (⌦)

R⌦

|ru|Ndx✓R

⌦

|u|q|x|N (log

aR

|x| )�

dx

◆N

q

.

When q = � = N , it is known that the optimal con-

stants F (⌦; a) are�N�1

N

�Nwhich are independent of

⌦ and are not attained both a = 1 and a = e.Note that, in general, we can not apply therearrangement technique for F (⌦; a) due tothe non-monotonicity of the potential function

|x|�N⇣log aR

|x|

⌘��. Furthermore the quotient E(u)

does not have the scale invariance under the scaling

u�(x) = ��N�1N u

✓⇣|x|aR

⌘��1

x

◆for � > 0, because

of the term ||ru||N .We prove the positivity and the attainability ofF (⌦; a) under some conditions of (q,�,⌦).


Dispersive Estimates for the StablyStratified Boussinesq Equations

Ryo TakadaTohoku University, JapanSanghyuk Lee

We consider the initial value problem for the 3DBoussinesq equations for stably stratified fluids with-out the rotational e↵ect. We establish the sharp dis-persive estimate for the linear propagator related tothe stable stratification. As an application, we givethe explicit relation between the size of initial dataand the buoyancy frequency which ensures the uniqueexistence of global solutions to our system. In par-ticular, it is shown that the size of the initial thermaldisturbance can be taken large in proportion to thestrength of stratification. This talk is based on thejoint work with Sanghyuk Lee (Seoul).

Some Improvements for a Classof the Ca↵arelli-Kohn-NirenbergInequalities

Futoshi TakahashiOsaka City University, JapanMegumi Sano

In this talk, we concern a weighted version of theHardy inequality:Z

⌦

|ru|p|x|�padx �✓N � p� pa

p

◆p Z

⌦

|u|p|x|p(a+1)

dx

for all u 2 C10

(⌦), where ⌦ is a smooth boundeddomain in RN (N � 3) with 0 2 ⌦, or ⌦= RN ,1 < p < N and �1 < a < N�p

p. This is a special

case of the more general Ca↵arelli-Kohn-Nirenberginequalities. On the whole space, we improve theinequality by adding a remainder term of the formof ratio of two weighted integrals. Also we derive aremainder term involving a distance from the mani-fold of the “virtual extremals”. Finally on boundeddomains, we prove the existence of remainder termsinvolving the gradient of functions. This talk isbased on a joint work with Megumi Sano (Osaka CityUniv.).

Div - Curl Estimate with CriticalPower Weight

Yohei TsutsuiShinshu University, Japan

We treat with div - curl estimates due to Coifman-Lions-Meyer-Semmes in Hardy spaces with powerweights. This inequality was applied to control theconvection term in the incompressible Navier-Stokes

equations. There is a critical degree of power weightconcerning the optimal L2-energy decay. Div - curlestimate with sub-critical power weights was alreadyknown. In this talk, we give an inequality with crit-ical power weight, using a real interpolation spacesof weighted Hardy spaces, but our estimate involvesquasi-Banach spaces.

On the Maximizing Problem Asso-ciated with Trudinger-Moser TypeInequalities

Hidemitsu WadadeKanazawa University / Institute of Science andEngineering, JapanMichinori Ishiwata

In this talk, we consider the existence and non-existence of maximizers associated with Trudinger-Moser type inequalities. Recently, A Trudinger-Moser inequality was derived by B. Ruf, JFA, 219(2005) for the two spacial dimension and extendedto the higher spacial dimensions by Y. Li-B. Ruf,Indiana UMJ, 57 (2008), which are in-homogeneoustype inequalities in the whole space. In the papersLi-B. Ruf, Indiana UMJ, 57 (2008) and M. Ishi-wata, Math. Ann. 351 (2011), the authors consid-ered the variational problems associated with theseTrudinger-Moser type inequalities and proved thatthe existence and non-existence results depending onthe exponents appearing in the exponential type in-tegrals. We revisit the existence and non-existenceproblems for the above inequalities and clarify thee↵ects of the norm-normalization to the structure ofthese variational problems.

Lp Estimates for Some Pseudo-Di↵erential Operators

Lu ZhangWayne State University, USAGuozhen Lu

We study the Lp estimates for a class of trilinearpseudo-di↵erential operators with flag symbols anda bi-parameter bilinear Calderon-Vaillancourt theo-rem. For the trilinear pseudo-di↵erential operators,the symbols are in the form of the product of twostandard symbols from the Hormander class BS0

1,0.Such operators are extensions from the trilinear op-erators with flag singularities, with the multipliersin the form of product of two Marcinkiewcz-Mikhlin-Hormander symbols. For the bi-parameter Calderon-Vaillancourt theorem, we take the symbols from thebi-parameter Hormander class BBSm

0,0 and and studythe Holder’s type estimate.


Special Session 120: Global Bifurcations and Complex Dynamics

Ivan Ovsyannikov, University of Bremen, Germany

The session is devoted to the topics related to global bifurcations, which emerge in the theoretical contextor various applications. It is well-known that the stable and unstable manifolds of fixed points (both incontinuous and discrete-time systems) are usually embedded into the phase space in a very complicatedway, forming, in particular, homoclinic or heteroclinic cycles. Moreover, small perturbations of such cyclesmay lead to the appearance of complex structures, such as finite or infinite sets of periodic orbits, invarianttori, formation of Smale horseshoes, strange attractors etc. At this session research talks on homoclinic andheteroclinic bifurcations are welcome, as well as the talks covering the study of the structures of stable andunstable invariant manifolds.

Bifurcations of Invariant ManifoldsNear a Non-Central Saddle-NodeHomoclinic Orbit

Pablo AguirreUniversidad Tecnica Federico Santa Maria, Chile

We investigate the role of the two-dimensional globalinvariant manifolds near a codimension-two non-central saddle-node homoclinic point in a three-dimensional vector field. The main question isto determine how the arrangement of global two-dimensional manifolds changes through the unfold-ing and how this a↵ects the topological organisationof basins of attraction. To this end, we compute therespective global invariant manifolds — rendered assurfaces in the three-dimensional phase space—, andtheir intersection curves with a suitable sphere, asfamilies of orbit segments with a two-point boundary-value-problem setup. As a specific example to workon, we consider a laser model with optical injectionwhich undergoes this codimension-two bifurcation.For this model vector field we present two-parameterbifurcation diagrams (with representative images) ofthe invariant manifolds and the relevant basins ofattraction near the codimension-two singularity. Inparticular, this combination of bifurcation analysisand identification of basin boundaries in phase spaceallow us to find conditions for multipulse behaviourin the laser model depending on the global manifoldsin open regions of parameter space and at the bifur-cations involved.

Pulses with Oscillatory Tails andA Homoclinic Banana in theFitzHugh-Nagumo System

Paul CarterBrown University, USABjorn Sandstede

It is well known that the FitzHugh-Nagumo systemexhibits stable, spatially monotone traveling pulses.Recently, it has been shown that this system also ad-mits traveling pulse solutions with oscillatory tails.We discuss analytical results regarding the existenceand stability of such pulses using geometric blow-uptechniques and singular perturbation theory, and weoutline a mechanism that explains the transition fromsingle to double pulses along a so-called homoclinicbanana that was observed in earlier numerical stud-ies.

Deconstructing the Stunning Com-plexity of Global Bifurcations in aFar-Infrared Raman Laser Model

Krishna PusuluriGeorgia State University, USAAndrey Shilnikov

A new computational technique based on the sym-bolic description of complex homoclinic and hete-roclinic bifurcations underlying the occurrence anddynamics of the Lorenz-like attractor has been em-ployed to an example system from nonlinear optics- 6D model of the optically pumped, far-infraredthree-level molecular laser. The bifurcation featureof this laser model is the coexistence of heteroclinicT-points of various kinds with accompanying self-similar fractal structures that they generate in theparameter plane - now the de-facto key signatures ofmost systems with the Lorenz attractor. Of specialinterest are quite unordinary codimension-two ho-moclinic bifurcations called inclination-switch, whichhas never been reported previously and can only oc-cur in multi-dimensional systems. By employing thelatest advancements in parallel computing techniques- OpenMP, CUDA, OpenACC and OpenMPI- weachieve significant performance improvements whichallow us to study the complex laser system at stun-ning resolutions.

Melnikov Processes and Chaos inRandomly Perturbed DynamicalSystems

Kazuyuki YagasakiKyoto University, Japan

We consider a wide class of randomly perturbed sys-tems subjected to stationary Gaussian processes andshow that chaotic orbits exist almost surely undersome degenerate condition, no matter how small therandom forcing terms are. This result is very contrastto the deterministic forcing case, in which chaotic or-bits exist only if the influence of the forcing termsovercomes that of the other terms in the perturba-tions. To obtain the result, we extend Melnikov’smethod and prove that the corresponding Melnikovfunctions, which we call the Melnikov processes, haveinfinitely many zeros, so that infinitely many trans-verse homoclinic orbits exist. We illustrate our the-ory for the randomly perturbed Du�ng oscillatorsubjected to the Ornstein-Uhlenbeck process.


Special Session 121: Recent Advancements in Computational MethodsInvolving Implicit or Non-parametric Interfaces

Catherine Kublik, University of Dayton, USARichard Tsai, KTH Royal Institute of Technology, Sweden and The University of Texas at Austin, USA

We are interested in gathering active researchers who are working on numerical methods for solving di↵er-ential or integral equations on manifolds with no explicit parameterizations (e.g. using level set or closestpoint methods). This session targets applications where the interface is evolving in time, where the solutionof a PDE is needed on some manifold, and is motivated by the recent need to work with unstructured pointclouds sampled from an underlying manifold.

The Dirichlet-To-Neumann Opera-tor with a Level-Set Function

Julien DambrineUniversity of Poitiers, FranceNicolas Meunier

The motion of surfaces with a velocity depending onthe Dirichlet-to-Neumann operator for a given ellip-tic problem appear in various practical applicationsranging from the motion of cells to the geometricaloptimisation of mechanical structures. The level-setframework is particularly interesting in this contextof moving surfaces. In this work we focus on thecomputation of the Dirichlet-to-Neumann operatorcalculation for the Laplace equation, following theideas developed in [1] intended for the computationof the bulk solution.

References

[1] Catherine Kublik, Nicolay M. Tanushev,Richard Tsai, An implicit interface boundaryintegral method for Poisson’s equation on arbi-trary domains, JCP, 2013.

On a Vector Field Embedding ofMultiphase Geometries

Elliott GinderHokkaido University, Japan

Tracking the evolution of multiphase geometries rep-resents a fundamental problem that arises in a varietyof scientific simulations. The phenomena under con-sideration can exhibit additional challenges to its nu-merical simulation, including the occurrence of topo-logical changes, volume constraints, and inertial ef-fects. We will introduce a method for treating theseissues and investigate its application in the simula-tion of 2D and 3D multiphase parabolic and hyper-bolic curvature flows. Volume constrained motionswill also be investigated through the use of minimiz-ing movements, and we will remark about aspectsrelated to our method’s numerical realization.

Implicit Interface Boundary Inte-gral Methods and Their Applicationto the Mullins-Sekerka Problem

Catherine KublikUniversity of Dayton, USARichard Tsai, Chieh Chen

We describe a simple formulation for integrating oversmooth curves or surfaces that are described implic-itly through a level set function or directly by theirclosest point mapping. Contrary to common prac-tice with level sets, the volume integrals derived fromour formulation coincide exactly with the boundaryintegrals. We use this formulation to simulate theMullins and Sekerka dynamics on unbounded do-mains via boundary integral methods.

Solve Geometric PDEs on Mani-folds Represented As Point Cloudsand Applications

Rongjie LaiRensselaer Polytechnic Institute, USA

In this talk, I will discuss our recent work of solv-ing geometric PDEs on manifolds sampled as pointclouds. These methods can achieve high order accu-racy and enjoy flexibility of solving di↵erent typesof equations on manifolds with possible high co-dimesion. We use the proposed methods to considerspecial designed geometric PDEs on point clouds,which provides us a bridge to link local and globalinformation. Based on this method, I will discuss afew applications to geometric understanding for pointclouds, including computation of LB eigen-systemsfor point clouds, extraction of global skeletons struc-ture from point clouds, extraction of conformal struc-tures from point clouds, and intrinsic comparisonsamong point clouds etc. In addition, our methodscan also be extended to solve PDEs on manifoldsonly represented as incomplete distance information.I will also discuss our results of this method for re-constructing and understanding distance data basedon solutions of Laplace-Beltrame equations.


A New Level Set Method for Spi-rals Evolving by Eikonal-CurvatureEquation and Its Application toSpiral Crystal Growth

Takeshi OhtsukaGunma University, Japan

Theory crystal growth by rotating spiral steps wasintroduced by Burton, Cabrera and Frank in 1951.One can find evolving spiral patterns on the crys-tal surface including singularities caused by collisionof several spirals or singular anisotropy of the mo-tion. To describe such a motion of spirals we in-troduce a simple level set method describing spiralswith a zero-level set of an auxiliary function minuspre-determined multi-valued function. Conventionallevel set method does not work well for spirals sincethe usual level set divide the domain into two dis-joint sets although spirals do not make such a divi-sion. Our method can be applied not only isotropicbut also anisotropic evolution. For anisotropic cur-vature with strong singularities, which is called ascrystalline curvature, we shall introduce a new ap-

proach. As an application of our method, we presentsome numerical results on vertical growth rates of thesurface by co-rotating spiral steps. In the theory ofcrystal growth, it is well known that if the centers ofco-rotating pair are close together, then the verticalgrowth rate increases from the single one. We ex-amine the growth rate and improve the estimate byBurton et al. numerically.

Using Evolving Interface Techniquesto Solve Network Problems

Yves van GennipUniversity of Nottingham, England

In recent years, ideas from the world of evolving in-terface PDEs have found their way into the arenaof graph and network problems. In this talk I willdiscuss how techniques based on the Allen-Cahnequation and the Merriman-Bence-Osher thresholddynamics scheme can be used to detect particularstructures in graphs, such as densely connected sub-graphs (clustering and classification) and bipartitesubgraphs.


Special Session 122: Variational Convergence and Degeneracies in PDEs:Fractal Domains, Composite Media, Dynamical Boundary Conditions

Maria Rosaria Lancia, Sapienza University of Rome, ItalyMaria Agostina Vivaldi,Sapienza University of Rome, ItalyRa↵aela Capitanelli, Sapienza University of Rome, Italy

In the last decades there has been an increasing interest in studying degeneracies in BVPs such as thosedue to highly irregular domains as in the case of fractal boundaries or interfaces, or due to the presenceof composite media . Other singularities arise when studying evolution problems with dynamical boundaryconditions in fractal domains. In all these cases it is important, also in view of numerical approximations,to approximate these wild geometries by smoother ones and to study the convergence of approximatingfunctions to the limit fractal one. This is obtained by variational convergence techniques. This session isdevoted to new results on this type of degeneracies.

On the E↵ective Boundary Condi-tions Throughquasi-Filling FractalLayers

Ra↵aela CapitanelliSapienza Univ. Roma, ItalyCristina Pocci

We study the periodic homogenization of the station-ary heat equation in domain with two connected com-ponents, separated by an oscillating interface definedon prefractal Koch type curves. The problem de-pends both on the parameter n, which is the index ofthe prefractal iteration, and ✏, that defines the peri-odic structure of the composite material. We discussabout the commutative nature of the limits in ✏ andn.

Numerical Approximation ofNon Local Venttsel Problems in(pre)fractal Domains

Simone CreoUniversity of Rome La Sapienza, ItalyM. Cefalo, M. R. Lancia, A. I. Nazarov, P.Vernole

We consider a di↵usion problem of parabolic typein a polygonal domain⌦ ✓ R2 having Koch typeboundary Kn with non local Venttsel boundary con-ditions. We prove the existence and uniqueness ofthe weak solution, as well as regularity results inweighted Sobolev spaces. We consider the numeri-cal approximation by finite element methods in spaceand finite di↵erences in time. Crucial tools in provinga priori error estimates of the numerical approxima-tion are the regularity properties of the weak solu-tion.

Elliptic Problems for the FractionalLaplacian with a Singular Nonlin-earity

Ida de BonisUniversity Giustino Fortunato, Italy

We study the existence, regularity and uniqueness ofsolution of the following nonlocal problem

(P )

⇢(��)su = F (x, u) in ⌦,

u = 0 in RN \ ⌦,

where ⌦ is a bounded smooth domain of RN , N > 2s,0

Trigonometrical Inequalities onRegular Lattices.

Anna Chiara LaiSapienza University of Roma, ItalyVilmos Komornik, Paola Loreti

Starting from Parseval’s formula for multiple Fourierseries, we consider non-harmonic Fourier expansionswith exponents belonging to sets that can be decom-posed in the finite union of translations of a fixedlattice. In the two dimensional case, this class of dis-crete sets includes well-known tilings of the plane,such as the honeycomb lattice and all the regularlattices. In this framework, we show Ingham typetrigonometrical inequalities. The talk is based on ajoint work with V. Komornik and P. Loreti.

Venttsel Boundary Value Problemsin Fractal Domains

Maria Rosaria LanciaS.B.A.I. Sapienza universita‘ di Roma, ItalyP. Vernole

Venttsel problems belong to the class of the so-calledboundary value problems with dynamical boundaryconditions. Problems with Venttsel type conditionshave applications to various fields of science and tech-nology e.g. water wave theory, models of heat trans-fer as well as engineering problems of hydraulic frac-turing. It is to be pointed out that these conditionsappear also under the name of e↵ective boundary


conditions (EBCs) for thin ferromagnetic layers orin laminar flows. Several natural and industrial pro-cesses lead to the formation of rough surfaces or oc-cur across them. Fractal boundaries and fractal lay-ers may be of great interest for those applications inwhich the surface e↵ects are enhanced with respectto the surrounding volume. The use of fractal ge-ometries is by now well di↵used in some industrialapplications; for instance with the increasing minia-turization of electronic chips and increasingly largerheat dissipation rates better design of cooling sys-tems are necessary. Fractal boundaries could also beused in catalytic converters as well as, in all those ap-plications in which di↵usion phenomena take place insmall volumes with large surfaces. Recently, we fo-cused on the study of linear, semilinear, quasilinearand non local evolution Venttsel problems in domainwith fractal boundaries and on their approximations.In this talk I present a survey of the results obtainedin collaboration with P.Vernole and A.Velez-Santiagoas well as some open problems.

Degenerate Elliptic Operators, HeatKernel Estimates and Harnack In-equalities.

Luisa MoschiniSapienza, University of Rome, Italy

We will deal with parabolic harnack inequalities, Li-ouville type theorems as well as heat kernel sharp twosided estimates in various context of degeneracies.Degeneracies will be given in some cases by distancesto smooth k dimensional surfaces, in these cases theyare also related to Scroedinger operators with criti-cal potentials, involving inverse square distances tothe same k dimensional surfaces. A brief approachto Schroedinger operators with fractional di↵usion isalso given as well as to the case of anisotropic degen-eracies, that is to the case in which di↵erent powersof the same distance function are involved in the var-ious derivatives.

On the Dynamics of Space-FillingAttractors

Umberto MoscoWorcester Polytechnic Institute, USA

We present a family of impulsive Cauchy initial valueproblems in the plane driven by irregular accelerationfields on discrete synchronized space time grids withattractors of infinite perimeter that asymptoticallyare space filling. The 1D dynamics of these attrac-tors spectrally interpolate the one dimensional andthe two dimensional Laplace operators on planar do-main. We also describe a fully discrete nonlinear selforganized criticality model with finite time extinctiondynamics, in which grid synchronization also plays animportant role.

Short Time Heat Di↵usion inBounded Domains with Discon-tinuous Transmission BoundaryConditions

Anna Rozanova-PierratCentraleSupelec, FranceClaude Bardos, Denis Grebenkov

We consider a heat problem with discontinuous dif-fusion coe�cients and discontinuous transmissionboundary conditions with a resistance coe�cient. Forall bounded (✏,� )-domains⌦ ⇢ Rn with a d-setboundary (for instance, a self-similar fractal), we findthe first term of the small-time asymptotic expansionof the heat content in the complement of ⌦, and alsothe second-order term in the case of a regular bound-ary. The asymptotic expansion is di↵erent for thecases of finite and infinite resistance of the boundary.The derived formulas relate the heat content to thevolume of the interior Minkowski sausage and presenta mathematical justification to the de Gennes‘ ap-proach. The accuracy of the analytical results is il-lustrated by solving the heat problem on prefractaldomains by a finite elements method.

Dirichlet Boundary Conditions forDegenerate and Singular NonlinearParabolic Equations

Marta StraniUniversite Paris Diderot, IMJ-PRG, FranceFabio Punzo

We study existence and uniqueness of solutions to aclass of quasilinear degenerate parabolic equations,in bounded domains⌦ ⇢ RN . We show that thereexists a unique solution which satisfies possibly in-homogeneous Dirichlet boundary conditions. To thispurpose some barrier functions are properly intro-duced and used.

Fractal Snowflake Domain Di↵usionwith Boundary and Interior Drifts

Alexander TeplyaevUniversity of Connecticut, USAMichael Hinz, Maria Rosaria Lancia, PaolaVernole

We study evolution equations for a (elliptic measur-able coe�cients) di↵usion in the classical snowflakedomain in situations when there are di↵usion anddrift terms not only in the interior but also on thefractal boundary, which is a union of three copiesof the classical Koch curve. We prove that understandard conditions related Cauchy problems pos-sess unique strict solutions and explain in which sensethey solve a rigorous formulation of our problem. Asa second result we prove that functions that are in-trinsically Lipschitz on the snowflake boundary ad-


mit Euclidean Lipschitz extensions to the closure ofthe entire domain. Our methods combine the frac-tal membrane analysis, the vector analysis for localDirichlet forms and PDE on fractals, coercive closedforms, and the analysis of Lipschitz functions.

Magnetorheological Composites

Bogdan VernescuWPI, USAG. Nika

We consider a magnetorheological composite mate-rial: i.e a suspension of solid magnetizable particlesin a non-conducting viscous fluid in the presence of anapplied external magnetic field. The Maxwell equa-tions are coupled with the Stokes equations and therigid particle motion to capture the magnetorheolog-ical e↵ect. We obtain the e↵ective equations con-sisting of a coupled nonlinear system in a connectedphase domain as well as the new constitutive laws.Qualitative properties of the solution of this nonlin-ear system are studied.

Asymptotics for Venttsel Problemsin Non Divergence Form in Irregu-lar Domains

Paola VernoleRoma Sapienza, ItalyM.R. Lancia, V. Regis Durante

In this talk we study a boundary value problem for asecond order operator in non divergence form L withVenttsel’s boundary conditions

(P )

8>><

>>:

ut(t, P )� Lu(t, P ) = f(t, P ) in [0, T ]⇥Qut(t, P )� �Su(t, P ) + b(P )u(t, P ) = � @u

@nA+ f(t, P ) on [0, T ]⇥ S

@u@nA

= 0 on [0, T ]⇥ @Q \ Su(0, P ) = 0 in Q,

where Lu =P

3

i,j=1

aij@iju+a0

u, aij are symmetric,uniformly Lipschitz functions in Q satisfying suit-able ellipticity conditions and a

0

is a positive L1(Q)function, Q is the three-dimensional domain withlateral boundary S = F ⇥ [0, 1], where F is the Kochsnowflake;� S is the fractal Laplacian on S , b isa continuous strictly positive function on S, @u

@nAis the co-normal derivative across @Q to be definedin a suitable sense, f(t, P ) is a given function inC✓([0, T ];L2(Q,m)), ✓ 2 (0, 1) and m is the sumof the three-dimensional Lebesgue measure and of asuitable measure g supported on S .

Existence and uniqueness results are proved via asemigroup approach. From the point of view of nu-merical analysis it is also crucial to study the corre-sponding approximating (prefractal) problems in thedomains Qh, where {Qh}h2N is a sequence of increas-ing (invading) domains approximatingQ, with lateralsurface Sh = Fh ⇥ [0, 1] , the corresponding approxi-mating polyhedral surfaces, where Fh is a prefractalcurve approximating F . To this aim the asymptoticbehavior, as h ! 1, of the approximating solutionsis studied. As to the asymptotic behavior of the so-lutions, the presence of the time derivative in theboundary conditions requires, as a natural functionalsetting for these problems, suitable varying Hilbertspaces. Moreover since the energy forms are not sym-metric we use the results of Ma-Rockner, Rockner-Zhang and Toelle in order to prove the convergenceof the forms, of the associated semigroups and henceof the solutions. Crucial tools to prove the conver-gence of the energy forms are density results for theenergy spaces V (Q,S).

A Second Order Elliptic Problemwith Unnatural Boundary Condi-tion Arising in Tensile StructuresDesign

Giuseppe ViglialoroUniversity of Cagliari, Italy

This talk is concerned with a mathematical prob-lem modeling the equilibrium of a thin membranestructure (tensile structure), with rigid and cableboundaries. The general formulation is expressedby means of a second order elliptic PDE in terms ofthe membrane shape and its stress tensor; contrarily,the boundary constrains take into account a singu-lar condition which makes the general analysis morecomplex. We discuss some partial results (both bythe theoretical and numerical point of view) as wellas present open questions.

P-Laplaceans and Mass TransportProblems

Maria Agostina VivaldiSBAI, Sapienza Universita di Roma, Italy

This talk deals with p-Laplaceans type problems onboth Euclidean domains with fractal boundaries andfractal structures such as the Sierpinski Gasket. Weuse the approach of the variational convergences toinvestigate the asymptotic behavior of the solutionsand the connection with the Kantorovich potentialfor the transport problem. Some numerical aspectsare also discussed.


Contributed Session 1: ODEs and Applications

Tikhonov Regularization of a Singu-larly Perturbed Two-Point Bound-ary Value Problem

Sukla AdakIndian Institute of Technology Madras, IndiaArindama Singh

In this paper, we consider a linear second order two-point boundary value problem where a small param-eter is multiplied with the highest order derivative.Since solutions of such a problem do not converge uni-formly in the interval of integration, it may be con-sidered as an ill-posed problem. We would like to ex-plore the possibility whether a regularization methodsuch as Tikhonov’s can be used for solving this prob-lem. It is found that the solution of the regularizedproblem converges to the original solution. When thenoisy data is available and it is known, then a prioriparameter choice rule can be given. Shishkin meshgives an approximate length of a boundary layer oncethe location of layer is roughly known. The numericalexperiments confirm that the computational solutionis improved on a Shishkin mesh by Tikhonov’s regu-larization technique.

Robustness of Isolated Non-SaddleSets

Hector BargeUniversidad Complutense de Madrid, SpainJ.M.R. Sanjurjo

Isolated non-saddle sets arise in a natural way whendealing with the qualitative study of ODEs. Locally,these sets exhibit much of the properties of attractorsand repellers. However the global picture of the flowmay be dramatically di↵erent. In this talk we willintroduce this kind of objects and we will focus ontheir continuation properties. It turns out that, inspite of the fact that attractors and repellers are ro-bust under small pertubations, the property of beingnon-saddle is not. We will see some examples illus-trating this fact and we will see that the continuationof this property has much to do with the continu-ation of some topological properties of the originalnon-saddle set. Some of the results presented herewere obtained in collaboration with J.M.R. Sanjurjo.

Matrix- Valued Di↵erential Equa-tions with Spectral Singularities

Serifenur CebesoyAnkara University, TurkeyElgiz Bairamov

The main aim of this paper is to obtain the Jostsolutions and some spectral properties of a secondorder matrix nonself-adjoint di↵erential equation onthe whole axis. In this study, we investigate the ana-

lytical properties and asymptotic behaviors of theseJost solutions. Then, we find continuous spectrum ofthe operator L generated by matrix-valued di↵eren-tial equation and we get that the operator L has afinite number of spectral singularities and real eigen-values.

Existence Results of Solutions forNonlinear Boundary Value Prob-lems on an Infinite Interval

Fulya Yoruk DerenEge University, TurkeyNuket Aykut Hamal

This study is concerned with a boundary value prob-lem of a second order di↵erential equation on an in-finite interval. Here, under some assumptions on thenonlinearities, the existence results of solutions areestablished by using some fixed point theorems.

On the Connectedness of the At-tainabvle Set of DiscountinuousQuantum Stochastic Di↵erentialInclusions

Dauda DikkoUniversity of Ibadan, NigeriaE. O. Ayoola

Using the notion of quantum stochastic di↵erentialinclusion, we establish a non-commutative generali-sation which shows that the attainable sets of dis-countinuous quantum stochastic di↵erential inclu-sions are connected. This result remain valid withinthe framework of the Hudson-Parthasarathy formu-lation of quantum stochastic di↵erential inclusions.

Bifurcational and Topological Meth-ods for the Global Qualitative Anal-ysis of Low-Dimensional DynamicalSystems

Valery GaikoNAS Belarus, Belarus

We carry out the global qualitative analysis of low-dimensional polynomial dynamical systems. First,using new bifurcational and topological methods,we solve Hilbert’s Sixteenth Problem on the maxi-mum number of limit cycles and their distributionfor the 2D general Lienard polynomial system andHolling-type quartic dynamical system. Then, apply-ing a similar approach, we study 3D polynomial sys-tems and complete the strange attractor bifurcationscenario for the classical Lorenz system connectingglobally the homoclinic, period-doubling, Andronov–Shilnikov, and period-halving bifurcations of its limitcycles which is related to Smale’s Fourteenth Prob-lem.

CONTRIBUTED SESSION 1 395

Improvements to the AveragingTheory of Periodic Di↵erentialEquations

Isaac GarciaUniversity of Lleida, Spain

We consider a family of T -periodic analytic di↵eren-tial equation in⌦ ⇢ Rn of the form

x = F (t, x;�," ) =X

i�1

Fi(t, x;�) "i,

where � 2 Rp are the parameters of the family and,for all i, the functions Fi are analytic and T -periodicin the t variable. Here, " is the small real perturba-tion parameter.We consider the Poincare displacement map at timeT , that is, d(z,�," ) = x(T ; z,�," ) � z where, foreach z 2 ⌦, we denote x(t; z,�," ) the solutionof ODE with initial condition x(0; z,�," ) = z. .Clearly, the zeros of d are initial conditions for theT -periodic solutions of the di↵erential equation. Ex-pand d(z,�," ) =

Pi�1

fi(z;�) "i, and denote the ↵-

th component of the averaged function fi by f [↵]i

with ↵ 2 {1, . . . , n}. Consider now the ideals

I [↵] = hf [↵]i (z;�) : i 2 Ni in the Noetherian ring

R{z,�}(z0,�⇤

)

formed by all the real analytic func-tions at (z

0

,�⇤).We show the relationship between the cardinality ofthe minimal basis of the nontrivial ideals I [↵] andthe maximum number of isolated T -periodic solu-tions that can bifurcating from a fixed z

0

2 ⌦ for� = �⇤ and |"| ⌧ 1.

First Integrals, Inverse IntegratingFactors and Lie Groups Admittedby N-Th Order Autonomous Sys-tems

Yanxia HuNorth China Electric Power University, Peoples Repof China

The methods of obtaining first integrals and inverseintegrating factors of n-th autonomous systems usingone-parameter Lie groups admitted by the systemsare discussed and given. Some su�cient conditionson the existence of the inverse integrating factors ofcertain classes of n-th order autonomous systems arepresented, and the explicit forms of the inverse inte-grating factors can be shown. Simultaneously, severalrelated examples are given to illustrate the feasibilityand the e↵ectiveness of the proposed methods.

Spread of CTV Using Meta Popula-tion ODEs

Stephen IppolitoASTA, USA

The spread of the Cirtus tristeza virus(CTV) hasbeen studied in Eastern Spain with the goal of de-termining the dynamics of the spread. Results donot support patterns in which the spread necessarilyfollowed from adjacent trees which were already in-fected. The suggested mechanism for the results wasinter- and intra-plot spread. Following a series of at-tempts to model this problem we consider a BayesianODE meta-population model where each meta pop-ulation indicates a probability of infection at a par-ticular time.

Oscillation of a Class of Delay Dif-ferential Equations with Impulses

Fatma KarakocAnkara University, Turkey

In this talk we present a class of linear delay di↵er-ential equations with impulses. The equation whichwe introduce includes continuous argument as wellas piecewise constant argument. Su�cient conditionsfor the oscillation of the solutions are obtained.

Cordial Volterra Integral Equations

Melusi KhumaloUniversity of Johannesburg, So AfricaZ. W. Yang

nt Cordial Volterra integral equations (CVIEs), as-sociated with a noncompact cordial Volterra integraloperator, have been prevalent in recent years since anumber of real problems incorporate delayed historyinformation. In this paper, we investigate some prop-erties of cordial Volterra integral operators influencedby a vanishing delay. It is shown that to replicate alleigenfunctions �, � = 0 or R(�)> 0, the vanishingdelay must be a proportional delay. For such a lin-ear delay, the spectrum, eigenvalues and eigenfunc-tions of the operators and the existence, uniquenessand solution spaces of solutions are presented. For anonlinear vanishing delay, we show a necessary andsu�cient condition such that the operator is com-pact, which also yields the existence and uniquenessof solutions to CVIEs with the vanishing delay


Stability Analysis of ImpulsiveSwitched Singular Systems withTime-Delay

Humeyra KiyakUniversity of Waterloo, CanadaMohamad S. Alwan, Xinzhi Liu

In this work, we consider the impulsive switched sin-gular systems with time-delay. The focus here is toaddress the problem of exponential stability of thesystem which consists of both stable and unstablesubsystems. The novelty of this work is to establishsome new su�cient conditions that guarantee the ex-ponential stability of the system. The methodologyof multiple Lyapunov function and the average dwell-time switching signal are used to achieve this goal.It has been observed that exponential stability of thesystem subject to perturbing impulses is guaranteedif the total activation time of stable subsystems islarger than that of the unstable ones. Numerical ex-amples with simulations are presented to clarify theproposed result.

Some Results on a Third Order Dif-ferential Equation with a PiecewiseConstant Argument

Mehtap LafciAnkara University, TurkeyHuseyin Bereketoglu, Gizem S. Oztepe

In this work, we consider a third order di↵erentialequation with a piecewise constant argument . Westudy existence and uniqueness of solutions of thisequation and obtain some conditions that guaranteequalitative properties such as oscillation, nonoscilla-tion and periodicity.

Periodic Solutions for Delay Di↵er-ential Equations with NonnegativityConstraints

David LipshutzBrown University, USARuth Williams

Dynamical system models with delayed feedback andnonnnegativity constraints arise in a variety of ap-plications in science and engineering. In some ap-plications, oscillatory behavior is critical to the wellfunctioning of the system. As a prototype for suchmodels, we consider a one-dimensional delay di↵er-ential equation with a nonnegativity constraint. Inparticular, the equation has discontinuous dynamicsat zero. We study the existence, uniqueness and sta-bility of slowly oscillating periodic solutions and illus-trate our findings using a model for a simple geneticcircuit. This is joint work with Ruth Williams.

Center Cyclicity of a Family ofQuintic Polynomial Vector Fields

Susanna MazaUniversitat de Lleida, SpainI. A. Garcıa, J. Llibre

We present a method for studying the Hopf cyclic-ity problem for the non-degenerate centers withoutthe necessity of solving previously the Dulac complexcenter problem associated to the larger complexifiedfamily. As application we analyze the Hopf cyclic-ity of the centers of the quintic polynomial familywritten in complex notation as

z = iz + zz(Az3 +Bz2z + Czz2 +Dz3).

The center problem for this family has been solved byJ. Llibre and C. Valls, but the Hopf cyclicity is onlystated for the easier case of having a focus at z = 0.Hence we will restrict our attention on the cyclicityproblem of the center at z = 0 and our results arestated below.Theorem. The following statements hold.(a) Any nonlinear center at the origin of family hasHopf cyclicity at most 6 when we perturb it insidethis family.(b) There are perturbations of the linear center z =iz inside family producing 6 limit cycles bifurcatingfrom the origin.This is a work already published in J. Di↵erentialEquations, 258, (2015), no.6, 1990-2009.

Dynamics and Integrability ofQuadratic Three-Dimensional Poly-nomial Vector Fields Having anInvariant Paraboloid

Marcelo MessiasSao Paulo State University - UNESP, BrazilAlisson de Carvalho Reinol

Invariant algebraic surfaces are frequently observedin vector fields or systems of ordinary di↵erentialequations arising in mathematical modeling of nat-ural phenomena. In this work we concider quadraticpolynomial vector fields defined in R3 having an el-liptic paraboloid as an invariant algebraic surface.We give the normal form for these type of vectorfields and state some results related to their integra-bility and to the existence of first integrals, Darbouxinvariants and exponential factors. For certain pa-rameter values, we prove the existence of a limit cy-cle contained in the invariant paraboloid and showthe ocurrence of centers and homoclinic orbits. Wealso investigate their dynamical behavior restrictedto the invariant paraboloid, showing the occurrenceof an interesting type of bifurcation, leading to theexistence of infinitely many orbits homoclinic to apoint at infinity, after compactification. To illustratethe possible applications of the obtained results, we


study the well-known Rabinovich system in the caseit has invariant paraboloids and perform a detailedanalysis of the dynamics and bifurcations of this sys-tem restricted to these invariant algebraic surfaces.

Volterra Integral Equations Relatedto Ordinary Di↵erential Equationswith Nondecreasing Nonlinearities

Wojciech MydlarczykWroclaw University of Science and Technology,Poland

We give a survey of the results on the solutionsto the Volterra type integral equations in the con-volution form u(t) =

R t

ak(t � s)g(u(s))ds, where

a = �1 or 0, the function k � 0 is locally inte-grable and the function g � 0 is continuous, nonde-creasing with g(0) = 0. It is easily seen that oneof the solutions is u(t) ⌘ 0, called a trivial solu-tion. However, from the point of view of applica-tions only nontrivial solutions are interested. Wefocus on the criteria for the existence of the posi-tive solutions and discuss conditions under which thesolutions experience the blowing up behavior. Theclass of the equations under consideration include theimportant class of the fractional integral equationsu(t) =

R t

a(t� s)↵�1g(u(s))ds, ↵ > 0. The considered

problems originate in discussing the classical initialvalue problem for the ordinary di↵erential equationu(n)(t) = g(u(t)), u(a) = u‘(a) = · · · = u(n�1)(a) = 0and the famous Osgood condition.

Uniform Asymptotic Stability andExponential Stability for a High-Dimensional Half-Linear Di↵eren-tial System

Masakazu OnitsukaOkayama University of Science, Japan

In this talk, we deal with a high-dimensional half-linear di↵erential system. In the special case, it be-comes the n dimensional linear system x = A(t)xwhere A(t) is an n⇥n continuous matrix and x is an ndimensional vector. It is known that the zero solutionof the linear system is uniformly asymptotically sta-ble if and only if it is exponentially stable. However,in the case of nonlinear systems, uniform asymptoticstability does not imply exponential stability. De-spite the fact that the high-dimensional half-linearsystem is nonlinear, will uniform asymptotic stabil-ity guarantee exponential stability? In this talk, wegive the answer to this question.

Boundedness for the General Semi-linear Du�ng Equations Via theTwist Theorem

Daxiong PiaoOcean University of China, Peoples Rep of China

We consider the boundedness of all solutionsfor the periodic semilinear equation x“ + !2x + (x, t) = 0, where (x, t) does not necessarily sat-isfy the so called polynomial-like growth conditionlim|x|!+1 xm (m)(x, t) = 0 for some finite m. Usu-ally this condition is needed in the references aboutboundedness problems of semilinear Du�ng equa-tions. Two cases of resonance and non-resonance areconsidered respectively.* Joint work with Yiqian Wang, Zhiguo Wang, LeiJiao and Xiao Ma

Collocation Method for SolvingSingular ODEs and Higher IndexDAEs

Ewa WeinmuellerVienna University of Technology, Austria

We deal with a numerical solution of BVPs in ODEswhich can exhibit singularities. Such problems haveoften the following form:

y0(t) =1t↵

M(t)y(t) + f(t, y(t)),

t 2 (0, 1], ↵ � 1, b(y(0), y(1)) = 0.

The search for e�cient numerical methods to solvethe above BVP is strongly motivated by numer-ous applications from physics, chemistry, mechanics,ecology, or economy. In particular, problems posedon infinite intervals are frequently transformed to afinite domain taking above form with ↵ > 1. Also,research activities in related fields, like di↵erential al-gebraic equations (DAEs) or singular Sturm-Liouvilleeigenvalue problems benefit from techniques devel-oped for singular BVPs.Polynomial collocation stays robust and shows ad-vantageous convergence properties in context of sin-gular problems. We illustrate how a collocation basedopen domain Matlab code bvpsuite can be used tosolve BVPs from applications, complex Ginzburg-Landau equation, density profile in hydrodynamics,and generalized Korteweg-de Vries equation.Finally, we consider higher index DAEs. Higher in-dex DAEs constitute a really challenging class ofproblems due to the involved di↵erentiation whichis a critical operation to carry out numerically. Wepresent a least-squares variant of the collocationmethod to deal with DAEs and discuss its perfor-mance by means of numerical experiments.


When Will a Formal Finite-Di↵erence Expansion BecomesReal?

Chiang Yik ManHong Kong University of Science and Technology,Hong KongShaoji Feng

It is a classical formula that one can expand a higher-order finite di↵erence as an infinite sum of derivativesin a formal sense. We shall show that such a formalexpansion becomes an equality when we restrict our-selves to meromorphic functions of small order (in thesense of Nevanlinna theory). The result is related torecent progress in complex function theory related todi↵erence operators.


Contributed Session 2: PDEs and Applications

Nonuniform Dependence on InitialData for the Whitham Equation

Mathias ArnesenNorwegian University of Science and Technology,Norway

We consider the Cauchy problem for the Whithamequation on the torus and on the real line and provethat the solution map for the Whitham equation isnot uniform in Hs(T) or Hs(R) for s > 3

2

. Thisis done by constructing two sequences of solutionsin Hs(T) (Hs(R)), converging to the same limit atthe initial time while the distance at any later timeis bounded below by a positive constant. The re-sult is also extended to a wide class of Whitham-type equations by considering more general disper-sive terms, covering, amongst other equations, frac-tional Korteweg-de Vries equations and the capillaryWhitham equation.

Existence of Positive Solution forBoundary Value Problem of NonLinear Fraction Di↵erential Equa-tion

Muhammad Ibrahim Badsha ZadaChinese Academy of Sciences, Peoples Rep of ChinaAnwarud Din

This paper is mainly concerned with the existenceof solution for some non-linear fractional di↵erentialequation with boundary conditions. The uniqueness,existence and multiplicity of the solution have beentried here, using Banach contraction principle andfixed point theorems. The process is based upon thereduction of the given problem to the Equivalent fred-holim integral equation and Greens function.

Quasicrystal Lattice Solitons inNLSM Systems

Mahmut BagciIstanbul Kavram Vocational College, TurkeyIlkay Bakirtas, Nalan Antar

The properties of crystal and quasicrystal materials‘surface are not necessarily the same as those of thebulk material. In the other words, although a ma-terial is centro-symmetric and the dynamics in thismaterial can be described by the cubic nonlinearSchrodinger equation (NLSE), there are quadraticpolarization e↵ects at surface of the material.

The nonlinear wave dynamics in quadratic (�(2)) ma-terials can be described by generalized NLSE withcoupling to a mean term (NLSM system). The NLSMsystem with an external potential is given by

iuz +12

�u+ |u|2 u� ⇢�u� V (x, y)u = 0,

�xx + ⌫�yy = (|u|2)xxwhere u(x, y, z) is the normalized amplitude of theelectric field, �(x, y, z) is the normalized static field(mean term), ⇢ is a coupling constant and ⌫ is theanisotropy coe�cient of the material. V (x, y) is anexternal optical lattice.In this study, we have investigated the quadratic non-linearity e↵ects on the quasicrystal lattice (Penrosetiling) solitons by the use of NLSM system as model.Using a fixed numerical scheme, we numerically ob-tained the soliton solutions of the NLSM systemswith Penrose type lattices. The linear and nonlinear(in)stabilities of these solitons have been examinedby direct computations of the NLSM systems.

Analysis of the Parallel SchwarzMethod for the Solution of Chainsof Particles in the Solvation Model

Gabriele CiaramellaUniversity of Geneva, SwitzerlandMartin J. Gander

A new class of Schwarz methods was recently pre-sented in the literature for the solution of solvationmodels, where the electrostatic energy contributionto the solvation energy can be computed by solving asystem of elliptic partial di↵erential equations; see [1]and related references. Numerical simulations haveshown an unusual convergence behaviour of Schwarzmethods for the solution of this problem, where eachparticle corresponds to a subdomain: the conver-gence of the Schwarz methods is independent of thenumber of particles [1], even though there is no coarsegrid correction. Despite the successful implementa-tion of Schwarz methods for this solvation model,a rigorous analysis for this unusual convergence be-haviour is required, since no theoretical results aregiven in the corresponding literature.


In this talk, we analyze the behaviour of the Schwarzmethod for the solution of a chain of particles andshow that its convergence does not depend on thenumber of particles. We use three di↵erent tech-niques to prove this result: the first technique isbased on a Fourier expansion of the error and onthe analysis of transfer matrices, the second con-sists in an application of the maximum-principle andthe third is a variational approach that allows us toregard the Schwarz method as an alternating projec-tion method.

References

[1] Cances et al., Domain decomposition for implicitsolvation models, J. of Chem. P. (2013).

Existence of a Unique Solution tothe Nonlinear Poisson Equation

Diane DennyTexas A&M University-Corpus Christi, USA

We consider the nonlinear Poisson equation� u =f(u) � 1

|⌦|R⌦

f(u)dx, where u(x0

) = u0

at a givenpoint x

0

2 ⌦. We prove that if f and its first deriva-tive f 0 are su�ciently small, then a unique classicalsolution u exists under periodic boundary conditions.

Nonlinear Picone’s Identity forP-Biharmonic Operator and ItsApplications

Gaurav DwivediIIT Gandhinagar, India

Classical Picone’s identity plays an important rolein proving several qualitative result in the theory ofsecond order elliptic PDE. Recently the classical Pi-cone’s identity has been extended to deal with prob-lems concerning operators such as p-Laplace opera-tor, polyharmonic operator etc. In this paper, weshall establish a nonlinear analogue of Picone’s iden-tity to deal with problems concerning p-biharmonicoperator. As an application of the Picone’s identity,we establish Hardy type inequality, Sturmian com-parison principle, Caccioppoli inequality and someother qualitative results.

Envelope Solutions to PDEs De-pending of Two Disjoint Sets ofVariables

Maria Lewtchuk EspindolaFederal University of Paraiba, Brazil

There are a lot of applications for the envelope solu-tions to PDEs, the hypersurfaces that enclose one ofthe families of the hypersurfaces given by the com-plete solutions. The development and discussion of

the existence of envelope solutions to PDEs that de-pends of two disjoint sets of variables are the mainpurpose of this research. As an example it is consid-ered the canonical variables describing a mechanicalsystem at the phase space in Hamiltonian Analyti-cal Mechanics. As one of the possible extensions itcan be discussed the development and the analysesof the existence of envelope solutions to the varia-tional PDEs that involves functional depending oftwo disjoint sets of variables. As it occurs in Hamil-tonian Analytical Mechanics applied to field theorieswhere the dependence is of the field functions and thecanonical variables represented by the density mo-menta.

Generalized Solution to the P-Laplace PDE: u2

x

uxx

+ 2ux

uy

uxy

+u2y

uyy

= 0

Maria Lewtchuk EspindolaFederal University of Paraiba, Brazil

The solutions to the p-Laplace partial di↵erentialequation u2

xuxx+2uxuyuxy+u2

yuyy = 0 are enlargedin this article. For this purpose the Monge methodfor uniforme partial di↵erential equations reduces theabove equation to a Monge system. This system re-sults in a PDE of first order of the type f(p, q) = 0.A method developed in previous papers by Espindolato the PDE f(p, q) = 0, furnishes a general solutionwhich is a generalized solution of the p-Laplace as itcontains one an arbitrary function.

Logarithmic Approximation of Non-Linear Ill-Posed Problems withNumerical Experiments

Matthew FuryPenn State Abington, USA

Various approximation methods, including the quasi-reversibility method, have been applied in order toregularize the ill-posed Cauchy problem du/dt = Au,t 2 (0, T ), u(0) = x where A is an unbounded linearoperator in a Banach space X and x 2 X. In thecase that A is positive, self-adjoint in Hilbert space,Boussetila and Rebbani apply a logarithmic approxi-mation A� = � 1

pTln(�+e�pTA), � > 0, p � 1 which

induces a mild error order as compared with that ofprevious methods. In this presentation, we will out-line the regularization of certain non-linear problemsdu/dt = A(t)u + h(t, u(t)), t 2 (0, T ), u(0) = x us-ing Boussetila and Rebbani’s perturbation in bothHilbert space and Banach space, where the opera-tors A(t) may be non-constant. In this case, the so-lution v�(t) of the corresponding well-posed problemconverges in an appropriate norm to a supposed so-lution u(t) of the original problem for each t 2 [0, T ].Finally, we demonstrate applications of the theory tothe backward heat equation and other partial di↵er-ential equations through numerical experiments in-cluding a finite-di↵erence method.


Double-Di↵usive Convection forNanofluids in MHD

Urvashi GuptaPanjab University, IndiaJyoti Sharma, Veena Sharma

The e↵ect of vertical magnetic field on a horizontallayer of electrically conducting nanofluid, heated andsoluted from below is studied analytically and nu-merically. For the analytical study, valid approxima-tions are made in the complex expression for Rayleighnumber to get useful and interesting results. Thecritical wave number increases whereas the frequencyof oscillation (for the bottom heavy configuration)decreases when the Chandrasekhar number increases.For top heavy nanofluids, oscillatory motions are notpossible and the instability through stationary con-vection increases with an increase in the nanoparti-cle concentration at the upper boundary of the fluidlayer. The destabilizing e↵ect of the higher concen-tration of nanoparticles at the top is so high thatthe magnitude of applied magnetic field must be in-creased so as to neutralize the e↵ects of nanoparti-cles or the temperature at the lower boundary mustbe decreased. The numerical results for alumina-water nanofluid are studied using MATHEMATICAsoftware. The heat transfer mode is oscillatory forbottom heavy nanofluids. The Soret parameter haslargely destabilizing e↵ect except around a small por-tion of the critical wave number whereas the e↵ect ofDufour parameter is destabilizing.

Di↵erential Equations in the Love-lock Modified Gravity Theory

Sudan HansrajUniversity of KwaZulu Natal, So AfricaNK Dadhich, B Chilambwe

The recent discovery by LIGO of gravitational wavesand the implied existence of black holes has servedas yet another verification of the success of Einstein’sgeneral theory of relativity (GR). The result was con-jectured by Einstein on studying the equations gov-erning the coalescence of a binary black-hole system.This discovery further emphasises the need for the-oretical explorations to accompany work on the ob-servational front. Despite its major advantages, GRhas failed to explain certain anomalous cosmologicalphenomena such as the late time accelerated expan-sion of the universe. Modified theories hold promisein these areas. In particular, we investigate the pureLovelock theory of gravity which has the advantageof reducing to GR in the special first order case. TheLovelock Lagrangian consists of terms quadratic inthe Riemann tensor, Ricci tensor as well as the Ricciscalar though remarkably the equations of motionturn out to be of second order only. This is in con-trast to other gravity theories such as f(R) theorywhere fourth order derivatives appear. The advan-tage of Lovelock theory is that the gains of GR areretained. Ordinary GR is well known for producing ahighly coupled system of nonlinear partial di↵erential

equations that are ten in number in the worst caseand these have been studied thoroughly over the lastcentury. The Lovelock equations are far more compli-cated involving both higher curvature terms involv-ing the spacetime dimension d as well as the orderof the Lovelock polynomial N . We have succeededfor the first time in writing the full Lovelock equa-tions for the gravitational behaviour of a sphericallysymmetric perfect fluid. Previously, only black-holesolutions were investigated. We are now in a posi-tion to construct models of the interiors of massiveobjects and to analyse their evolution in a di↵erentgravity regime. To date we have established that thefamous Schwarzschild interior solution is universal inthat the field equations are independent of d and N .Additionally, we have studied the case of an isother-mal fluid sphere (PRD Vol 93 to appear) analogousto the Einstein case where a density and pressurefall-o↵ is according to an inverse square law. A linearbarotropic equation of state is automatically built in.We indicate further cases that are being studied andtheir potential to serve as models of compact objectsin the universe. The major di�culty in this study isthe complexity of the systems of partial di↵erentialequations. To work around this, assumptions suchas the existence of symmetries, are made. These pre-scriptions on mathematical grounds then compromisethe physical behaviour that is already observed. Re-solving this tension between mathematical e�cacyand observations is a cornerstone of research in grav-ity theories.

The Inviscid Limit for the Landau-Lifshitz Equation

Chunyan HuangCentral University of Finance and Economics,Peoples Rep of ChinaZihua Guo

We prove that in dimension three and higher theLandau-Lifshitz equaiton is globally well-posed forsmall initial data in the critical Besov space uni-formly with respect to the Gilbert damping parame-ters. Then we show the global solution converges tothat of the Schroedinger maps as the Gilbert damp-ing term vanishes.

Investigating Bi-Stability of Com-bustion Waves

Zhejun HuangUNSW Canberra, AustraliaHarvinder Sidhu, Isaac Towers, Zlatko Jo-vanoski, Vladimir Gubernov

When high temperature is applied to a material atone end by an ignition source, there is a possibilitythat a flame front is formed and this front propagatesdown the material. The propagating front is knownas a combustion wave. We investigate properties ofcombustion waves for a competing two-stage exother-mic reaction scheme.Previous studies for the adiabatic case have shownthe existence of bi-stability – fast (combustion waves


with high speeds) and slow (low speeds) solutionbranches co-exist for the same parameter values. Wereport the influence of heat loss on the existence ofbi-stability. It is demonstrated that bi-stability canstill exist for the non-adiabatic case if the heat lossparameter is below a certain threshold value. Thetransition between the fast and slow branches can beobserved by perturbing the temperature profile forboth adiabatic and non-adiabatic cases. Includingheat loss into the system has a ’stabilizing’ e↵ect onthe bi-stability behavior. In other words, for the non-adiabatic case the solutions do not ‘jump‘ from onesolution branch to the other unless a strong perturba-tion is applied to the system. Our results have clearimplications for fire safety and industrial processes.

Conditional Lie-Backlund Symme-tries and Functionally General-ized Separable Solutions of (n+1)-Dimensional Evolutions Equations

Muhammad KhanInstitute of Business Management, PakistanAmjad Faizan

We propose a generalization of the notion of invari-ance of (n+1)�dimensional partial di↵erential equa-tions with respect to Lie-Backlund vector field. Suchgeneralization , known as conditional Lie-Backlundsymmetries plays a vital role in constructing newansatz that enables us to construct functionally gen-eralized separable solutions of evolution equationswhich are defined on the invariant subspaces gener-ated by the functionals. Using invariant subspaces,we reduce the PDE into a system of time depen-dent ODEs. A complete list of invariant subspacesof (2 + 1)�dimensional nonlinear di↵usion equationwith convection and source term is provided. Suchgeneralization results in a number of new solutionswhich cannot be constructed by means of the sym-metry reduction procedure.

Inertial Manifolds for SemilinearParabolic Equations Which Do NotSatisfy the Spectral Gap Condition

Anna KostiankoUniversity of Surrey, England

We report on resent results concerning the existenceof inertial manifolds for semilinear parabolic equa-tions which do not satisfy the spectral gap conditions.The list of problems includes the 3D Cahn-Hilliardproblem with periodic boundary conditions, the so-called modified Leray-↵ model which regularizes the3D Navier-Stokes equations with periodic boundaryconditions and 1D reaction-di↵usion advection prob-lems.

Asymptotic Behavior for Rayleigh’sProblem Based on Kinetic Theory

Hung-Wen KuoNational Cheng Kung University, Taiwan

We investigate the dynamics of the gas bounded byan infinite flat plate which is initially in equilibriumand set at some instant impulsively into uniform mo-tion in its own plane. We use the BGK model equa-tion to describe intermolecular collisions and assumethe di↵use reflection to describe the interaction ofthe gas with the boundary. The Mach number ofthe plate is assumed to be small so that we can lin-earize the equation as well as the boundary condition.We establish the asymptotic expansions of the micro-scopic density function and the macroscopic velocityfor small Knudsen numbers. The exact first two or-der terms of our asymptotic expansion of solutionsprovide a point of view to see the long time behaviorfor Rayleigh’s problem.

A Nekhoroshev Type Theorem forthe Beam Equation with Deriva-tive Nonlinear Perturbation on theTorus

Chunyong LiuDalian University of Technology, Peoples Rep ofChina

In this paper, we prove a Nekhoroshev type theoremfor the nonlinear beam equation with convolution po-tential under periodic boundary conditions. Moreprecisely, we prove that if the initial datum is ana-lytic in a strip of width 2⇢ > 0 whose norm on thisstrip is equal to ✏, then if ✏ is small enough, the so-lution of the nonlinear beam equation above remainsanalytic in a strip of width ⇢/2, with norm boundedon this strip by C✏ over a very long time interval of

order ✏��| ln ✏|�

, where 00 are positive constants de-pending on � and ⇢.

Approximation of Non-ClassicalShocks by Shadow Waves, En-tropies and Wave Interactions

Marko NedeljkovUniversity of Novi Sad, Yugoslavia

Our aim is to present a tool for dealing withnon-classical solutions to conservation law systemscontaining delta functions or some other irregularobjects. We call it Shadow Waves (SDWs, forsort), with the first reference [M. Nedeljkov, ShadowWaves: Entropies and Interactions for Delta andSingular Shocks, ARMA, 2010]. They are inspiredby the well known Wave Front Tracking algorithm.SDWs are possibly unbounded nets of piecewise con-stant functions for fixed time depending on a smallparameter "..They have distributional limits as " !0 and the typical examples are delta and singularshocks. The main advantage of SDWs is (a) possibil-


ity to use the Lax entropy pairs as an admissibilitycriteria and (b) easy dealing with wave interactions.We will give basic properties of SDWs (1-D case first)with several examples of their usefulness concerningthe above points (a) and (b), and some of their ap-plications in n-D conservation law systems.

Wave Propagation in HeterogeneousGeological Media with Cracks

Viktoria SavatorovaUNLV, USAAlexey Talonov

Wave propagation through heterogeneous materialsis of the utmost importance in various fields includingstructural dynamics, geophysical studies, earthquakeengineering, oil exploration. Focusing on geologicalmedia, it is important to take into account such pe-culiarities of structure as soil layering, material het-erogeneity, and cracks. The mathematical descrip-tion of wave motion in heterogeneous media involvesthe solution of partial di↵erential equations (PDEs)with variable coe�cients. The aim of this talk is topresent our results for multiscale modeling of wavepropagation in geological materials with cracks. Weconsider multiscale model of heterogeneous mediumconsisting of randomly distributed micro-cracks (mi-cro level) and periodic (layered) structure of hetero-geneities at a higher scale (meso level). We justifyusing of averaging technique to perform the upscalingfrom micro scale (cracks) to mesoscale (layers), andthen from mesoscale to macro (wave length) scale.As a result of upscaling we derive homogenized waveequation with e↵ective coe�cients, and investigatethe influence of heterogeneities on micro and mesolevels on the speed of propagation of waves, and theirattenuation. E↵ective characteristics of the mediumare determined from the solution of the problem de-fined on the periodic cell with taking into accountthe distribution of cracks.

Exact Solutions of a Non-LinearPDE Arising in Biological Processes

Muhammad ShabeerQatar University, QatarMuhammad Tahir Mustafa

The existence of travelling wave solutions of a classof non-linear PDEs, arising in biological processes,was shown in [J. D. Murray, Mathematical BiologyI: An Introduction, Springer-Verlag, NewYork, 2002].Applying the modified tanh method, we provide ex-amples of travelling wave solutions of these non-linearPDEs.

A Numerical Study of Free Con-vection Heat and Mass Transfer ina Rivlin-Ericksenian ViscoelasticFlow Past an Impulsively StartedVertical Plate with Variable Tem-perature and Concentration

Veena SharmaHimachal Pradesh University, IndiaRajneesh Kumar, Ibrahim A. Abbas, RadheShyam

In this paper, an analysis of the e↵ect of vis-coelasticity on the natural free convective unsteadylaminar heat transfer fluid flowing past an impul-sively started vertical plate with variable surfacetemperature and mass concentration is considered.The Rivlin-Ericksen model is employed to simulatethe rheological liquids. The transformed two-pointboundary value problem is solved numerically usingthe Galerkin method for solving the partial di↵er-ential equations relevant to the physical problem forthe weak formulation. Numerical results are obtainedto study the influence of viscoelasticity parameter,surface temperature power law exponent and surfaceconcentration power law on the velocity, tempera-ture, concentration fields ,the local skin-friction, theNusselt number and the Sherwood number; which aregraphically presented and discussed. It is found thatincrease in viscoelasticity parameter accelerates thevelocity profiles, but has a little influence on the tem-perature and concentration fields; enhances the shearstress for X > 0.28 and reduces local Nusselt num-ber and local Sherwood number for X > 0.51 in theboundary layer. It is also depicted that increase inpower law exponents decelerate the velocity profiles,reduces temperature and the concentration fields ofthe di↵using species; but increases the surface shearstress for X > 0.6.

Reaction-Di↵usion Systems with(p(x, t), q(x, t))-Growth and L1 Datafor Image Processing

Kehan ShiHarbin Institute of Technology, Peoples Rep ofChinaDazhi Zhang, Zhichang Guo, Jiebao Sun,Boying Wu

In this paper we investigate a class of reaction-di↵usion systems coupled with p(x, t)-Laplacianequation and q(x, t)-Laplacian equation that arisefrom the study of image denoising problem and im-age decomposition problem. We extend the notion ofentropy solution for reaction-di↵usion systems withtime-dependent variable exponents. The existenceand uniqueness of entropy solutions for this problemwith L1 initial data is established.


Higher Order Boundedness of theHarmonic Projection Operator onSolutions to the Dirac-HarmonicSystem

Guannan ShiHarbin Institute of Technology, Peoples Rep ofChinaShusen Ding, Donna Sylvester, Yuming Xing

In this paper, higher order estimates for the harmonicprojection operator applied to solutions of the Dirac-harmonic equations are established and the higherintegrability of the projection operator is proven. Asapplications, new highter Sobolev imbedding inequal-ities of the projection operator and di↵erential formsare constructed.

Potential Method in the Theoryof Double Porosity ThermoelasticMaterials

Merab SvanadzeIlia State University, Rep of Georgia

The theory of thermoelasticity of double porosity ma-terials, as originally developed for the mechanics ofnaturally fractured reservoirs, has found applicationsin many branches of civil engineering, geotechnicalengineering, technology and biomechanics.We shall consider the coupled linear theory of ther-moelasticity for materials with double porosity struc-ture. The boundary value problems (BVPs) of thesteady vibrations are investigated. The fundamentalsolution of system of equations of steady vibrations isconstructed. The Green’s formulas and the represen-tations of general solution of equations of steady vi-brations are obtained. The basic properties of planewaves and the radiation conditions for regular vec-tor are established. The uniqueness theorems of theinternal and external BVPs of steady vibrations areproved. The basic properties of surface (single-layerand double-layer) and volume potentials are estab-lished. The existence of regular solution of the BVPsby means of the potential method (boundary integralequation method) and the theory of singular integralequations are proved.

A Range of Reynolds Number forStationary Solution of the ViscousIncompressible Fluid Flow Down anInclined Plane

Kyoko TomoedaInstitute for Fundamental Sciences, Setsunan Uni-versity, JapanYoshiaki Teramoto

We consider the two dimensional motion of a vis-cous incompressible fluid flowing down an inclinedplane with an angle of inclination ↵, under the ef-fect of gravity. The fluid motion is governed bythe Navier-Stokes equations with the free boundary

conditions. This problem contains two dimension-less quantities: Reynolds number andWeber number.When the Reynolds number and the angle ↵ is suf-ficiently small, Nishida-Teramoto-Win (1993) provedthe global existence of periodic solutions with an ex-ponential decay rate for su�ciently small initial data.To obtain a specific range of this “su�ciently smallReynolds number”, we examine the spectra of thecompact operator arising the linearized problem. Inthis talk we discuss about a Range of the Reynoldsnumber and Weber number, when the linear operatorhas a non-zero spectral value.

Existence of Viscosity Solutions ofSingularly Perturbed Problem

Ram Baran VermaIIT Gandhinagar, IndiaJagmohan Tyagi

In this presentation, we will present the existence oftwo viscosity solutions of the following singularly per-turbed problem

(� ✏2M+

�,⇤(D2u) = f(x, u) in ⌦

u = 0 on @⌦,(1)

where f is a non-Lipschitz function and ⌦ is a smoothbounded domain in Rn, n > 2. Here we will use infregularization process to make the nonlinearity lo-cally lipschitz continuous. That will allow us to ap-ply the result that are available for locally Lipschitznonlinearity.

A Stationary Core-Shell Assemblyin a Ternary Inhibitory System

Chong WangGeorge Washington University, USAXiaofeng Ren

A ternary inhibitory system motivated by the tri-block copolymer theory is studied as a nonlocal ge-ometric variational problem. The free energy of thesystem is the sum of two terms: the total size of theinterfaces separating the three constituents, and alonger ranging interaction energy that inhibits micro-domains from unlimited growth. In a particular pa-rameter range there is an assembly of many core-shells that exists as a stationary set of the free energyfunctional. The cores form regions occupied by thefirst constituent of the ternary system, the shells formregions occupied by the second constituent, and thebackground is taken by the third constituent. Theconstructive proof of the existence theorem revealsmuch information about the core-shell stationary as-sembly: asymptotically one can determine the sizesand locations of all the core-shells in the assembly.The proof also implies a kind of stability for the sta-tionary assembly.


Mathematical Analysis of a Class ofIntegro-Di↵erential Equations

Weiqing XieCal Poly Pomona, USA

A nonlinear integro-di↵erential equations related tothermoelasticity will be studied. Mathematical anal-ysis to this special nonlinear di↵erential equationswill be discussed and analyzed.

Permanence of Di↵usive Modelsfor Three Competing Species inHeterogeneous Environments

Benlong XuShanghai Normal University, Peoples Rep of ChinaZhenzhang Ni

In this talk, we address the question of the long termcoexistence of three competing species whose dynam-ics are governed by the partial di↵erential equations.We obtain criteria for permanent coexistence in aLotka-Volterra system modeling the interaction ofthree competing species in a bounded habitat whoseexterior is lethal to each species. It is also proved thatif the inter-competing strength is very weak, the sys-tem is always permanent, provided that each singleone of the three species can survive in the absence ofthe two other species.

The Existence and The Non-Existence of Global Solutions ofa Free Boundary Problem

Rong YinNantong University, Peoples Rep of ChinaWanghui Yu

We study a free boundary problem of parabolic equa-tions with a positive parameter ⌧ included in the co-e�cient of the derivative with respect to the timevariable t. This problem arises from some reaction-di↵usion system. We prove that, if ⌧ is large enough,the global solution exists while, if ⌧ is small enough,the solution exists only in finite time.

A Bessel Type Pantograph EquationArising in a Cell Growth Model

Ali Ashher ZaidiLahore University of Management Sciences (LUMS),PakistanBruce van Brunt, Graeme Wake

A simple model for cell growth and division into↵ > 1 daughter cells is given by the functional pde

� @2

@x2

(D(x)n(x, t)) +@@x

(g(x)n(x, t))

+@@t

n(x, t) + bn(x, t) = b↵2n(↵x, t).

Here, n denotes the number density of cells of size xat time t , D is dispersion, g is the growth rate, andb is the division rate. (“Size“ is usually measured bymass or DNA content.) The di↵erential equation issupplemented by the condition

n(x, 0) = n0

(x),

where n0

is the initial cell size distribution, and theboundary condition

� @@x

(D(x)n(x, t)) + g(x)n(x, t) = 0,

at x = 0. The problem is thus of the initial-boundaryvalue type. There is a paucity of analytical solutiontechniques for these problems; however, it is possibleto solve the problem for some simple cases of inter-est. Although the leading order long time asymptoticbehaviour of solutions to these problems is knownfor fairly general cases, the higher order terms arerelatively unexplored. The exact solutions yield thehigher order long time asymptotic behaviour of so-lutions for the special cases and may provide someinsight into more general cases.

A Doubly Degenerate Di↵usionModel for Multiplicative Noise Re-moval

Zhenyu ZhouHarbin Institute of Technology, Peoples Rep ofChinaZhichang Guo, Jiebao Sun, Dazhi Zhang,Boying Wu

Multiplicative noise removal is a challenging task inimage processing. In this talk, the problem is ad-dressed by using a doubly degenerate di↵usion model,which is analyzed with respect to the existence of theweak solution. Experimental results illustrate e↵ec-tiveness and e�ciency of the proposed model.


Contributed Session 3: Modeling, Math Biology and Math Finance

Global Stabilizing Feedback Law fora Problem of Biological Control ofMosquito-Borne Diseases

Maria Soledad AronnaSchool of Applied Mathematics, FGV/Rio, BrazilPierre-Alexandre Bliman, Flavio C. Coelho,Moacyr A.H.B. da Silva

The use of bacteria Wolbachia is a promising methodpresently considered to block the transmission ofdengue, zika and chikungunya viruses. Systematicprocedures for introduction of mosquitos infected bythe bacteria in a healthy population are still to bestudied. This is a central question, with heavy im-pact on the cost and e�ciency.The aim of the present study is the synthesis of amethod allowing the reduction of the number of intro-duced mosquitos, and consequently the cost, withoutputting at risk the success of the infestation (some-thing that could happen e.g. if the initial size ofthe population has been underestimated). Using thefact that measurements are completed during thewhole release process, techniques from the theoryof control of dynamical systems are used to definethe quantity to be introduced. The original systemis shown to have two stable equilibria, correspond-ing to Wolbachia-free and complete infestation situa-tions. A simple feedback law is proposed and shownto have the capacity to asymptotically settle the bac-teria. Up to our knowledge, this is the first attemptto use feedback for introduction of Wolbachia withina population of arthropods.The techniques are based on the theory of monotonesystems, recently extended by D. Angeli, E. Sontagand E. Enciso to analyze the asymptotic behavior ofinput-output monotone systems closed by negativefeedback. Due to bistability, the considered input-output system has multivalued static characteristics,but the existing results are unable to prove almost-global stabilization, so ad hoc analysis has to be used.

Stock Pricing with New DynamicFractal Analysis Approach

Alireza BahiraieSemnan University and University of Malaya, Iran

Fractal analyzing of continuous processes have re-cently emerged in literatures in various domains. Ex-istence of long memory in many processes includingfinancial time series has been evidenced via di↵er-ent methodologies in many literatures in the pastdecade. This has inspired many recent literatureson quantifying the fractional Brownian motion (fBm)characteristics of financial time series. This presen-tation questions the accuracy of commonly appliedfractal analyzing methods on explaining persistentor anti-persistent behavior of time series understudy.Rescaled range (R/S) and power spectrum techniques

produce fractal dimensions for daily returns of twelvehundred S&P100 stocks from the most well per-formed firms are estimated. Zipf’s law generates lin-ear and logarithmic power-law distribution plots toevaluate the validity of estimated fractal dimensionson prescribing persistent and anti-persistent charac-teristics with less ambiguity. Findings of this studyrecommend a more thoughtful approach on classify-ing persistent and anti-persistent behaviors of finan-cial time series by utilizing existing fractal analyzingmethods.

Roots of Some Trinomial Equa-tions with Applications to FinancialProblems

Vanessa BottaUNESP, Brazil

The main purpose of this paper is to determine thebehavior of the roots of a kind of trinomial equa-tion that appears in certain financial mathematicsproblems. In addition, we present the regions of thecomplex plane where these roots are located.

Spot Dynamics in a Plant HairInitiation Model

Victor F. Brena-MedinaCentro de Ciencias Matematicas, UNAM CampusMorelia, MexicoMichael J. Ward

Patch location dynamics of an initiation process ina plant root hair cell at a sub-cellular is thoroughlyanalysed. An earlier model proposed by Payne andGrierson captures key features of an interacting smallG-protein family so-called Rho of Plants (ROPs).These proteins are in charge of promoting certainprotuberances on root hair cells, which are crucialfor nutrients uptake from the soil and anchorage,for instance. Auxins are a class of hormones thatare known to take part on the morphogenesis ofplants. As experimental observations show that afast auxin flow is heterogeneously distributed alongthe cell at the ROPs di↵usive scale, auxin catalysisis modelled as a spatially dependent coe�cient con-trolling dominant cubic terms. Such a model con-sists of a generalised two-component Schnakenbergreaction-di↵usion system, which is set up in a non-homogeneous domain. Upon considering a more re-alistic cell geometry, a two-dimensional root hair cellgathers the essential ingredients that allows to rigor-ously analyse whether shape and form are relevantfor patch location dynamics. Numerical bifurcationanalysis, as well as time numerical simulations, andthe theory of semi-strong interactions are performedin order to shed light on the understanding of dy-namical root hair morphogenesis.


Estimation of Natural Mortalityfor Isochronal Fish : Applicationto Sandfish in the Eastern CoastalWaters of Korea

Giphil ChoPusan National Univerisy, KoreaIl Hyo Jung

Studies of Estimating natural mortality play an im-portant role in analyzing fisheries stock assessmentand management, but current methods of estima-tion do not allow size-dependent instantaneous rateof natural mortality rates based on e↵ective fecun-dity, growth and catch simultaneously. We proposestage-structure model that represent size-dependentnatural mortality rates for isochronal fish becauselarger fish have fewer predators than smaller ones andnatural mortality of larvae and juveniles is knownto decrease with increasing body size. Here werepresent an example of application to the sand-fish (Arctoscopus japonicus) in eastern coastal wa-ters of Korea. E↵ective fecundity was estimated byconsidering von bertalan↵y growth equation, logis-tic equation of maturity and an exponential equa-tion of fecundity with total size, which were derivedfrom otolith and gonad analyses of sandfish collectedfrom eastern coastal waters of Korea from 2005 to2008. We assume that CPUE (Catch Per Unit E↵ort)for each year is direct proportion to average biomassfor each year. As assuming various initial value, wecan estimate optimal size-dependent natural mortal-ity corresponding to the largest coe�cient of deter-mination (R2) to account for interaction formula ofassumption. We compared the results of previousstudies with our result.

About New Mathematical Ap-proaches for Inferring and Fore-casting the Dynamics of Regula-tory Network Models-A NumericalStudy

Ozlem DefterliCankaya University, TurkeyArmin Fugenschuh,Gerhard-Wilhelm Weber

The analysis and anticipation of real-world complexphenomena, appearing frequently in the areas of lifeand environmental sciences, engineering sciences andfinance, based on experimental data and environmen-tal measurements is a challenging research problemof mathematical modeling.In this study, we focus on such complex systems thatappears in system biology, namely genetic regula-tory networks. Based on the given experimental dataand environmental measurements, the interactions ofeach gene with the others in a metabolic and geneticstructure have to be identified clearly and the in-fluences need to be predicted. Furthermore, antici-pating the future behaviour of such complex systemsrequires the study of their discrete dynamics basedon various numerical schemes.

Here, our model is a gene-environment networkwhose dynamics are represented by a class of time-continuous systems of ordinary di↵erential equa-tions based on the information provided by gene-expression levels and certain levels of environmen-tal factors. The model contains unknown parame-ters to be optimized by considering the data at thesample times. Accordingly, time-discrete versions ofthat model class is studied and improved by higher-order explicit numerical methods. These numericalschemes are used to generate corresponding gene-expressions values for further time levels. We per-form illustrative examples with simulations both foran artificial data set and also for a real-world dataset. Furthermore, new extensions of this study willbe mentioned briefly by considering new mathemat-ical tools for the concept of uncertainty in the data.

Modeling TCP, Recurrence, andSecond Cancer Risks Induced byProton Radiation

Andrew DhawanQueen’s University, CanadaV. S. K. Manem

In the past few years, proton therapy has taken cen-ter stage to treat various tumor types, with increas-ing success due to volume-conforming dosing. How-ever, the price of success may be the appearance ofa secondary malignancy post-irradiation. The pri-mary contribution of this study is to investigate thetumor control probability (TCP), relapse time, andthe corresponding secondary cancer risks induced byproton beam radiation therapy. We incorporate tu-mor relapse kinetics into the tumor control proba-bility framework, and calculate the associated sec-ond cancer risks, using the well-known initiation-inactivation-proliferation (IIP) formalism. We usethe available in-vitro data for the linear energy trans-fer (LET) dependence of cell killing and mutation in-duction parameters. TCP, relapse and radiation in-duced second cancer risks are evaluated for protons inthe clinical range of LETs. We show that comparedto photon therapy, proton therapy markedly reducesthe risk of secondary malignancies, and for equiva-lent dosing regimens achieves better tumor controlas well as a reduced primary recurrence outcome, es-pecially within a hypo-fractionation regimen. Thisstudy may serve as a framework for further work inthis field, and, elucidates proton induced TCP andthe associated secondary cancer risks, not previouslyreported in the literature.


Derivation of Windage Jump toCompensate Barrel Angular Rateson Flightpath of a Projectile

Fatih GeridonmezTubitak Sage, Turkey

This study describes the algorithm used in order tocompensate the e↵ect of muzzle angular rates on tra-jectory of a projectile. Starting with aerodynamicjump, the derivation of the windage jump is ex-plained. The Windage Jump model is used in bal-listic model of a projectile to compensate the e↵ectof the crosswind on its flightpath. In ballistic the-ory, the wind-shear e↵ect needs to be computed andadded to the velocity vector of the projectile. Thestarting point in the derivation is the generalizedaerodynamic jump e↵ect on the spin stabilized pro-jectiles due to cross wind.

Introduction of an InterventionTerm to the Universal Law for Tu-mor Growth

Ja↵ar Ali Shahul HameedFlorida Gulf Coast University, USASamantha Burns, Menaka Navaratna

Recent work in computational biology has led to thedevelopment of a universal mathematical model oftumor growth based on metabolism. Generally, thegrowth of tumor in terms of mass is represented asdmdt

= f(m, t). In this paper, we introduce an in-tervention term, c, to this nonlinear computationalmodel in order to analyze tumor growth in adversecellular conditions, i.e. dm

dt= f(m, t) � c. We as-

sess the intervention term, c, as a constant and as afunction of time. Our simulations suggest that tu-mor treatment does not, in general, cause a mono-tonic decrease in mass and this finding is reflectedin experimental data from the cancer literature. Ourfindings have potential applications for the develop-ment of cancer treatment by demonstrating that theduration and quantity of treatment can be optimizedto suppress tumor growth, prevent metastasis, andprepare for tumor excision.

Population Dynamics with Immi-gration

Dan HanUniversity of North Carolina at Charlotte, USAStanislav A. Molchanov, Joseph M. Whit-meyer

We present population dynamic models by construct-ing Galton-Watson processes with immigration onthe lattice Zd. We use not the forward but the Kol-mogorov backward equation, which is simpler and atthe same time gives more detailed information on thelimiting process. Asympototic analysis is given forthe number of population as time tends to infinity.And we use these to prove a local central limit theo-

rem result. The goal of the present paper is to provethe existence of a stationary limiting distribution fora wide class of branching random walks or processesin Zd. Joint work with Dr. Stanislav A. Molchanovand Dr. Joseph M. Whitmeyer.

Analytical Issues in Pulsatile BloodFlow: Arterial Sti↵ness and PulseWave Velocity

Simona HodisTexas A&M University-Kingsville, USAMair Zamir

Vascular compliance is a major determinant of wavepropagation within the vascular system, hence themeasurement of pulse wave velocity (PWV) is com-monly used clinically as a method of detecting vas-cular sti↵ening. The accuracy of that assessment isimportant because vascular sti↵ening is a major riskfactor for hypertension. PWV is usually measured bytiming a pressure wave as it travels from the carotidartery to the femoral or radial artery and estimatingthe distance that it travelled in each case to obtainthe required velocity. A major assumption on whichthis technique is based is that the vessel wall thick-ness h is negligibly small compared with the vesselradius a. The extent to which this assumption issatisfied in the cardiovascular system is not knownbecause the ratio h/a varies widely across di↵erentregions of the vascular tree and under di↵erent patho-logical conditions.In this study we show that an expansion for smallh/a of the classical solution of this problem does notin fact lead to valid results for values of h/a that arenot infinitely small. An alternative solution for largevalues of h/a is presented, together with a method ofpatching the two solutions.

Understanding Pollution withWiener-Hopf Lattice Factoriza-tions

Hermen Jan HupkesLeiden University, NetherlandsE. Augeraud-Veron

We study optimal control problems with time delaysposed on lattices, which can be used to weigh thecosts and benefits of utilizing polluting agents to en-hance crop yields. The conditions defining optimalstrategies turn out to be Hilbert-space valued func-tional di↵erential equations of mixed type (MFDEs).We develop tools such as exponential dichotomiesand Wiener-Hopf factorizations for such systems todetermine whether optimal strategies can retain theiroptimality under small variations in their initial con-ditions. Complications are caused by the fact thatthe modelling state space is only half of the naturalmathematical state space.


Stoneley Waves at an Interface Be-tween Thermoelasticdi↵usion SolidHalf Spaces

Rajneesh KumarKurukshatra University, Kurukshatra, India

This paper is concerned with the study of propaga-tion of Stoneley waves at the interface of two dis-similar isotropic thermoelastic di↵usion medium inthe context of generalized theories of thermoelasticity(Lord-Shulman,1967 and Green-Lindsay,1972). Thefrequency equation of Stoneley waves is derived in theform of a determinant by using the boundary condi-tions. The dispersion curves giving the phase veloc-ity and attenuation coe�cients with wave numberare computed numerically. Numerically computedresults are shown graphically to depict the di↵u-sion e↵ect alongwith the relaxation times in thermoe-lastic di↵usion solid half spaces for thermally insu-lated and impermeable boundaries, respectively. Thecomponents of displacement, stress, and temperaturechange are presented graphically for two dissimilarthermoelastic di↵usion half-spaces. Several cases ofinterest under di↵erent conditions are also deduced.

Modeling and Simulation of Carrier-Based Aircraft Approach and Land-ing Based on Touchdown Precision

Jiaming LinBeihang University, Peoples Rep of ChinaTing Yue, Lixin Wang

To establish the relationship between control systemdesign and landing precision, a dynamics model inte-grated with 6-DOF nonlinear model of F/A-18, Auto-matic Carrier Landing System (ACLS), ship airwakeinfluence and typical carrier deck motion is presented.The ACLS parameters are firstly designed for thepowered approach flight condition according to fre-quency domain requirements. Touchdown errors andits dispersions then can be obtained through closedloop simulations with random initial settings of car-rier motion and turbulence characteristics, which willbe a significant evaluation index of carrier-based air-craft landing performance. The simulation result in-dicates the validity of the established model, anddemonstrates the di↵erent e↵ects of turbulence, gustand deck motion on touchdown accuracy. This ana-lytical method also proves its availability and conve-nience in ACLS design and landing safety research.

Modeling the Trade-O↵ BetweenTransmissibility and Contact inInfectious Disease Dynamics

Chiu-Ju LinThe Ohio State University, USAKristen A. Deger, Joseph H. Tien

Symptom severity a↵ects disease transmission bothby impacting contact rates, as well as by influencingthe probability of transmission given contact. Thisinvolves a trade-o↵ between these two factors, as in-creased symptom severity will tend to decrease con-tact rates, but increase the probability of transmis-sion given contact (as pathogen shedding rates in-crease with symptom severity). This paper exploresthis trade-o↵ between contact and transmission givencontact, using a simple compartmental susceptible-infected-recovered type model. Under mild assump-tions on how contact and transmission probabilityvary with symptom severity, we give su�cient, bio-logically intuitive criteria for when the basic repro-duction number varies non-monotonically with symp-tom severity. Multiple critical points are possible.We give a complete characterization of the region inparameter space where multiple critical points arelocated in the special case where contact decreasesexponentially with symptom severity. We consider amulti-strain version of the model with complete cross-immunity and no super-infection. We prove that thestrain with highest basic reproduction number drivesthe other strains to extinction. This has both evo-lutionary and epidemiological implications, includingthe possibility of an intervention paradoxically result-ing in increased disease prevalence.

Adsorption of Flexible PolymerChains on a Surface

Paulo MartinsFederal University of Mato Grosso, BrazilJoao Plascak, Michael Bachmann

Polymer chains undergo a continuous adsorp-tion/desorption transition onto a flat surface in sucha way that for T < Tc the system is in the adsorbedphase and for T > Tc the polymer is desorbed, Tc be-ing the critical temperature. In three dimensions, theprecise values of the corresponding critical exponentsremain still an open question. It has been noted thattheir values are not only size dependent, but alsodepend on the precise location of Tc. As a result,the known estimates of the critical exponent covera broad range of values. Additionally, most studiesonly consider good solvent conditions, in which themonomer-monomer interaction is negligible. In thepresent work we estimate critical quantities for dif-ferent solvent conditions showing that, in analogy tomagnetic phase transitions, finite-size-scaling meth-ods furnish quite good results by taking into accountcorrections to scaling.


Delay Di↵erential Equations withDynamical Systems: Numeric andAnalytic

Fathalla RihanUnited Arab Emirates University, United ArabEmirates

In this paper, we show the consistency of delay dif-ferential equations with biological systems with mem-ory, in which we present a class of mathematical mod-els with time-lags in immunology, physiology, epi-demiology and cell growth. We also incorporate opti-mal control parameters into a delay model to describethe interactions of the tumour cells and immune re-sponse cells with external therapy. We then studyparameter estimations and sensitivity analysis withdelay di↵erential equations. Sensitivity analysis is animportant tool for understanding a particular model,which is considered as an issue of stability with re-spect to structural perturbations in the model. Weintroduce a variational method to evaluate sensitiv-ity of the state variables to small perturbations inthe initial conditions and parameters appear in themodel. The presented numerical simulations showthe consistency of delay di↵erential equations withbiological systems with memory. The displayed re-sults may bridge the gap between the mathematicsresearch and its applications in biology and medicine.

Sympathetic Inhibition: Under-standing a Dynamic Paradox WhenInhibitors Promote Future Expres-sions

Jonathan RowellUniversity of North Carolina at Greensboro, USA

An inhibitor is traditionally conceived as a substancethat has a negative a↵ect on the rate of changein the concentration of a particular state variableand/or that variable‘s equilibrium level, and thereare many mechanisms by which suppression can oc-cur. With some types of inhibitors, however, thereare secondary e↵ects which may actually boost ex-pression to levels not normally achieved in systemsin which the inhibitor is entirely absent. By convert-ing expressible factors from a free to stored state, thissympathetic inhibition creates a potential reserve forfuture expression. When the inhibitor is removedfrom the system, this stored concentration is free torevert back to its regular state, leading to a largebloom of expression/variable concentration.

Feedback Linearization and Its Dy-namics in a Prey Predator Model

Anuraj SinghGraphic Era University, India

The stabilization problem through feedback lin-earization of a prey-predator model with is discussed.Feedback linearization is a technique through whichcomplexity can be controlled and system may behavein a desired fashion. By approximate linearizationapproach, a feedback control law is obtained whichstabilizes the closed loop system. On the other hand,by suitable change of coordinates in the state space, afeedback control law is obtained. This feedback con-trol renders the complex nonlinear system to be linearcontrollable system such that the positive equilibriumpoint of the closed-loop system is globally asymptoti-cally stable. Numerical experiments substantiate theanalytical findings.

Notes of Mathematical Model inQuorum Sensing

Bismark Owusu TawiahUniversity of L’Aquila, Italy and University of Silesiain KatowiceSarangam Majumdar

In this present work, we discuss a mathemati-cal model of quorum sensing mechanism of bacte-ria.Quorum sensing is density dependent behaviourwhich is widespread in bacteria like Vibrio Fis-cheri.This biochemical process is regulated by thequorum sensing molecules.To understand this bio-logical phenomenon, we use a nonlinear dynamicalsystem and study the stability analysis based on ex-periment.

Global Attractivity of PositiveAlmost Periodic Solution for a Den-sity Dependent Predator-Prey Sys-tem with Mutual Interference andCrowley-Martin Response Function

Jai TripathiCentral University of Rajasthan, India

A nonautonomous density dependent predator-preymodel with mutual interference and Crowley-Martinresponse function is proposed and studied. The dy-namics of the system is analyzed mainly from thepoint of view of permanence, extinction, stability, ex-istence and uniqueness of a positive almost periodicsolution. It is also shown that the the obtained per-manence conditions are only su�cient but not neces-sary. The su�cient conditions are derived for glob-ally attractive unique positive solution by construct-ing suitable Lyapunov functional. It is shown thatsu�cient conditions obtained for globally attractiveunique positive solution depend on both the predatordensity dependent death rate and Crowley-Martincoe�cient. The obtained analytical results are illus-trated with the help of numerical examples.


Mathematical Modeling, Analysisand Monte Carlo Simulation ofEbola Epidemics

Thomas TuluHarbin Institute of Technology, Peoples Rep ofChinaBoping Tian

Ebola virus disease is a fatal disease which becomethe headache of the whole world by now. As its fatal-ity rate is the highest, an urgent solution(measure)has to be taken to control the disease. In this articlewe built a new mathematical model to study the dy-namics of Ebola. Besides, as the spread of Ebolavirus is random process simulation is done usingMonte Carlo method(algorithm). Key words: Math-ematical modeling, Monte Carlo, Epidemic model,Ebola

Cellular Blebs: Modelling CellularMovement Through Dynamic Mem-brane Protrusions

Thomas WoolleyUniversity of Oxford, EnglandRuth E. Baker, Eamonn A. Ga↵ney, JamesM. Oliver, Sarah L. Waters, Alain Goriely

Human muscle undergoes an age-related loss in massand function. Preservation of muscle mass depends,in part, on stem cells, which navigate along muscle fi-bres in order to repair damage. Critically, these stemcells have been observed to undergo a new type ofmotion that uses cell protrusions known as “blebs“,which protrude from the cell and permit it to squeezein between surrounding material.In order to understand this dynamic motion we havecreated a model which combines a stochastic protru-sion model with a mechanical membrane model. Theresulting multiscale framework allows us to link easilygenerated experimental data on trajectories to me-

chanical properties of the membrane, which are muchmore di�cult to explore experimentally. Through in-vestigating multiple extensions of this model we findnumerous results concerning size, shape and limitingfactors of blebs based movement.

Multibody Dynamic Characteristicsof Oblique-Wing Aircraft in theWing Morphing Process

Zijian XuBeihang University, Peoples Rep of ChinaTing Yue, Lixin Wang

The multibody dynamic modeling and simulation ofan oblique-wing aircraft in the wing morphing pro-cess is studied. When the wing is oblique and nolonger symmetric, wing area, moments of inertia,aerodynamics and center of gravity alter consider-ably, and these will change the dynamic character-istics of this aircraft. Six degree of freedom nonlin-ear dynamic equations of oblique-wing aircraft in thewing morphing process are derived. On conditionthat the angular velocity of oblique wing is small andthe unsteady aerodynamic e↵ect can be ignored, theresponses in di↵erent angular velocities are numeri-cally simulated by quasi-steady aerodynamic suppo-sition. The simulation results show that the dynamicresponse exhibits large variations in wing morphingprocess. Meanwhile, the aircraft dives quickly andcannot achieve a new state of level flight withoutcontrol. Furthermore the rules of how dynamic char-acteristics are a↵ected by the shift of center gravityposition and variation of aerodynamics are investi-gated. It can be concluded that the key factor whichinfluences the dynamic response in the wing morph-ing process is the variation of aerodynamics resultingfrom asymmetric wing.


Contributed Session 4: Control and Optimization

Pontryagin Maximum Principle forOptimal Nonpermanent ControlProblems on Time Scales

Loic BourdinLimoges University, FranceEmmanuel Trelat

Pontryagin maximum principle (in short, PMP) is afundamental result of the optimal control theory. Inits historical statement, the control of the dynamicalsystem is considered as permanent, that is, the con-trol is authorized to be modified at any real time. Inmany problems, it follows that achieving the optimaltrajectory requires a permanent modification of thecontrol. However, such a request is not conceivablein practice for human beings, even for mechanicalor numerical devices. Therefore, piecewise constantcontrols (also called sampled-data controls or digitalcontrols), for which only a finite number of mod-ifications is authorized, are usually considered inAutomatic and Engineering. Sampled-data controlsare one example of nonpermanent controls. Anotherexample concerns dynamical systems whose trajec-tories go across noncontrolled areas (like a mobilephone or a GPS device going under a tunnel).

In this talk, we will present a new version of the PMPthat can be applied to optimal nonpermanent controlproblems. This result was recently obtained in [1]and is stated with the help of the time scale calculustheory. Numerous properties about optimal perma-nent controls are well-known in the literature (such asthe continuity of the Hamiltonian, or the saturationof the control constraints set when the Hamiltonianis a�ne, etc.). In this talk, we will be interested inthe preservation (or not) of these classical propertieswhen we consider nonpermanent controls. Finally,in the linear-quadratic case, we will state that theoptimal sampled-data controls converge to the opti-mal permanent control when the distances betweenconsecutive sampling times uniformly tend to zero.

References

[1] L. Bourdin and E. Trelat. Optimal sampled-data control, and generalizations on time scales.Mathematical Control and Related Fields,6(1):53-94, 2016.

Multiplicity and Stable Manifoldsfor Time-Delay Systems: FurtherRemarks on the Rightmost Root

Islam BoussaadaParis Saclay University and IPSA, FranceIslam Boussaada, Silviu Iulian Niculescu,Hakki Unal

Multiple spectral values in dynamical systems are of-ten at the origin of complex behaviors as well as un-stable solutions. However, the starting point of thistalk is an example of delay system where the maximalmultiplicity of an appropriate delay-dependant neg-ative spectral value leads to a negative spectral ab-scissa and, as a consequence, the asymptotic stabilityof the corresponding steady state solution holds. Inalgebraic terms, the manifold corresponding to such amultiple root defines a stable manifold for the steadystate. It will be shown that such a multiple spectralvalue is nothing but the rightmost root. Motivated bythe potential implication of such a property in controlsystems applications, this study is devoted to betterexplore the connexion between those manifolds.

Su�cient Conditions for L1-Minimization in Celestial Mechanics

Zheng ChenUniversity Paris-Sud, FranceJean-Baptiste Caillau, Yacine Chitour

In this talk, we consider the L1-minimization prob-lem for celestial mechanics. First, some basic prop-erties of the extremal flow including the existence ofbang, singular, and chattering solutions will be pre-sented as consequences of Pontryagin maximum prin-


ciple. Then, we will deal with the su�cient optimal-ity conditions for bang-bang extremals with regularswitching points. The ability to establish and ver-ify the su�cient conditions is a fundamental issue inoptimal control theory. The crucial idea for establish-ing such conditions is to construct a parameterizedfamily of extremals such that a bang-bang extremalwith regular switching points can be embedded into afield of broken extremals. Two no-fold conditions forthe canonical projection of the parameterized familyof extremals are devised. For the scenario that theendpoints are fixed, these no-fold conditions are suf-ficient to guarantee that the bang-bang extremal islocally minimizing. If the final point is not fixed butlies on a smooth submanifold, another su�cient con-dition involving the geometry of the target subman-ifold is established. Finally, two numerical examplesin celestial mechanics will be shown to verify thesesu�cient conditions.

P-th Moment and Almost SureStability of Neutral StochasticSwitched Nonlinear Systems

Caixia GaoUniversity of Massachusetts Amherst, USAHaibo Gu

As one of the major issues in the study of controltheory, the stability of stochastic system has stirredsome initial research interest. Since � stability con-tains exponential stability and polynomial stability,it has a wide applicability. However, there are fewresearch results about p-th moment and almost sure � stability for neutral stochastic switched nonlin-ear systems. In this paper, we attempt to investigatep-th moment and almost sure � stability of neu-tral stochastic switched nonlinear systems. Firstly weintroduce Razumikhin-type theorems and Lyapunovmethods. By the aid of Lyapunov-Razumikhin ap-proach, we obtain the p-th moment � stability ofneutral stochastic switched systems. In order to es-tablish the criterion on almost surely � stabilityof neutral stochastic switched systems, the Holderinequality, Burkholder-Davis-Gundy inequality andBorel-Cantelli’s lemma are utilized in this paper. Fi-nally one numerical example is provided to demon-strate the e↵ectiveness of the proposed results.

Chains of Minimal Image Sets CanAttain Arbitrary Length

Byungik KahngUniversity of North Texas at Dallas, USA

It is known that the maximal invariant set of a contin-uous iterative dynamical system in a compact Haus-dor↵ space is equal to the intersection of its forwardimage sets, which we will call the first minimal imageset. In this talk, we discuss the corresponding rela-tion for a class discontinuous self maps that are onthe verge of continuity, or topologically almost con-tinuous endomorphisms. We prove that the iterativedynamics of a topologically almost continuous endo-morphisms yields a chain of minimal image sets that

attains a unique transfinite length, which we call themaximal invariance order, as it stabilizes itself atthe maximal invariant set. We prove the converse,too. Given ordinal number ⇠, there exists a topolog-ically almost continuous endomorphism f on a com-pact Hausdor↵ space X with the maximal invarianceorder ⇠.

Uncountable Set of Real NumbersHaving No Uncountable Subset ofthe First Category in RPaula KempMissouri State University, USA

In this paper, it is shown that the statement, Ev-ery Uncountable set of Real Numbers R has an un-countable subset which is of the first category in Ris independent of the axioms of the ZFC Set Theory,(Zermelo-Fraenkel with the Axiom of Choice). Also,an example of a nonmeasurable set is given.

Chain Sequences of Some FiniteClasses of Classical OrthogonalPolynomials

Pradeep MalikUniversity of Petroleum and Energy Studies,Dehradun, India

In this work, we find chain sequences for Laguerrepolynomials and some finite classes of classical or-thogonal polynomials (COP). Chain sequences aremuch useful in the classification of the birth anddeath process.

The Stochastic Linear QuadraticControl Problem with Singular Es-timates

Hermann MenaUniversity of Innsbruck, Austria

We study an infinite dimensional finite horizonstochastic linear quadratic control (SLQ) problem inan abstract setting. We assume that the dynamics ofthe problem are generated by a strongly continuoussemigroup, while the control operator is unboundedand the multiplicative noise operators for the stateand the control are bounded. We prove an optimalfeedback synthesis along with well-posedness of theRiccati equation for the finite horizon case. In ad-dition, we investigate a numerical framework for theSLQ problem, in particular, the convergence of theRiccati operators.


A Computed Feedforward Com-pensation and Robust DynamicsForce Feedback Control for a 6DOFStewart-Type Nanoscale Platform

Yung TingChung Yuan Christian University, Taiwan

In this article, a new computed feedforward compen-sation for a 6DOF Stewart-type nanoscale platformis investigated to enhance the tracking performancesof the system due to the nonlinear hysteresis, creepe↵ect and drifting disturbance of the piezoelectricactuators, which are the main driving resource ofthe system. Exponentially Weighted Moving Aver-age (EWMA) method has been widely used in pro-cess control and verified its capability of overcomingsystematic change and drift disturbance. A doubleExponentially Weighted Moving Average (dEWMA)instead is used in combination of a dynamic inversehysteresis Preisach model for the feedforward com-puted controller. Besides using feedforward control,an improved robust dynamics force feedback con-troller, which is based on the idea of combining theclassical impedance control and predictive force con-trol, is also proposed to face with unacknowledgedinteracting forces in nano-cutting applications. Thealgorithm provides a way to increase the robustnessof force control scheme with respect to a variation ofthe environment characteristics. The experiment isperformed to evaluate the e↵ectiveness and robust-ness of the proposed controller above for in-feed andcross-feed motion of the 6DOF Stewart-type nano-scale platform.

Internal Model Control with Run-To-Run EWMA for Speed Controlof a Co-Planar Stage Driven byPiezoelectric Motors

Yung TingChung Yuan Christian University, TaiwanMark Leorna

In this article, speed control for a co-planar stagedriven by a bimodal piezoelectric motor is inves-tigated. Such a motion control system is subjectto disturbance such as friction, preload and oper-ating temperature rise. Especially, operating tem-perature rise is an essential problem of using piezo-electric motor, but very few research works ad-dress this topic in depth. Exponentially WeightedMoving Average (EWMA) method has been widelyused in process control and verified its capabilityof overcoming systematic change and drift distur-bance. Besides using EWMA for speed control, itis attempted to map the EWMA method into a run-to-run (RtR) Internal Model Control (IMC) struc-ture to achieve a RtR-IMC adaptive control scheme(RtR-IMC EWMA). Based upon this control struc-ture, a PI controller is added in the feedforward path

and proved to be able to deal with the ramp dis-turbance determined in practical experiment at vari-ant operating temperature. Friction causes dead-zone area, a dead-zone compensator is thus designedbased on a predictive friction model in order to re-duce the friction e↵ect. Associated with the frictioncompensator, performance of several control meth-ods including a general PID controller with opti-mal gains, RtR-IMC EWMA, and RtR-IMC EWMAwith PI controller (RtR-IMC EWMA+PI) are ex-amined. From experiment, the proposed RtR-IMC EWMA+PI control is superior to other meth-ods. Such a new adaptive control scheme is easy toestablish and provides flexibility of adding suitablecontroller to enhance system robustness.

Limit Theorems for One Class ofErgodic Markov Chains

Nezihe TurhanUniversity of North Carolina at Charlotte, USAStanislav Molchanov

In this talk, I will start with giving an intuitive back-ground on the limit theorems for Markov chains.Since my work includes both discrete and continuous-time Markov chains, I provide some preliminary workon both cases. Later on, I will briefly explain twomethods, namely Doeblin method and Martingaleapproximation, to prove the Central Limit Theoremfor the Loop Markov chains. Subsequently, I in-troduce three models of Loop Markov chain, and Iprove the Central Limit Theorem for a special classof functionals. As an example, I talk about RandomNumber Generators (RNGs) which are appropriateapplications of Loop Markov chains. Lastly, I anal-yse convergence to the stable limiting distributions,where we consider not a special class functionals butarbitrary ones.

Optimal Harvesting Control of aDi↵usive Population Model withSize Random Growth and Dis-tributed Recruitment

Qiangjun XieHangzhou Dianzi University, Peoples Rep of ChinaZe-Rong He, Zhaosheng Feng

We investigate an optimal harvesting control prob-lem for a spatial di↵usion population system, whichincorporates individual’s random growth of size anddistributed style of recruitment. The existence anduniqueness of nonnegative solutions to this practicalmodel are shown by means of Banach’s fixed pointtheorem, and the continuous dependence of the pop-ulation density on the harvesting e↵ort is analyzed.The optimal harvesting strategies are established vianormal cone and adjoint techniques. Some conditionsare presented to assure that there is only one optimalpolicy.


Contributed Session 5: Scientific Computation and NumericalAlgorithms

Stability and Convergence of a Vec-tor Penalty Projection Scheme forthe Incompressible Navier-StokesEquations with Moving Body.

Adrien DoradouxBordeaux University, FranceVincent Bruneau, Pierre Fabrie

This work is devoted to the study of a Vector PenaltyProjection scheme to solve the incompressible vis-cous flow around a moving body in the case wherethe interaction of the fluid forces on the solid canbe neglected. The velocity inside the solid regionis enforced using a penalization technique. Thestability of the system is shown using energy es-timates on each equation of the splitting. Thisresult additionally leads to a bound of the veloc-ity in a Nikolskii space which is useful to provethe strong convergence of the velocity through Si-mon’s compactness Lemma. We therefore showthat there exist u 2 L1(]0;T [,L2) \ L2(]0;T [,H1)and p 2 W�1,1(]0;T [,L2

0

) limit of the scheme whenthe time step and the corection parameter that en-sures a small velocity divergence tend to 0. Moreover,the couple (u, v) is a weak solution of the penalizedNavier-Stokes equations. Finally, studying the con-vergence when the penalty parameter that enforcesthe velocity inside the body tends to 0, we give a newproof of the existence of weak solutions to the Navier-Stokes problem with a no sleep boundary conditionon the moving body boundary.

RBF Solution of Natural Convec-tion of Nanofluids in a Cavity

Bengisen Pekmen GeridonmezTED University, Turkey

In this study, natural convection in a unit square cav-ity filled with a nanofluid is solved numerically uti-lizing the multi quadric radial basis function pseudospectral (MQ RBF-PS) in space domain and di↵er-ential quadrature method (DQM) in time domain.The governing dimensionless equations are solved interms of stream function, temperature and vortic-ity. In cavity, thermally insulated top and bottomwalls are maintained while the left and right wallsare at constant temperatures. Numerical solutionspresent the average Nusselt number variation as wellas streamlines, isotherms and vorticity contours. Theproblem parameters, Rayleigh number Ra and solidvolume fraction � are varied as 103 Ra 106 and0 � 0.2, respectively. It is found that the heattransfer is enhanced in presence of nanoparticles.

Magneto-Convection in BinaryNanofluids: a Revised Model

Urvashi GuptaPanjab University, IndiaJyoti Sharma, R. K. Wanchoo

The paper presents the e↵ect of vertical magneticfield on double-di↵usive nanofluid convection withthe assumption that nanoparticle flux is zero alongthe boundaries of the layer. The nanofluid layer in-corporates the e↵ect of Brownian motion and ther-mophoresis due to the presence of nanoparticles ande↵ect of Dufour and Soret parameters due to the pres-ence of solute. Normal mode technique and singleterm Galerkin method are used to solve the conser-vation equations related to the system.For the ana-lytical study, valid approximations are made in thecomplex expression for the Rayleigh number to getuseful and interesting results. Due to the inclusion ofmagnetic field, Lorentz force term is added in the mo-mentum equation, which results in strong stabilizinge↵ects of the magnetic field parameter (the Chan-drasekhar number) on the fluid layer. Oscillatorymotions are not possible and hence mode of convec-tion is invariably through stationary mode. Binarynanofluids are found to be much less stable than regu-lar fluids. Numerical computations are carried out forwater based nanofluids to analyze solutal e↵ects onthe stability of the system using the software Math-ematica. Higher conductivity and density of metallicnanofluids make them less stable as compared to non-metallic/semiconducting nanofluids.

Computable Error Estimates forMonte Carlo Finite Element Ap-proximation of Elliptic PDE withRough Lognormal Di↵usion Coe�-cients

Eric HallUniversity of Massachusetts Amherst, USAHakon Hoel, Mattias Sandberg, AndersSzepessy, Raul Tempone

The Monte Carlo (and Multi-level Monte Carlo) fi-nite element method can be used to approximate ob-servables of solutions to di↵usion equations with log-normal distributed di↵usion coe�cients, e.g. model-ing ground water flow. Typical models use lognormaldi↵usion coe�cients with Hølder regularity of orderup to 1/2 almost surely. This low regularity impliesthat the high frequency finite element approximationerror (i.e. the error from frequencies larger than themesh frequency) is not negligible and can be largerthan the computable low frequency error. We address


how the total error can be estimated by the com-putable error and propose goal-oriented estimates forthe pathwise Galerkin and expected quadrature er-rors that are derived using easily validated assump-tions.

Discontinuous Galerkin Methodwith Weighted Numerical Flux forTime-Dependent Equation

Jia LiHarbin Institute of Technology, Peoples Rep ofChinaBoying Wu, Jiebao Sun, Shengzhu Shi

This article studies the numerical scheme by usingweighted numerical flux for solving time-dependentequation. The numerical flux is a vital item fordiscontinuous Galerkin methods, that it is the keyfor the solvability and properties of scheme. Theweighted flux (up-wind biased flux) this article stud-ies is a special form of numerical fluxes. Unlike up-wind and central flux taking the value of one sideor average of two side at the nodes, it takes theweighted value of two side, which made the scheme bemore adaptive for complex equations and have bet-ter convergence. So weighted flux has better prop-erties than others in some ways. This article devel-ops a numerical scheme to solve heat equation bydiscontinuous Galerkin methods applies the weightedflux, and studies the stability analysis and error esti-mate. Meanwhile, numerical examples are consistentwith the error estimate, which proves the validity ofscheme.

Semi-Lagrangian Numerical Meth-ods for Systems of Time-DependentPartial Di↵erential Equations

Nikolai LipscombUniversity of North Carolina Wilmington, USA

Semi-Lagrangian methods are numerical methods de-signed to find approximate solutions to particulartime-dependent partial di↵erential equations (PDEs)that describe the advection process. We proposesemi-Lagrangian one-step methods for numericallysolving initial value problems for two general systemsof partial di↵erential equations:

8>>>><

>>>>:

@y1

@t+ !

@y1

@x= f

1

(t, x, y1

, . . . , yn), y10 = y

1

(0, x),

......

......

@yn@t

+ !@yn@x

= fn(t, x, y1, . . . , yn), yn0 = yn(0, x),

where (t, x) 2 [0,1)⇥ [a, b] and ! can take on one ofthree cases:

1. ! 2 R,2. ! = !(t, x), |!| < 1,3. ! = !(t, x,y), |!| < 1,

and8>>><

>>>:

@u@t

+ u@u@x

+ v@u@y

= f(t, x, y, u, v), u0

= u(0, x, y),

@v@t

+ u@v@x

+ v@v@y

= g(t, x, y, u, v), v0

= u(0, x, y),

where (t, x, y) 2 [0,1)⇥ [a, b]⇥ [c, d].Along the characteristic lines of the PDEs, we willuse ordinary di↵erential equation (ODE) numericalmethods to solve the PDEs. The main benefit of ourmethods is that we have managed to achieve high or-der local truncation error through the use of Runge-Kutta methods along the characteristics. In addi-tion, we have established numerical analysis prece-dents for semi-Lagrangian methods applied to sys-tems of PDEs: stability, convergence, and maximumerror bounds.

Transformation Optics BasedFinite-Di↵erence Time-DomainMethod for Solving the MaxwellEquations

Jinjie LiuDelaware State University, USAMoysey Brio, Jerome V. Moloney

Transformation optics (TO) is an elegant coordinatetransformation technique to design the metamate-rial invisibility cloak. An important feature of theTO technique is the invariance of the Maxwell equa-tions after a coordinate mapping, while the trans-formed material becomes electrically and magneti-cally anisotropic.Recently, we have developed a stable TO basedmesh refinement method to numerically solve theMaxwell’s equations. The TO technique is appliedto the Maxwell equations to achieve local mesh re-finement, by mapping a structured non-orthogonalgrid to a rectangular mesh. Then the transformedanisotropic Maxwell equations are solved using astable anisotropic Finite-Di↵erence Time-Domain(FDTD) method. With TO based local mesh refine-ment, the computational e�ciency is significantly im-proved. In comparison to other subgridding FDTDmethods, one of the major advantages of our methodis the stability property of the numerical methodsapplied to the anisotropic Maxwell equations, whileother subgridding FDTD methods often su↵er fromthe late-time instability problems. In this talk, wewill discuss the recent progress in the TO based nu-merical algorithm for solving Maxwell equations andits applications.


Conservative Finite-Di↵erenceScheme for Computer Simulationof the Field Optical Bistability inSemiconductor

Maria LoginovaLomonosov Moscow State University, RussiaVyacheslav A. Trofimov, Vladimir A.Egorenkov

We consider femtosecond laser pulse interaction with2D semiconductor under the condition of field op-tical bistability occurrence. This is very promisingphenomenon for the creation and developing of all-optical data processing. The optical bistability isbased on semiconductor absorption coe�cient depen-dence from strength of an electric field induced bya laser radiation. The laser pulse interaction witha semiconductor is described by the set of nonlin-ear di↵erential equations concerning the electric fieldpotential, and a free electron concentration, and ion-ized donor concentration. Laser pulse propagation inthe semiconductor is described by either a nonlinearequation with respect to its intensity or the nonlinearSchrodinger equation. For a set of these equations wedevelop conservative finite-di↵erence scheme. Thisfinite-di↵erence scheme is nonlinear one. To realizeit we propose an original two-step iteration process,which is also conservative one on each iteration andallows to provide a calculation on long time intervalwith high accuracy. Therefore, our finite-di↵erencescheme possesses an asymptotic stability property.This is very important feature because we have to doa calculation during very long time interval. An ap-proach for the nonlinear finite-di↵erence scheme real-ization can be easy generalized for multidimensionalproblem.

Numerical Study of Magneto-Thermal Convection of Ferromag-netic Fluids in the Presence of HallCurrents in a Porous Medium

Veena SharmaHimachal Pradesh University, IndiaAnukampa Thakur, Urvashi Gupta

In this study, the e↵ect of Hall currents on the onsetof convection in a porous medium layer saturated byan electrically conducting ferromagnetic fluid heatedfrom below using linear stability analysis is inves-tigated. Darcy law for the ferromagnetic fluid isused to model the momentum equations for a porousmedium. The employed model incorporates the ef-fects of polarization force and body couple. Ferro-magnetic fluid behaves as a homogenous continuumand exhibits a variety of interesting phenomenon.Hall currents are likely to be important in flows oflaboratory plasmas as well as in many geophysicaland astrophysical situations. The coupled partialdi↵erential equations governing the physical problemare reduced to a set of ordinary di↵erential equationsusing normal mode technique. These equations aresolved analytically for stress free boundaries and nu-

merically using the software Mathematica for rigid-free boundaries by obtaining approximate solutionsusing Galerkin method. For the case of stationaryconvection, it is found that the magnetic field andmagnetization have a stabilizing e↵ect as such theire↵ect is to postpone the onset of thermal instability;whereas Hall currents are found to hasten the same.The medium permeability hastens the onset of con-vection under certain conditions.

Split-Step Compact Finite Di↵er-ence Method for Cubic-QuinticComplex Ginzburg-Landau Equa-tions

Shanshan WangNanjing University of Aeronautics and Astronautics,Peoples Rep of ChinaLuming Zhang

Split-step compact finite di↵erence method is pro-posed for the cubic-quintic complex Ginzburg-Landau (CQ CGL) equations both in one dimen-sion and in multi-dimensions. The orginal CQ CGLequations are separated into two nonlinear subprob-lems and one or several linear ones by the split-stepmethod. The linear subproblems are solved by thecompact finite di↵erence schemes. As the nonlinearsubproblems cannot be solved exactly as usual, theRunge-Kutta method is applied and the total accu-racy order is not reduced. Extensive numerical exper-iments are carried out to examine the performance ofthis method for nonlinear Schrodinger equations, thecubic complex Ginzburg-Landau equation, and theCQ CGL equations.

Split Bregman Method for Mini-mization of Convex Image Segmen-tation Model Based on Local andGlobal Intensity Fitting Energy

Yunyun YangHarbin Institute of Technology Shenzhen GraduateSchool, Peoples Rep of ChinaYi Zhao, Boying Wu

In this paper we apply the split Bregman methodto minimize our proposed convex image segmenta-tion model based on local and global intensity fittingenergy, which can be used to segment more generalimages more accurately. We emphasize several im-portant theoretical results including the convergenceresult of our proposed model and present proofs forthem. We also show several experimental resultsto demonstrate the segmentation accuracy and ef-ficiency of the proposed model.


The Numerical Method for Time-Fractional Convection-Di↵usionProblems with High-Order Accu-racy

Wenjuan YaoHarbin Institute of Technology, Peoples Rep ofChinaBoying Wu, Jiebao Sun, Jia Li

In this paper, we consider the numerical methodfor solving the two-dimensional fractional convection-di↵usion equation with a time fractional derivative oforder ↵ (1 < ↵ < 2). By combining the compact dif-

ference approach for spatial discretization and the al-ternating direction implicit (ADI) method in the timestepping, a compact ADI scheme is proposed. Theunconditional stability and H1 norm convergence ofthe scheme are proved rigorously. The convergenceorder is O(⌧3�↵ + h4

1

+ h4

2

), where ⌧ is the tempo-ral grid size and h

1

, h2

are spatial grid sizes in the xand y directions, respectively. It is proved that themethod can attain (1 + ↵) order accuracy in tempo-ral for some special cases. Numerical results are pre-sented to demonstrate the e↵ectiveness of theoreticalanalysis.


Contributed Session 6: Bifurcation and Chaotic Dynamics

Flip Bifurcation for P-PeriodicMaps

Muna Abu AlhalawaBirzeit University, IsraelHenrique Oliveira

In this talk we extend the theory of period doublingbifurcation in one dimensional autonomous mapsto one dimensional non-autonomous periodic maps.We deduce the invariance of the bifurcation equa-tions, including the degeneracy, non-degeneracy andtransversality conditions under cyclic permutationsof composition of maps.Therefore, the results in this work establish that thedefinitions of bifurcation with eigenvalue �1, i.e., theflip D

1

and degenerate flip bifurcations with codi-mensions two and µ, D

2

andDµ resp., with µ natural,in one dimensional p-periodic non-autonomous itera-tion of families of maps are well posed. We prove thatthe bifurcation equations of the double compositionof the maps defining the p-periodic non-autonomousmaps are invariant, thus proving the conditions onthe flip bifurcations.We also give a catalog and some examples of the oc-currence of the degenerate type of bifurcations in pe-riodic non-autonomous maps with codimension 2 .The new results extend the invariance of the bifur-cation conditions of the bifurcations with eigenvalue+1, i.e., the Aµ class in Arnold’s classification, innon-autonomous p-periodic maps generated by pa-rameter depending families with p maps. It is provedthat the conditions of degeneracy, non-degeneracyand transversality are invariant relative to cyclic or-der of compositions for any natural number µ.

Mode-Matching Approach forAcoustic Scattering in FlexibleWalled Bifurcated Waveguides

Muhammad AfzalCapital University of Science and Technology,Islamabad, Pakistan

This paper deals with the propagation and scat-tering of acoustic wave through a bifurcated flexi-ble channel and discontinuity. The orthogonal andnon-orthogonal duct modes across the interface arematched via the continuity of pressure and normalvelocities. It enables to determine the amplitudesof scattering duct modes. However, there appearsome oscillations in normal velocities which can beremoved by using the Lanczos filters to extract therequired useful information. Further, the accuracyof algebra can be seen through the conserve poweridentity , and that is also useful to insights the prob-lem physically. It is interesting to note that thefundamental duct mode incident, which contains thecharacteristics of rigidly bounded duct, is scatteredthrough the bifurcation of flexible channel and thediscontinuity of structure.

Probability Density Functions ofStochastic Dynamics and The PathIntegration Methods

Linghua ChenNorwegian University of Science and Technology,NorwayEspen Robstad Jakobsen, Arvid Naess

In this talk we present new results for Fokker-Planckequations with unbounded coe�cients: solvabilityand the strong convergence of numerical schemes inL1 space. The solution of such equations correspondsto the evolution of probability density functions ofstochastic di↵erential equations (SDE). The latterhas a large number of applications in various areas:including physics, economics, and finance. Existenceand uniqueness of a mild solution is derived. Onthe numerical side, we study the so-called discretepath integration method which produces approxi-mate probability density functions for the solutionsof corresponding SDEs. We prove that this schemestrongly converges in L1 norm, uniformly in any fi-nite time horizon. Specifically, we use the conceptof dissipative operators, combined techniques fromsemigroup and PDE theory, as well as methods fromstochastic analysis.This is a joint work with Arvid Naess and EspenRobstad Jakobsen.


Chaotic Behaviours of the ShiftMap on the Generalized M-SymbolSpace and Their Topological Conju-gacy

Tarini DuttaGauhati University, India

This paper deals with some illuminating results inconnection with various chaotic behaviors of the for-ward shift map �+ on the generalized one-sided sym-bol spaceP

+

m,m(� 2) 2 N . We prove that

(i) the shift map �+ onP

+

m is Devaneychaotic, exact Devaney Chaotic, mixing De-vaney Chaotic, weak mixing Devaney Chaotic,Auslander-Yorke Chaotic and generically �-Chaotic;

(ii) �+ onP

+

m is topologically conjugate to themap fm(x) = mx(mod1) on the circle S1;

(iii) some characteristic features of the spaceP+

m, viz, perfectness, connectedness, etc. arefruitfully investigated and

(iv) a few applications and open problems onperiod-doubling bifurcation as well as Hopf-bifurcation are demonstrated for further re-search work.

Bifurcation Analysis for a BilliardProblem in Nonlinear and Nonequi-librium Systems

Tomoyuki MiyajiMeiji University, Japan

We study a four-dimensional dynamical system de-fined by a system of ordinary di↵erential equations.It is a mathematical model of self-propelled motionsof a camphor disk floating on a water surface. A cam-

phor disk moves as if it is a billiard ball: it repeatsa uniform motion and reflection. There are some dif-ferences from the mathematical billiards in which aparticle exhibits a uniform motion and a completelyelastic reflection. A camphor disk changes the di-rection of motion without collision at the boundaryand the reflection is not completely elastic. More-over, when the domain is square-shaped, orbits of themathematical model eventually tends to a limit cycleas time passes. In this study, we consider behavior ofthe system in a rectangular domain. It is known bynumerical study that there exists an attractor, whichmay be periodic, quasi-periodic, or chaotic depend-ing on the aspect ratio of the rectangle. We applymethods of dynamical systems and bifurcation the-ory for understanding bifurcations of the system. Wereveal that the Hopf-Hopf bifurcation of the rest stateis the organizing center of a complicated bifurcationstructure of the system.

POSTER SESSION 421

Poster Session

Mathematical Modeling of HER2Signaling Pathway: Implications forBreast Cancer Therapy

Sameed AhmedUniversity of South Carolina, USAXinfeng Liu, Hexin Chen

The cancer stem cell hypothesis states that thereis a small subset of tumor cells, called cancer stemcells (CSCs), that are responsible for the prolifer-ation and resistance to therapy of tumors. CSCshave the ability to self renew and di↵erentiate toform the nontumorigenic cells found in tumors. Over-expression of human epidermal growth factor recep-tor 2 (HER2) plays a role in regulation of CSC pop-ulation in breast cancer. Current cancer therapy in-cludes drugs that block HER2, however, patients candevelop anti-HER2 drug resistance. Downstream ofHER2 is nuclear factor B (NFB). The aberrantregulation of NFB leads to cancer growth, whichmakes it a promising target for cancer therapy, es-pecially for those who have developed resistance toanti-HER2 treatment. Our collaborator’s lab has dis-covered that IL1, which is downstream of HER2, isresponsible for NFB activation, thus making it a po-tential target for cancer treatment. We have devel-oped a mathematical model to represent this new sig-naling pathway, and simulations of the model matchthe some of the results from the lab. We will use themathematical model to make predictions for di↵erentscenarios, and it will be updated and expanded basedupon new experiments.

A Basis for Improving NumericalWeather Prediction in the Gulf Areaby Assimilating Doppler Radar Ra-dial Winds

Mohamed-Naim AnwarUnited Arab Emirates University, United ArabEmiratesF.A. Rihan, C.G. Collier

Numerical Weather Prediction (NWP) is consideredas an initial-boundary value problem: given an es-timate of the present state of the atmosphere, themodel simulates (forecasts) its evolution. Specifica-tion of proper initial conditions and boundary condi-tions for numerical dynamical models is essential inorder to have a well-posed problem and subsequentlya good forecast model (A well-posed initial/boundaryproblem has a unique solution that depends continu-ously on the initial/boundary conditions). The goalof data assimilation is to construct the best possibleinitial and boundary conditions, known as the analy-sis, from which to integrate the NWP model forwardin time. In this paper, we describe an approach toassimilate Doppler radar radial winds into a high res-olution NWP model using 3D-Var system. We dis-cuss the types of errors that occur in radar radial

winds. Some related problems such as nonlinearityand sensitivity of the forecast to possible small er-rors in initial conditions, random observation errors,and the background states are also considered. Thetechnique can be used to improve the model fore-casts, in the Gulf area, at the local scale and underhigh aerosol (dust/sand/pollution) conditions.

Continuous Simulation of the Dy-namical Forces Acting on CervicalSpine

Murat BakirciOld Dominion University, USAAshley Lara

Cervical syndrome is a range of symptoms which aredue to the degenerative changes of the cervical spine,and cervical muscles. It is quite necessary to considerthe strength of the cervical muscles because somepeople with weak muscles in their neck region suf-fer from cervical syndrome. In particular, when rid-ing in a car for extended periods of time, they su↵erawful headache attacks, because their head vibrates(oscillates) with the vibrations of the car on the road,since their head is not attached sti✏y enough to theshoulders. To a first approximation, a human bodycould be modeled as a translational mechanical sys-tem and so it can be considered as a combination ofdi↵erent sub-parts. When dealing with the relativepositions of the human body parts or relationshipbetween those parts such as anterior, proximal, ordistal movement, a translational mechanical systemapproximation is su�cient enough to model the en-tire system. In this study, dynamical forces acting oncervical spine in response to the vertical movementof human body has been investigated. The extendedmodel is obtained by applying Newton’s law to eachbody parts and the obtained set of di↵erential equa-tions has been solved numerically.

Simulation of the ElectroosmoticFlow Induced by AC Electric Fieldin a 2D Rectangular Micropore

Murat BakirciOld Dominion University, USAGeorge Pratt

Ions and molecules passing through a micropore mustinteract with the pore walls. Often the pore wallshave a non-zero surface charge density which a↵ectsionic concentrations within the pore. This exposes amathematical di�culty that is usually treated withnumerical techniques. A charged surface in contactwith an ionic solution attracts ions of opposite charge(counterions) and repel ions of like charge (coions),and form electric double layer (EDL) in the vicinityof the charged surface. Within the EDL, the concen-tration of the counterions is much higher than thatof coions. Hence, the EDL plays an important role


in ionic mass transport in micropores such as a neg-atively charged conical micropore has been shown totransport positive ions with higher selectivity whichmeans that most of the ionic current through the mi-cropore is carried by positive ions. In this research,AC electroosmotic flow through a micropore underthe e↵ect of discrete surface potential has been inves-tigated. An electrokinetic model that combines thePoisson-Boltzmann distribution for ions with Stokesequation for the fluid flow in the absence of a hy-drostatic pressure gradient has been developed andnumerically solved for the given computational do-main.

Dynamical Analysis of ConnectedNeuronal Motifs with OpenAcc andOpenMPI

Sunitha BasodiGeorgia State University, USAKrishna Pusuluri, Andrey Shilnikov

Large scale analysis of the dynamical behavior ofCentral Pattern Generators (CPGs) formed by neu-ronal networks of even small sizes is computation-ally intensive and grows exponentially with networksize. We have developed a suite of tools to exhaus-tively study the behavior of such networks on modernGPGPU accelerators using the directive based ap-proach of OpenAcc. We also achieve parallelizationacross clusters of such machines using OpenMPI. Di-rective based approaches simplify the task of portingserial code onto GPUs, without the necessity for ex-pertise in lower level approaches to GPU program-ming, such as CUDA and OpenCL. 3-cell neuronalCPGs have been explored previously using variousGPGPU tools [1]. As motifs form the building blocksof larger networks, we have employed our frameworkto study 4-cell CPGS and two connected 3-cell motifs.We discuss the performance improvements achievedusing this framework and present some of our results.

The Impact of Resource Abundanceon Pathogen Invasion Risk

Rebecca BorcheringUniversity of Florida, USAJ.M. Flynn, S.E. Bellan, J.R.C. Pulliam, S.A.McKinley

Territorial animals share a variety of common re-sources, which can be a major driver of conspecificencounter rates. We investigate how changes in re-source density influence the rate of encounters be-tween individuals in a population. We develop amodel of resource selection by consumers on a spatialresource landscape and estimate changes in encounterrates as a function of resource availability. Using sim-ulations and asymptotic analysis we show that therelationship between resource availability and con-sumer encounter rate is nonmontonic. We also findthat the maximum distance at which consumers areable to detect resources greatly influences the ex-

pected consumer encounter rate. We discuss thesetheoretical results in the context of a jackal popula-tion which has access to a seasonally varying numberof carcasses and their subsequent vulnerability to ra-bies virus outbreaks.

Dynamics of Target Mediated DrugDisposition in PKPD

Jonghyuk ByunBusan National Univerisity, KoreaIl Hyo Jung

Target Mediated drug disposition (TMDD) is amodel which concerns the relation between adrug and its target (receptor) in vivo or vitro.TMDD predicts Pharmacokinetics & Pharmacody-namics (PKPD) behavior of various drugs suchas monoclonal antibodies(mAbs). We introducetwo compartment TMDD model and simplify theTMDD with quasi equilibrium, quasi steady stateor Michaleis-Menton models. Furthermore, we alsopresent strong and weak points of those models andhow parameters a↵ect the concentration of drug, re-ceptor and drug-receptor complex in TMDD.

A Mathematical Analysis on theTransmission Dynamics of NeisseriaGonorrhoeae

Christine CraibUniversity of North Carolina at Wilmington, USA

In this project, we analyze an epidemiological modeldescribing the transmission of gonorrhea, with a coresexual activity class and a noncore sexual activityclass. We discuss the behavior of the model aroundthe two equilibrium points, a disease-free equilib-rium and a coexistence equilibrium. The focus ofthe project is to identify equilibrium points, analyzethe stability of these points, and discuss the results interms of the epidemiological model. Ultimately, thegoal of the project is to find conditions of an endemicstate, and the conditions that ensure the eradicationof gonorrhea.

Interaction of Scroll Waves in anExcitable Medium

Nirmali Prabha DasIndian Institute of Technology Guwahati, IndiaSumana Dutta

Spiral waves occur in systems ranging from biology toastrophysics, fluids to superconductors. Scroll wavesare the three dimensional counterparts of spiral wavesthat rotate around a one dimensional phase singular-ity, known as filament. Their presence in the car-diac tissues is many times the cause arrythmia thatfinally lead to heart failure. So the interaction ofscroll waves may have far-reaching consequences oncardiac activity. In fluids and liquid crystals, there isevidence of vortex interaction leading to interestingphenomena like filament reconnection. If likewise,

POSTER SESSION 423

scroll rings interact and reconnect, then small ringsmay merge and form large ones that will have en-hanced life-times. If this happens in heart tissues, itwill ensure a long life of the filaments which in turnwill have a detrimental e↵ect on cardiac health. Thework reported here is motivated by these concerns.Here, we report the first experimental evidence ofscroll wave reconnection. Our results demonstratethat when two scroll rings are brought close enough,they can either attract each other, and reconnect toform a large scroll ring, or they can repel so that theyrupture on touching the boundaries. We also carryout simple numerical simulations that helps explainthe filament behavior in our experiments.

New a Priori Estimates for Mean-Field Games with Congestion

David Evangelista da Silveira JuniorKAUST, Saudi ArabiaDiogo Gomes

We present recent developments in crowd dynamicsmodels (e.g. pedestrian flow problems). Our for-mulation is given by a mean-field game with con-gestion. We start by reviewing earlier models andresults. Next, we develop our model. We establishnew a priori estimates that give partial regularity ofthe solutions. Finally, we discuss numerical results.

Relativistic Compact Objects witha Linear Equation of State

Megandhren GovenderDurban University of Technology, So Africa

In this work we present a general framework for gen-erating exact solutions to the Einstein field equa-tions for static, anisotropic fluid spheres in comov-ing, isotropic coordinates obeying a linear equationof state of the form pr = ↵⇢ � �. We show that allpossible solutions can be obtained via a single gener-ating function defined in terms of one of the grav-itational potentials. The physical viability of oursolution-generating method is illustrated by model-ing a static fluid sphere describing a strange star.

Stability and Uniqueness of SlowlyOscillating Periodic Solutions toWright’s Equation

Jonathan JaquetteRutgers University, USAJean-Philippe Lessard, Konstantin Mis-chaikow

Jones‘ conjecture states that when ↵ > ⇡, then thereexists an unique slowly oscillating periodic solution(SOPS) to Wright’s equation: y‘(t) = �↵y(t�1)[1+y(t)]. While Jones proved the existence of SOPS in1962, the conjecture has yet to be proven in whole.To prove uniqueness, it su�ces to show that all SOPSto Wright’s equation are asymptotically stable. By

developing asymptotic bounds on the behavior ofSOPS and then estimating their Floquet multipli-ers, in 1991 Xie was able to prove the conjecture for↵ > 5.67. We sharpen Xie’s result by using rigor-ous numerics to estimate the Floquet multipliers ofSOPS to Wright’s equation. In this manner we areable to make progress on a long standing conjecturein delay di↵erential equations.

Interaction Energy Between Col-loidal Particles

Wai-Ting LamYeshiva University, USA

One of the central problems on colloids is their stabil-ity. That is, whether under known conditions (suchas concentration) the system will coagulate or remainindefinitely stable.Mathematically, the stability depends on the detailsof the pairwise energy as a function of separation ofcolloidal particles. One approach to obtain that en-ergy is to solve Poisson-Boltzmann equation (PB),which gives the charge density and electrostatic po-tential of the solution surrounding the colloidal par-ticles. PB is a di↵erential equation that has an ex-act known solution only for one-dimensional geome-tries. In the case of general interest, the problemis three-dimensional and not amenable to analyticalsolutions. In this paper we solve the PB equation ap-proximately via variational methods. We introducethe density and corresponding electrostatic poten-tial parametrically and minimize the PB functionalwith respect to the parameters. We also find use-ful polynomial approximations for the parameters asfunctions of separation and boundary conditions. Fi-nally, we produce energy-separation curves and con-clude with the stability properties predicted by thisapproach.

Some Numerical Aspects on CrowdMotion - the Hughes Model

Roberto Machado VelhoKAUST, Saudi ArabiaDiogo A. Gomes

We study the Hughes model for crowd motion andwe present a numerical approach to solve it.The model comprises a Fokker-Planck equation cou-pled with an Eikonal with Dirichlet or Neumanndata:

(⇢t(x, t)� div(⇢(1� ⇢)2Du) = �⇢,|Du(x)|2 = 1

(1�⇢)2 .

The Fokker-Planck equation gives the evolution ofthe crowd density ⇢. The Eikonal equation deter-mines the optimal direction of movement.We establish a priori estimates for the solution andidentify a shock formation mechanism. We illustratethe existence of congestion, the breakdown of themodel, and trend to equilibrium.


Our proposed numerical method explores the adjointstructure in this system and we illustrate throughsome examples. The method is valid in arbitrary di-mension, conserves of mass and positivity.

On the Existence of Limit Cycles ina 3D Piecewise Linear Van Der PolLike Memristor Oscillator

Marcelo MessiasSao Paulo State University - UNESP, BrazilAnderson Luiz Maciel

In this work we study a mathematical model for aVan der Pol like oscillator formed by three funda-mental electronic elements: one memristor, one ca-pacitor and one inductor. The model is given bya discontinuous piecewise linear system of ordinarydi↵erential equations, defined on three zones in R3,given by |z| 1, the internal zone, and |z| � 1,the external zones. We show that the z–axis is filledwith equilibrium points of the system and analyzethe linear stability of the equilibria in each zone. Weprove that, due to the existence of this line of equi-libria, the phase space is foliated by parallel invariantplanes transversal to the z–axis, so the dynamics isessentially two-dimensional and no chaotic behaviormay occur. We also determine numerically the occur-rence of nonlinear oscillations, given by the existenceof limit cycles belonging to the invariant planes andpassing by two of the three zones or passing by thethree zones. These periodic orbits arise due to homo-clinic and heteroclinic bifurcations, obtained varyingone of the parameters of the studied model. Theanalytical and numerical results obtained extend theanalysis presented in the literature

Dynamics Analysis of a Inflamma-tory Cytokines Model with StateDependent Impulsive E↵ect in Au-toimmune Disease

Anna ParkPusan National University, KoreaIl Hyo Jung

An autoimmune disease is a chronic inflammatorydisease and triggered by abnormal immune response.It occurs when the immune system attacks its owntissue and consequently destroys and damages tis-sue. This is related to a break down in immune tol-erance. An autoimmune disease is characterized bythe breakdown of tolerance of the pro-inflammatorycytokine and anti-inflammatory cytokine. Our pro-pose is to restore the immune tolerance by usingcontrol strategy with state dependent impulsive ef-fect. We present a two-variable model for the inter-actions between pro-inflammatory cytokine and anti-inflammatory cytokine. Our result illustrate how fea-sibility condition protects pro-inflammatory cytokineincrease. Numerical simulation imply that the pres-ence of pulses makes the dynamic behavior more com-plex.

On the Computation of ExpansionEntropy

Eric RobertsUniversity of California, Merced, USA

Despite the rich history in the field of nonlinear dy-namics and chaos, an exact definition for the termchaos is still very much up for debate. In this poster,we investigate a recently proposed entropy-based def-inition of chaos which measures the di↵erence be-tween the exponential growth rate of volume ex-pansion in an arbitrary region with the exponentialgrowth rate at which volume escapes. Implementa-tion of this method could have a large impact in widearray of applications, including parameter control forchaos detection. We advocate for this new, broadlyapplicable, more computationally feasible measurefor complexity and explore certain numerical com-putational aspects of this expansion entropy for dis-crete mappings. Namely, by way of calculating thesingular values of the iterated Jacobian mapping, weuse expansion entropy to help extend the conceptsof turnstile and transport into 3D volume preservingmaps.

An Analytic Approach to the Designand Payback Time of PV Plants

Angela SciammettaUniversita degli Studi di Messina, ItalyGabriele Bonanno, Salvatore De Caro, Tom-maso Scimone, Antonio Testa

Grid connected photovoltaic plants for residential ap-plications are normally designed through determin-istic methods on the basis of system requirementsas: PV array peak power, available budget, or avail-able space. Unfortunately, these approaches give onlysuboptimal results. More e↵ective design techniquesare those oriented to fulfill given performance targetsas minimum pay-back time, minimum life cycle cost,or minimum annual electrical energy cost. However,they require very complex optimization techniques,especially if the plant encompasses energy storagesystems. An analytic approach is presented to op-timally sizing the photovoltaic module array and theenergy storage system in a grid-connected generatorserving a set of residential loads, on the basis of an-nual average solar radiation data and load demand.The configuration with the shorter pay-back time isidentified through a mathematical model, describingthe variation of the annual energy cost as a functionof the plant configuration. The proposed method canbe easily modified to take into account di↵erent op-timization targets, as maximum life-cycle profit.

POSTER SESSION 425

Numerical Investigation of the Hier-archical Stability of the CaledonianSymmetric Four Body Problem

Anoop SivasankaranKhalifa University of Science Technology and Re-search, United Arab EmiratesMuhammad Shoaib

The Caledonian Symmetric Four Body Problem(CSFBP) is a restricted four body problem developedby Steves and Roy (1998) with a symmetrically re-duced phase space which can be applied to study thestability and evolution of symmetric quadruple stellarclusters and exo-planetary systems. Sivasankaran,Steves and Sweatman (2010) developed a global regu-larization scheme for CSFBP that consists of adaptedversions of several known regularisation transforma-tions such as the Levi-Civita-type coordinate trans-formations; that together with a time transformation,removes all the singularities due to colliding pairs ofmasses. Using this newly developed numerical al-gorithm, we numerically investigate the relationshipbetween the hierarchical stability of the system andthe analytical stability parameter characterised bythe Szebehely constant C

0

, which is a function of thetotal energy and angular momentum of the system.We also show that for C

0

� 0.046 all the CSFBPsystems will be hierarchically stable for all time.

Generalized Ordinary Di↵erentialEquations and Measure FunctionalDi↵erential Equations

Patricia TacuriSao Paulo State University, BrazilM. Federson, J. Mesquita, M. Frasson

We introduce the equations called neutral measurefunctional di↵erential equations and we prove thatthese equations can be also related with a class ofabstract generalized ordinary di↵erential equations(GODEs). Then, using these correspondence, we areable to prove existence and uniqueness of solutionsand continuous dependence results for neutral mea-sure functional di↵erential equations.

Bayesian Logistic Modeling of Dis-ease Dispersal with Partial Di↵er-ential Equation

Kim TaekyoungChonnam National University, KoreaTaekyoung Kim and Ilsu Choi

Bayesian inference provide good parameter esti-mates, parsimonious descriptions of observed data,predictions for missing data and forecast of futuredata, a computational structure for model estimationselection and validation. Partial di↵rential equationmodels give describing disease dispersal with pop-ulation process and have been used extensively toevaluate the e↵ects of spatial variation on popula-tions. The transmisson is focused in disease preva-lence and control with new kinds of virus and bacte-ria. We consider basic finite di↵erence methods andspatial-temporal Bayesian logistic models for spatial-temporal binary data which are observed on visitinghospital and apply the methods with an example ofoutcomes of Noro-virus on Korea peninsula. We dis-cuss dispersal, disease invasions, and di↵usion spatialpattern as well as interaction between space and time.

Symbolic Computation and Abun-dant Traveling Wave Solutions toModified Burgers’ Equation

Muhammad YounisUniversity of the Punjab, PakistanM. Younis, S.T.R.Rizvi, A. Sardar

In this article, the novel G’/G expansion scheme issuccessfully applied to construct the abundant trav-elling wave solutions to the modified Burgers equa-tion with the aid of symbolic computation. The ap-plied scheme is reliable and useful which gives moreand general exact travelling wave solutions than theother existing schemes. These obtained solutions arein the form of hyperbolic, trigonometric and rationalfunctions including solitary, singular and periodic so-lutions as well, which have many other potential anduseful applications in physical science and engineer-ing. Most of these solutions are new and some havealready been constructed using di↵erent techniques.Additionally, the constraint conditions, for the exis-tence of the solutions are also listed.


Student Paper Competition Session

Picone’s Identity for P-BiharmonicOperator and Its Applications

Gaurav DwivediIndian Institute of Technology Gandhinagar, India

In this article we prove the nonlinear analogue ofPicone’s identity for p-biharmonic operator. As anapplication of our result, we show that the Morse in-dex of the zero solution to a p-biharmonic boundaryvalue problem is 0. We also prove a Hardy type in-equality and Sturmian comparison principle. We alsoshow the strict monotonicity of the principle eigen-value and linear relationship between the solutionsof a system of singular p-biharmonic system. In theend, we establish a Caccioppoli type inequality forp-biharmonic operator.

A Mathematical Model of the Hu-man Papillomavirus (HPV) with aCase Study in Japan

Arielle GaudielloUniversity of Central Florida, USA

The human papillomavirus (HPV) is a sexually trans-mitted infection prominent among young adults inmost countries. The disease typically clears in twoyears naturally, but causes increased risk of develop-ment of cancerous cells post-infection. We develop anordinary di↵erential equation model including vacci-nation of the female classes and investigate existenceand stability of the disease-free equilibrium and en-demic equilibrium. We also discuss applications torecent issues arising in Japan, where government-based vaccination programs have terminated and vac-cination has drastically decreased.

Inertial Manifolds for SemilinearParabolic Equations Which Do NotSatisfy the Spectral Gap Condition

Anna KostiankoUniversity of Surrey, England

We report on resent results concerning the existenceof inertial manifolds for semilinear parabolic equa-tions which do not satisfy the spectral gap conditions.The list of problems includes the 3D Cahn-Hilliardproblem with periodic boundary conditions, the so-called modified Leray-↵ model which regularizes the3D Navier-Stokes equations with periodic boundaryconditions and 1D reaction-di↵usion advection prob-lems.

On a Homogeneous Evolution Equa-tion: Lax Pair and Peakon Solutions

Priscila Leal da SilvaUniversidade Federal do ABC, Brazil

In this talk we discuss integrability and peakon solu-tions of a family of homogeneous evolution equations.Recursion operators are obtained and two membersrelated to KdV-type equations are shown to be com-pletely integrable using a Lax representation and theexistence of an infinite number of conserved quanti-ties. Conditions for the existence of peakon solutionsare then given.

Dark Soliton Linearization of the 1DGross-Pitaevskii Equation: Long-Time Dynamics

Numann MalikBrown University, USA

Consider the nonlinear Schrodinger equation

i@tu(t, x) + @2

xu(t, x)� 2u(t, x)(|u(t, x)|2 � 1) = 0

in spatial dimension one, subject to the non-vanishing boundary conditions |u(t, x)| ! 1as |x| ! 1. Linearizing around the black soli-ton q(x) = tanh(x) yields an evolution equationi@t ~w = L~w +O(w2) where w = w(t, x) is a complexvalued perturbation and

L =

�@2

x + 2 2�2 @2

x � 2

�+

�4 �22 4

�sech2(x).

We describe the long-time dynamics by carrying outa spectral out a spectral and Fourier analysis of Lto derive a formula for the propagator e�itL usingthe resolvent kernel. This is performed via a natu-ral distorted Fourier transform obtained from explicitsquared Jost solutions f to Lf = Ef.

Global Existence and AsymptoticStability of Solutions to a Two-Species Chemotaxis System

Masaaki MizukamiTokyo University of Science, Japan

Nowadays, there are a lot of lives in the high civ-ilization developed by mathematics, physics, chem-istry, biology, and so on. One of the things whichplay an important role in their lives is chemotaxis.Chemotaxis is the directed movement in response toa chemoattractant. This talk is concerned with a two-species chemotaxis system with logistic source. Thissystem describes a situation in which multi popula-tions react on single chemoattractant. The main re-sult asserts boundedness of solutions and asymptotic

STUDENT PAPER COMPETITION SESSION 427

stability for the system with “any” chemical di↵u-sion. This result improves the previous results for“slow” or “non”-di↵usive chemoattractant by Negre-anu and Tello in 2014, 2015. This work will be thefirst step to study a two-species chemotaxis systemwith Lotka-Volterra competitive interaction.

Slow Motion for the One-Dimensional Swift-HohenbergEquation

Matteo RinaldiCarnegie Mellon University, USA

The behavior of solutions of the Swift–Hohenbergequation in a bounded interval I ⇢ R with periodicboundary conditions is studied. Combining resultsfrom �–convergence and ODE theory it is shown thatsolutions that start L1–close to a jump function v, re-main close to v. This can be achieved by regardingthe equation as the L2–gradient flow of a given en-ergy functional and studying the asymptotic behav-ior of solutions of its Euler–Lagrange equation. Thelinearization of such equation provides almost sharpestimates on the tail of the associated energy.

Reaction-Di↵usion Approximationfor Understanding Pattern Forma-tions Through Non-Local Interac-tions

Yoshitaro TanakaMeiji University, Japan

Motivated by the problem of studying the relation-ship between non-local interaction and local dynam-ics in pattern formation, we analyze the followingmathematical model with the Mexican-hat interac-tion:⇢ut = duuxx + (J ⇤ u)u+ f(u), in T⇥ {t > 0},u(x, 0) = u

0

(x), on T,

where u(x, t) is the theoretical concentration at theposition x at time t, T = [�L,L] with periodic condi-tions, J 2 L1(T) is a kernel satisfying

RT J(x)dx = 0,

f is a C1 function from R to R, and du > 0 is a dif-fusion coe�cient.

Unified Weighted Poincare Inequal-ities in Metric Measure Space andApplications

Huiju WangNorthwestern Polytechnical University, China

In this paper we establish unified weighted Poincareinequalities in metric measure spaces. As applica-tions, we obtain weighted higher order Poincare in-equalities in the Euclidean space and stratified Liegroups, respectively, in which a new class of higherorder Poincare inequalities in the Euclidean space isgiven.

Global Wellposedness of CubicCamassa-Holm Equations

Qingtian ZhangPenn State University, USA

Cubic Camassa-Holm equation is a model for shallowwater dynamic. I will discuss the Cauchy problem ofcubic Camassa-Holm equation. We prove the globalexistence of entropy weak solution to this problem inspace H1 with its derivative in BV. The stability anduniqueness of entropy weak solution are obtained inW 1,1.

List of Contributors

Abdelrazec, Ahmed (SS 37), 132Abe, Ken (SS 61), 208Abernathy, Kristen (SS 29), 105Abernathy, Zach (SS 29), 105Abram, William (SS 44), 153Abramov, Rafail (SS 19), 74Abudaib, Mufid (SS 57), 189Acosta, Sebastian (SS 62), 212Adak, Sukla (CS 1), 394Adami, Riccardo (SS 33), 119Adams, Ronald (SS 81), 274Adem, Abdullahi (SS 57), 189Afzal, Muhammad (CS 6), 419Agam, Oded (SS 53), 181Aguirre, Pablo (SS 120), 388Agusto, F (SS 47), 163Ahmed, Mamoon (SS 52), 178Ahmed, Nasiruddin (SS 72), 244Ahmed, Sameed (PS), 421Ahmen, Md Salik (SS 32), 115Ahn, Inkyung (SS 57), 189Aiki, Toyohiko (SS 78), 263Airey, Dylan (SS 44), 153Aissiouene, Nora (SS 84), 284Akbar, Noreen (SS 57), 189Akbar, Noreen (SS 97), 323Akhunov, Timur (SS 38), 136Alacam, Deniz (SS 85), 286Alhalawa, Muna Abu (CS 6), 419Ali, Yasir (SS 97), 323Ali, Zakaria (SS 57), 189Almeida, Joao (SS 22), 85Alotibi, Manal (SS 108), 361Alvin, Lori (SS 44), 153Alvin, Lori (SS 80), 270Ambartsoumian, Gaik (SS 62), 212Anco, Stephen (SS 57), 190Anco, Stephen (SS 93), 309Anderson, Alexander (SS 59), 199Anderson, Patrick (SS 103), 344Antonelli, Paolo (SS 56), 186Anwar, Mohamed-Naim (PS), 421Anwar, Mohamed-Naim (SS 88), 297Arbogast, Todd (SS 108), 361Archibald, Rick (SS 114), 377Arciero, Julia (SS 85), 287Arioli, Gianni (SS 81), 274Aristotelous, Andreas (SS 67), 229Arnesen, Mathias (CS 2), 399Arnold, Anton (SS 76), 257Aronna, Maria Soledad (CS 3), 406Aronna, Maria Soledad (SS 72), 244Arredondo, John (SS 31), 112Asfaw, Te↵era (SS 113), 375Astudillo, Maria (SS 43), 149Athanassoulis, Agissilaos (SS 69), 237Athanassov, Zhivko (SS 14), 54

Attouchi, Amal (SS 65), 223Auzinger, Winfried (SS 76), 257Avalos, George (SS 25), 93Avalos, George (SS 35), 126Aydogmus, Ozgur (SS 96), 319Azimi, Mahdi (SS 74), 250Aziz, Asim (SS 57), 190Aziz, Taha (SS 57), 190

Bachir, Mohammed (SS 72), 244Bae, Hyeong-Ohk (SS 61), 208Bae, Hyeong-Ohk (SS 95), 316Bae, Myoungjean (SS 28), 101Bae, Soohyun (SS 73), 248Bae, Soohyun (SS 95), 316Bagci, Mahmut (CS 2), 399Bahiraie, Alireza (CS 3), 406Bai, Xueli (SS 104), 350Baigent, Stephen (SS 15), 58Bakhtin, Yuri (SS 41), 143Bakirci, Murat (PS), 421Bakker, Lennard (SS 31), 112Ban, Jung-Chao (SS 10), 42Banisch, Ralf (SS 103), 344Barbagallo, Annamaria (SS 52), 178Barchiesi, Marco (SS 8), 36Barge, Hector (CS 1), 394Barker, Blake (SS 81), 274Barnsley, Michael (SS 44), 153Barnsley, Michael (SS 80), 270Barza, Sorina (SS 52), 178Basanta, David (SS 101), 337Basanta, David (SS 66), 225Basodi, Sunitha (PS), 422Baustian, Falko (SS 16), 62Baustian, Falko (SS 29), 105Beheshti, Shabnam (SS 94), 313Bell, Jonathan (SS 85), 287Bell, Jonathan (SS 87), 295Benevieri, Pierluigi (SS 54), 184Benevsova, Barbora (SS 68), 233Benfield, Michael (SS 111), 371Bengochea, Abimael (SS 31), 112Benson, Brian (SS 17), 68Benzoni-Gavage, Sylvie (SS 33), 119Berezovski, Mihhail (SS 78), 263Berry, Tyrus (SS 88), 297Berz, Martin (SS 81), 274Bessaih, Hakima (SS 64), 219Best, Janet (SS 85), 287Bhatt, Ashish (SS 76), 257Bhattacharya, Souvik (SS 51), 174Bhattarai, Santosh (SS 58), 196Birnir, Bjorn (SS 75), 254Birnir, Bjorn (SS 79), 268Biswas, Animikh (SS 9), 39Blessing, David (SS 89), 300

428

LIST OF CONTRIBUTORS 429

Bloshanskaya, Lidia (SS 108), 361Blot, Joel (SS 72), 244Bociu, Lorena (SS 111), 371Bociu, Lorena (SS 43), 149Bognar, Gabriella (SS 63), 215Bona, Jerry (SS 1), 8Bona, Jerry (SS 38), 136Bona, Jerry (SS 4), 24Bonacini, Marco (SS 27), 98Bonetto, Federico (SS 44), 153Borchering, Rebecca (PS), 422Borisyuk, Alla (SS 101), 337Boronski, Jan (SS 44), 153Boronski, Jan (SS 80), 270Bose, Chris (SS 103), 344Botta, Vanessa (CS 3), 406Bou-Rabee, Nawaf (SS 82), 278Boullu, Lois (SS 34), 123Bourdin, Loic (CS 4), 412Boussaada, Islam (CS 4), 412Branicki, Michal (SS 88), 297Braverman, Elena (SS 5), 28Braverman, Elena (SS 77), 259Bray, William (SS 91), 303Brena-Medina, Victor F. (CS 3), 406Brian, Will (SS 44), 154Brooks, Bernard (SS 64), 219Browne, Cameron (SS 47), 163Browne, Cameron (SS 51), 174Bruell, Gabriele (SS 107), 358Bruschi, Simone (SS 60), 204Bruzon, Maria (SS 109), 365Bruzon, Maria (SS 57), 190Buckingham, Robert (SS 38), 136Buerger, Reinhard (SS 5), 28Bui-Thanh, Tan (SS 114), 377Bui-Thanh, Tan (SS 48), 165Burgos-Garcia, Jaime (SS 31), 112Burgos-Garcia, Jaime (SS 89), 300Byeon, Jaeyoung (SS 2), 12Byun, Jonghyuk (PS), 422

Cacace, Simone (SS 118), 381Cacciafesta, Federico (SS 33), 119Cacciafesta, Federico (SS 69), 237Caginalp, Carey (SS 73), 248Caginalp, Gunduz (SS 3), 19Caginalp, Gunduz (SS 43), 149Caginalp, Reggie (SS 22), 85Cagnetti, Filippo (SS 27), 98Cagnetti, Filippo (SS 8), 36Caldas, Andre (SS 83), 281Calleja, Renato (SS 64), 219Calleja, Renato (SS 89), 300Calvez, Vincent (SS 13), 50Calvez, Vincent (SS 5), 28Cameron, Maria (SS 82), 278Campos, Juan (SS 36), 129Candito, Pasquale (SS 71), 242Canic, Suncica (Plenary), 1

Cano-Casanova, Santiago (SS 16), 62Cano-Casanova, Santiago (SS 36), 129Cantrell, Robert Stephen (SS 36), 129Cantrell, Robert Stephen (SS 5), 28Canizo, Jose (SS 40), 140Cao, Xinru (SS 13), 50Capitanelli, Ra↵aela (SS 122), 391Caraballo, Tomas (SS 16), 62Caravenna, Laura (SS 56), 186Carlotto, Alessandro (SS 17), 68Carlson, Dean (SS 72), 244Carter, Paul (SS 120), 388Carter, Paul (SS 18), 71Carvalho, Alexandre (SS 60), 204Carvalho, Ana (SS 34), 123Casal, Alfonso (SS 16), 62Catellier, Remi (SS 41), 143Catrina, Florin (SS 71), 242Cavagnari, Giulia (SS 28), 101Cavalcanti, Marcelo (SS 43), 149Cavalcanti, Valeria Domingos (SS 43), 150Cebesoy, Serifenur (CS 1), 394Ceci, Shaun (SS 25), 93Celik, Emine (SS 108), 362Celik, Emine (SS 97), 323Chae, Myeongju (SS 95), 316Chamorro, Diego (SS 4), 24Chamorro, Diego (SS 91), 303Chan, David (SS 96), 319Chang, Chih-Hung (SS 10), 42Chang, Tongkeun (SS 73), 248Chang, Xiangke (SS 38), 136Chang, Xiangke (SS 93), 309Chang, Xiaoyuan (SS 49), 167Chang-Lara, Hector (SS 17), 68Charro, Fernando (SS 17), 69Chekroun, Mickael (SS 1), 8Chekroun, Mickael (SS 4), 24Chen, Geng (SS 111), 371Chen, Geng (SS 56), 186Chen, Hongqiu (SS 38), 136Chen, Hongqiu (SS 4), 25Chen, Jianqing (SS 107), 358Chen, Jing (SS 110), 368Chen, Jing (SS 96), 319Chen, Kuo-chang (SS 10), 42Chen, Kuo-Chang (SS 31), 112Chen, Linghua (CS 6), 419Chen, Linghua (SS 50), 172Chen, Qingshan (SS 4), 24Chen, Shaohua (SS 32), 115Chen, Tianlong (SS 32), 115Chen, Weitao (SS 102), 341Chen, Weitao (SS 67), 229Chen, Wenbin (SS 23), 87Chen, Yanlai (SS 102), 341Chen, Yanping (SS 6), 33Chen, Yuming (SS 37), 132Chen, Zheng (CS 4), 412Chen, Zheng (SS 102), 341


Chen, Zheng (SS 30), 109Chen, Zhi-You (SS 21), 80Cheng, Ching-Hsiao (SS 119), 385Cheng, Jingrui (SS 28), 102Cheng, Yougan (SS 59), 199Chern, Jann-Long (SS 119), 385Chern, Jann-Long (SS 95), 316Cheskidov, Alexey (SS 4), 25Chhetri, Maya (SS 29), 105Chhetri, Maya (SS 63), 215Childs, Lauren (SS 51), 174Chiodaroli, Elisabetta (SS 17), 69Chiodaroli, Elisabetta (SS 56), 186Chirilus-Bruckner, Martina (SS 33), 119Chkadua, Otar (SS 99), 333Cho, Giphil (CS 3), 407Cho, Manki (SS 113), 375Choe, Hi Jun (SS 73), 248Choe, Kwangseok (SS 95), 316Cho↵rut, Antoine (SS 56), 187Choi, Na Ri (SS 95), 317Choi, Wonhyung (SS 57), 190Chorin, Alexandre (SS 1), 8Chou, Ching-Shan (SS 101), 337Chou, Ching-Shan (SS 30), 109Choudhury, Sudipto (SS 106), 356Christen, Andres (SS 48), 165Christov, Ivan (SS 108), 362Chu, Cho-Ho (SS 94), 313Chugh, Renu (SS 44), 154Chumley, Timothy (SS 83), 281Chung, Yu-Min (SS 89), 300Ciaramella, Gabriele (CS 2), 399Cid, Jose Angel (SS 63), 215Cirant, Marco (SS 118), 381Claudel, Christian (SS 18), 71Cojocaru, Monica (SS 113), 375Cojocaru, Monica (SS 86), 293Colin, Mathieu (SS 33), 119Colin, Mathieu (SS 84), 284Collegari, Rodolfo (SS 64), 219Comerford, Mark (SS 44), 154Conus, Daniel (SS 75), 254Cook, Scott (SS 83), 281Correa, Wellington (SS 42), 147Correa, Wellington (SS 43), 150Correia, Maria (SS 83), 281Cosner, Chris (SS 16), 63Cosner, Chris (SS 5), 29Costa, David (SS 29), 106Costa, David (SS 54), 184Coville, Jerome (SS 12), 47Coville, Jerome (SS 5), 29Cox, Chris (SS 83), 282Coxe, Infeliz (SS 63), 215Craib, Christine (PS), 422Creo, Simone (SS 122), 391Crippa, Gianluca (SS 28), 102Crisan, Dan (SS 24), 90Crisan, Dan (SS 88), 297

Crismale, Vito (SS 68), 233Cristoferi, Riccardo (SS 27), 98Crooks, Elaine (SS 2), 12Crowdy, Darren (SS 38), 137Crowdy, Darren (SS 53), 181Cruz, Carlos Montalto (SS 62), 213Cui, Ming (SS 105), 353Cui, Renhao (SS 49), 167Cui, Shumo (SS 18), 71Curtis, Christopher (SS 33), 120Cyranka, Jacek (SS 81), 275

D’Aguı, Giuseppina (SS 71), 242D’Asero, Salvatore (SS 52), 178D’Onofrio, Luigi (SS 52), 179da Luz, Cleverson (SS 42), 147da Silva, Priscila (SS 57), 190da Silva, Priscila (SS 97), 324da Silva, Priscila Leal (SPC), 426da Silveira Junior, David Evangelista (PS), 423Dabas, Jaydev (SS 34), 123Dai, Mimi (SS 4), 25Dai, Mimi (SS 61), 208Dai, Wanyang (SS 24), 90Dal Maso, Gianni (SS 27), 98Dal Maso, Gianni (SS 8), 36Dambrine, Julien (SS 121), 389Dambrine, Julien (SS 4), 25Daoulatli, Moez (SS 35), 126Das, Nirmali Prabha (PS), 422Davis, Andrew (SS 48), 165Davoli, Elisa (SS 27), 98Davoli, Elisa (SS 8), 36de Bievre, Stephan (SS 33), 120de Bonis, Ida (SS 122), 391de Cristoforis, Massimo Lanza (SS 99), 334de Leenheer, Patrick (SS 15), 58de Leenheer, Patrick (SS 5), 29de Luca, Jayme (SS 77), 259de Luca, Lucia (SS 27), 99de Luca, Lucia (SS 68), 233de Oliveira, Hermenegildo (SS 108), 362de Oliveira, Valeriano (SS 72), 245de Queiroz, Olivaine (SS 54), 184de Soares Quitalo, Veronica R. A. (SS 17), 68de Teresa, Luz (SS 43), 150Deangelis, Donald (SS 5), 29Deconinck, Bernard (SS 38), 137Deconinck, Bernard (SS 45), 159Defterli, Ozlem (CS 3), 407del Portal, Francisco Ruiz (SS 80), 272Delfour, Michel (SS 35), 126delle Monache, Maria Laura (SS 18), 72Denes, Attila (SS 47), 163Denes, Attila (SS 97), 324Deng, Weihua (SS 6), 33Deng, Yiyang (SS 31), 113Denny, Diane (CS 2), 400Deren, Fulya Yoruk (CS 1), 394Dettmann, Carl (SS 44), 154


Dhawan, Andrew (CS 3), 407di Nunno, Giulia (SS 24), 90Diatta, Bassirou (SS 97), 324Diblik, Josef (SS 77), 259Diegel, Amanda (SS 23), 87Diehl, Joscha (SS 41), 143Dikko, Dauda (CS 1), 394Ding, Yuting (SS 49), 167Disconzi, Marcelo (SS 35), 126Dogan, Abdulkadir (SS 63), 215Dong, Bo (SS 102), 342Dong, Bo (SS 30), 109Doradoux, Adrien (CS 5), 415Dragoni, Federica (SS 28), 102Dragovic, Vladimir (SS 44), 154Du, Yihong (SS 2), 12Du, Yihong (SS 36), 129Duan, Lixia (SS 85), 287Dujardin, Guillaume (SS 69), 237Dunlop, Matthew (SS 114), 377Dunlop, Matthew (SS 48), 166Dutta, Tarini (CS 6), 420Duzgun, Fatma Gamze (SS 14), 54Dwivedi, Gaurav (CS 2), 400Dwivedi, Gaurav (SPC), 426Dyachenko, Sergey (SS 19), 74Dyachenko, Sergey (SS 45), 159

Edward, Julian (SS 43), 150Ehrnstrom, Mats (SS 33), 120Ehrnstrom, Mats (SS 38), 137Eisenberg, Marisa (SS 59), 200Eisenberg, Marisa (SS 66), 225el Arwadi, Toufic (SS 62), 212el Berdan, Nada (SS 65), 223Eldering, Jaap (SS 111), 371Elias, Jan (SS 2), 12Enatsu, Yoichi (SS 21), 80Enderling, Heiko (SS 66), 225Epshteyn, Yekaterina (SS 45), 159Ercole, Grey (SS 54), 184Espejo, Elio (SS 13), 50Espindola, Maria Lewtchuk (CS 2), 400Everett, Rebecca (SS 20), 78Everett, Rebecca (SS 96), 320

Fan, Guihong (SS 37), 132Fang, Jian (SS 15), 58Fang, Jian (SS 2), 13Fanhai, Zeng (SS 6), 33Farah, Luiz (SS 38), 137Fathi, Max (SS 17), 69Fatima, Aeeman (SS 57), 191Feireisl, Eduard (SS 2), 13Feireisl, Eduard (SS 61), 208Feldman, Mikhail (SS 28), 102Feltrin, Guglielmo (SS 92), 306Feng, Peng (SS 96), 320Feng, Wei (SS 12), 47Feng, Zhaosheng (SS 10), 42

Feng, Zhaosheng (SS 96), 320Ferone, Vincenzo (SS 92), 306Ferrario, Benedetta (SS 75), 254Festa, Adriano (SS 118), 381Fialho, Joao (SS 63), 216Figalli, Alessio (Plenary), 1Figalli, Alessio (SS 8), 36Figueiredo, Giovany (SS 54), 184Fonseca, Irene (Plenary), 2Fonseca, Irene (SS 27), 99Formica, Maria Rosaria (SS 65), 223Forni, Fulvio (SS 15), 58Franco, Daniel (SS 51), 175Freire, Igor (SS 57), 191Freire, Igor (SS 97), 325Frenod, Emmanuel (SS 84), 284Frenod, Emmanuel (SS 97), 325Friedman, Avner (SS 59), 200Frigon, Marlene (SS 63), 216Froyland, Gary (SS 1), 8Froyland, Gary (SS 103), 345Fujie, Kentarou (SS 13), 51Fujie, Kentarou (SS 78), 263Fujiwara, Hiroshi (SS 62), 212Fujiwara, Kazumasa (SS 14), 54Fukao, Takeshi (SS 78), 263Furno, Joanna (SS 44), 155Furtado, Marcelo (SS 54), 185Fury, Matthew (CS 2), 400

Gaiko, Valery (CS 1), 394Galanthay, Theodore (SS 5), 30Galenko, Peter (SS 3), 19Galenko, Peter (SS 53), 181Gallaher, Jill (SS 66), 226Gambino, Gaetana (SS 49), 167Gandarias, Maria (SS 109), 365Gandarias, Maria Luz (SS 57), 191Gao, Caixia (CS 4), 413Gao, Daozhou (SS 5), 30Gao, Yueyuan (SS 2), 13Garcia, Isaac (CS 1), 395Garcia-Huidobro, Marta (SS 58), 196Garrido-Atienza, Maria (SS 64), 220Gassiat, Paul (SS 41), 143Gatenby, Robert (SS 66), 226Gaudiello, Arielle (SPC), 426Gavish, Nir (SS 2), 13Gavish, Nir (SS 53), 181Gavrus, Cristian (SS 17), 69Geba, Dan (SS 38), 137Genoud, Francois (SS 45), 159Geredeli, Pelin Guven (SS 35), 126Geridonmez, Bengisen Pekmen (CS 5), 415Geridonmez, Fatih (CS 3), 408Gess, Benjamin (SS 74), 250Geyer, Anna (SS 33), 120Ghil, Michael (Plenary), 2Ghil, Michael (SS 1), 9Giannakis, Dimitrios (SS 88), 298


Gibson, Peter (SS 62), 212Gidea, Marian (SS 26), 95Gidea, Marian (SS 31), 113Gidoni, Paolo (SS 68), 233Gidoni, Paolo (SS 92), 306Gie, Gung-Min (SS 4), 25Ginder, Elliott (SS 121), 389Gkioulekas, Eleftherios (SS 4), 25Gkioulekas, Eleftherios (SS 9), 39Glotov, Dmitry (SS 16), 63Goddard II, Jerome (SS 16), 63Goddard II, Jerome (SS 29), 106Golden, Kenneth (SS 44), 155Gomes, Diogo (SS 118), 381Gomez-Castro, David (SS 16), 63Gomez-Castro, David (SS 65), 223Gomez-Serrano, Javier (SS 70), 239Goncharova, Elena (SS 72), 245Gonzalez, Jorge (SS 89), 301Goodrich, Christopher (SS 63), 216Goubet, Olivier (SS 3), 19Goudenege, Ludovic (SS 3), 19Govender, Megandhren (PS), 423Govinder, Kesh (SS 57), 191Govinder, Kesh (SS 97), 325Graber, Philip (SS 118), 381Graef, John (SS 63), 216Grammatico, Sergio (SS 118), 382Granados, Albert (SS 26), 95Grecksch, Wilfried (SS 41), 144Grecksch, Wilfried (SS 64), 220Green, Christopher (SS 70), 239Green, William (SS 91), 303Greenblatt, Michael (SS 91), 303Greenhalgh, Scott (SS 37), 133Grigorieva, Ellina (SS 72), 245Grothaus, Martin (SS 41), 144Grun, Gunther (SS 3), 20Gu, Xiang (SS 109), 365Guerra, Ignacio (SS 58), 196Gui, Guilong (SS 107), 358Guidetti, Davide (SS 97), 325Guidotti, Patrick (SS 36), 129Guirao, Juan L.G. (SS 64), 220Gulbudak, Hayriye (SS 51), 175Gulbudak, Hayriye (SS 96), 320Gumel, Abba (SS 51), 175Guo, Daniel (SS 43), 150Guo, Kanghui (SS 91), 303Guo, Qianqiao (SS 104), 350Guo, Yao (SS 86), 293Guo, Yuxiao (SS 49), 168Guo, Zhenlin (SS 23), 87Gupta, Urvashi (CS 2), 401Gupta, Urvashi (CS 5), 415Gupta, Vidushi (SS 34), 123Gwiazda, Piotr (SS 61), 208Gyllenberg, Mats (SS 15), 59

Hafstein, Sigurdur (SS 25), 93Hagen, Thomas (SS 25), 93Hagen, Thomas (SS 35), 127Hairer, Martin (Plenary), 3Hall, Eric (CS 5), 415Hall, Eric (SS 82), 278Haller, George (SS 1), 9Ham, Seheon (SS 119), 385Hamamuki, Nao (SS 28), 102Hameed, Ja↵ar Ali Shahul (CS 3), 408Hamm, Arran (SS 94), 313Hamouda, Makram (SS 4), 26Han, Dan (CS 3), 408Han, Daozhi (SS 23), 87Han, Fang (SS 85), 287Han, Jongmin (SS 21), 80Han, Xiaoying (SS 82), 278Han, Xiaoying (SS 96), 320Han, Zhiqing (SS 104), 350Hansen, Scott (SS 43), 150Hansraj, Sudan (CS 2), 401Haro, Alex (SS 89), 301Hartung, Ferenc (SS 77), 259Hashizume, Masato (SS 119), 385Haus, Emanuele (SS 69), 237Hayashi, Masayuki (SS 14), 54Hayrapetyan, Gurgen (SS 27), 99Hayrapetyan, Gurgen (SS 8), 37He, Xiaolong (SS 77), 259Hernandez, Marco (SS 9), 39Hersh, Reuben (SS 1), 9Hetzer, Georg (SS 29), 106Hetzer, Georg (SS 36), 130Hidalgo, Arturo (SS 16), 64Hilhorst, Danielle (SS 3), 20Himonas, Alex (SS 107), 358Himonas, Alex (SS 38), 137Hinow, Peter (SS 101), 338Hinow, Peter (SS 86), 293Hirn, Matthew (SS 91), 304Hirsch, Morris (SS 15), 59Hosek, Radim (SS 3), 20Hoang, Luan (SS 13), 51Hoang, Viet (SS 114), 377Hodis, Simona (CS 3), 408Hofmanova, Martina (SS 41), 144Hofmanova, Martina (SS 9), 39Holliman, Curtis (SS 38), 138Holm, Darryl (SS 1), 9Holmes, John (SS 38), 138Honda, Tatsuhiro (SS 94), 313Hong, Youngjoon (SS 2), 13Hooper, Patrick (SS 44), 155Hoshino, Gaku (SS 14), 54Hosoya, Yuhki (SS 72), 245Hristova, Yulia (SS 62), 213Hsia, Chun-Hsiung (SS 119), 386Hsia, Chun-Hsiung (SS 21), 80Hsiao, George (SS 99), 333Hsu, Sze-Bi (SS 15), 59


Hu, Changbing (SS 5), 30Hu, Qiaoyi (SS 93), 309Hu, Qingwen (SS 77), 260Hu, Yanxia (CS 1), 395Huang, Chunyan (CS 2), 401Huang, Hsin-Yuan (SS 21), 80Huang, Qiumei (SS 105), 353Huang, Sue (SS 44), 155Huang, Tao (SS 56), 187Huang, Zhejun (CS 2), 401Huanzhen, Chen (SS 105), 353Humphries, Tony (SS 77), 260Huo, Xi (SS 37), 133Huo, Xi (SS 86), 293Hupkes, Hermen Jan (CS 3), 408Hurtado, Paul (SS 5), 30Hwang, Sukjung (SS 73), 248Hyakuna, Ryosuke (SS 14), 55Hynd, Ryan (SS 17), 69

Iacopetti, Alessandro (SS 52), 179Iannizzotto, Antonio (SS 54), 185Iannizzotto, Antonio (SS 71), 242Ikeda, Hideo (SS 12), 47Ikeda, Hideo (SS 21), 81Ikeda, Kota (SS 12), 47Ioku, Norisuke (SS 119), 386Ippolito, Stephen (CS 1), 395Ippolito, Stephen (SS 89), 301Isett, Philip (SS 56), 187Iurlano, Flaviana (SS 27), 99Ivanov, Anatoli (SS 77), 260Ivey, Thomas (SS 111), 372Ivey, Thomas (SS 38), 138Izuhara, Hirofumi (SS 2), 13

Jadamba, Baasansuren (SS 113), 375Jain, Harsh (SS 20), 78Jain, Harsh (SS 66), 226James, Jason Mireles (SS 81), 276Jang, Bongsoo (SS 6), 34Jankowski, Krzysztof (SS 83), 282Jao, Casey (SS 33), 120Jaquette, Jonathan (PS), 423Jaquette, Jonathan (SS 81), 275Jenkinson, Michael (SS 19), 74Ji, Quanbao (SS 85), 288Jia, Hao (SS 117), 379Jiang, Jifa (SS 15), 59Jiang, Jifa (SS 16), 64Jiang, Weihua (SS 49), 168Jiang, Yi (SS 59), 200Jiao, Yujian (SS 6), 34Jin, Bangti (SS 105), 353Jin, Bangti (SS 6), 34Jin, Fei-Fei (SS 1), 9Jin, Haiyang (SS 13), 51Jin, Jiayin (SS 3), 20Jin, Wenlong (SS 18), 72Jing, Wenjia (SS 28), 103

Jordao, Thais (SS 91), 304Jorgensen, Palle (SS 44), 155Junge, Oliver (SS 103), 345Junxiang, Xu (SS 83), 282

Kabeya, Yoshitsugu (SS 21), 81Kahng, Byungik (CS 4), 413Kahng, Byungik (SS 83), 282Kajikiya, Ryuji (SS 54), 185Kalies, William (SS 81), 275Kalimeris, Konstantinos (SS 93), 309Kalisch, Henrik (SS 38), 138Kaminski, Yirmeyahu (SS 72), 245Kanagawa, Shuya (SS 100), 336Kanagawa, Shuya (SS 97), 325Kaneko, Yuki (SS 21), 81Kang, Hye-Won (SS 59), 200Kang, Jing (SS 107), 359Kang, Yun (SS 5), 30Kang, Yun (SS 51), 175Kano, Risei (SS 78), 264Kao, Chiu-Yen (SS 102), 342Karakoc, Fatma (CS 1), 395Karaliolios, Nikolaos (SS 83), 282Karlovich, Yuri (SS 99), 333Karolak, Aleksandra (SS 66), 226Karp, Lavi (SS 53), 182Kaschner, Scott (SS 44), 155Kaspar, David (SS 28), 103Kasti, Dinesh (SS 89), 301Katok, Anatole (Plenary), 4Kawan, Christoph (SS 103), 345Ke, Ruian (SS 101), 338Ke, Ruian (SS 110), 368Keesling, James (SS 44), 156Keesling, James (SS 80), 270Keller, Alevtina (SS 87), 295Kelly, David (SS 30), 109Kelly, David (SS 82), 278Kelly, David (SS 88), 298Kelly, Scott (SS 111), 372Kemp, Paula (CS 4), 413Kennedy, Benjamin (SS 77), 260Kennedy, Judy (SS 44), 156Kennedy, Judy (SS 80), 270Kepley, Shane (SS 89), 301Khalique, Chaudry Masood (SS 57), 191Khan, Adnan (SS 97), 325Khan, Akhtar (SS 113), 376Khan, Akhtar (SS 92), 306Khan, Farzana (SS 97), 326Khan, Muhammad (CS 2), 402Khumalo, Melusi (CS 1), 395Kieu, Chanh (SS 1), 9Kieu, Thinh (SS 108), 362Kikuchi, Hiroaki (SS 33), 121Kile, Jennifer (SS 19), 74Kim, Doyoon (SS 117), 379Kim, Eunjung (SS 66), 227Kim, Hyea Hyun (SS 73), 248


Kim, Hyejin (SS 64), 221Kim, Kwangjoong (SS 57), 191Kim, Namkwon (SS 95), 317Kim, Peter (SS 101), 338Kim, Peter (SS 59), 200Kim, Sungwhan (SS 62), 213Kim, Yangjin (SS 2), 14Kim, Yangjin (SS 66), 227Kimmerle, Sven-Joachim (SS 72), 246Kimura, Yoshifumi (SS 70), 239Kiyak, Humeyra (CS 1), 396Klobusicky, Joe (SS 19), 75Kobayashi, Ryo (SS 2), 14Koch, Hans (SS 81), 275Koeppe, Jeanette (SS 74), 250Kogan, Irina (SS 111), 372Kohr, Gabriela (SS 94), 313Kohr, Mirela (SS 99), 334Koltai, Peter (SS 103), 345Korotkevich, Alexander (SS 19), 75Korotkevich, Alexander (SS 45), 160Kosmatov, Nickolai (SS 63), 217Kostelich, Eric (SS 20), 78Kostianko, Anna (CS 2), 402Kostianko, Anna (SPC), 426Kosztolowicz, Tadeusz (SS 53), 182Kovacic, Gregor (SS 19), 75Kovacic, Gregor (SS 45), 160Kovats, Jay (SS 71), 242Kramer, Peter (SS 101), 338Kramer, Peter (SS 19), 75Kreml, Ondrej (SS 56), 187Kropielnicka, Karolina (SS 76), 257Kryven, Ivan (SS 78), 264Kuang, Yang (SS 37), 133Kuang, Yang (SS 59), 201Kublik, Catherine (SS 121), 389Kubo, Akisato (SS 14), 55Kucera, Petr (SS 61), 209Kucherenko, Tamara (SS 44), 156Kuhn, Charlotte (SS 68), 234Kumar, Rajneesh (CS 3), 409Kumazaki, Kota (SS 78), 264Kuo, Hung-Wen (CS 2), 402Kuperberg, Krystyna (SS 44), 156Kuperberg, Krystyna (SS 80), 271Kurata, Kazuhiro (SS 2), 14Kuroda, Takanori (SS 14), 55Kurosaki, Yoji (SS 14), 56Kuto, Kousuke (SS 21), 81

Labbe, Cyril (SS 41), 144Lafci, Mehtap (CS 1), 396Lai, Anna (SS 111), 372Lai, Anna (SS 122), 391Lai, Rongjie (SS 121), 389Lakoba, Taras (SS 45), 160Lam, Kei Fong (SS 3), 20Lam, King-Yeung (SS 15), 60Lam, King-yeung (SS 5), 31

Lam, Wai-Ting (PS), 423Lan, Kunquan (SS 63), 217Lanchier, Nicolas (SS 101), 338Lanchier, Nicolas (SS 96), 321Lancia, Maria (SS 122), 391Lankeit, Johannes (SS 13), 51Lankeit, Johannes (SS 78), 264Lanz, Aprillya (SS 96), 321Lauriere, Mathieu (SS 118), 382Laux, Tim (SS 17), 69Law, Kody (SS 114), 378Law, Kody (SS 75), 254Lawley, Sean (SS 101), 339Lazzaroni, Giuliano (SS 27), 99Lazzaroni, Giuliano (SS 68), 234Lazzo, Monica (SS 58), 196le Coz, Stefan (SS 69), 238Le, Dung (SS 16), 64Le, Dung (SS 32), 115Le, Trang (SS 51), 176Leach, Andrew (SS 19), 75Leander, Rachel (SS 74), 250Lee, Christina (SS 19), 76Lee, Seung-Yeop (SS 53), 182Lee, Wanho (SS 59), 201Legoll, Frederic (SS 76), 258Legoll, Frederic (SS 82), 279Leisman, Katelyn (SS 19), 76Leonardi, Filippo (SS 28), 103Leoni, Giovanni (SS 27), 99Lessard, Jean-Phillipe (SS 60), 205Lessard, Jean-Phillipe (SS 81), 276Leung, Yuk (SS 94), 313Li, Bao Qin (SS 94), 314Li, Bingtuan (SS 50), 172Li, Congming (SS 11), 44Li, Fengjie (SS 11), 44Li, Fengquan (SS 104), 350Li, Fengquan (SS 65), 224Li, Jia (CS 5), 416Li, Jia (SS 34), 124Li, Jia (SS 96), 321Li, Jiajia (SS 85), 288Li, Lei (SS 94), 314Li, Tianhong (SS 10), 43Li, Wan-Tong (SS 2), 14Li, Wan-Tong (SS 50), 172Li, Wuchen (SS 82), 279Li, Xiaodi (SS 77), 261Li, Xin (SS 94), 314Li, Xue-Mei (SS 41), 144Li, Yachun (SS 42), 147Li, Yan (SS 104), 350Li, Yao (SS 110), 368Li, Yao (SS 82), 279Li, Ye (SS 104), 351Li, Yi (SS 11), 44Li, Yuxiang (SS 13), 51Liang, Jianli (SS 83), 282Liao, Shijun (SS 79), 268


Lie, Victor (SS 52), 179Lim, Chjan (SS 15), 60Lin, Chiu-Ju (CS 3), 409Lin, Guo (SS 50), 172Lin, Jessica (SS 28), 103Lin, Jiaming (CS 3), 409Lin, Junshan (SS 98), 331Lin, Kevin (SS 1), 10Lin, Kevin (SS 82), 279Lin, Runchang (SS 102), 342Lin, Wei (SS 86), 294Lin, Ying-Chieh (SS 21), 81Lipscomb, Nikolai (CS 5), 416Lipshutz, David (CS 1), 396Lischke, Anna (SS 105), 353Liu, Bingchen (SS 32), 115Liu, Chunyong (CS 2), 402Liu, Di (SS 82), 279Liu, Honghu (SS 1), 10Liu, Honghu (SS 9), 40Liu, Jinjie (CS 5), 416Liu, Liu (SS 93), 310Liu, Ping (SS 109), 365Liu, Shibo (SS 92), 307Liu, Xiaochuan (SS 107), 359Liu, Xinfeng (SS 105), 354Liu, Xinfeng (SS 67), 229Liu, Xueyan (SS 63), 217Liu, Yansheng (SS 104), 351Liu, Yingjie (SS 105), 354Liu, Yu-Yu (SS 11), 44Liu, Yuan (SS 30), 110Livrea, Roberto (SS 71), 242Loginova, Maria (CS 5), 417Lombardo, Maria Carmela (SS 49), 168Lopez-Gomez, Julian (SS 36), 130Lorin, Emmanuel (SS 33), 121Lou, Jie (SS 86), 294Lou, Yifei (SS 98), 331Lou, Yijun (SS 77), 261Lou, Yuan (SS 15), 60Lou, Yuan (SS 5), 31Lucardesi, Ilaria (SS 27), 100Lucarini, Valerio (SS 1), 10Lucarini, Valerio (SS 103), 346Ludu, Andrei (SS 106), 356Lueddeckens, Jens (SS 74), 251Lunasin, Evelyn (SS 88), 298Luo, Lin (SS 109), 366Luo, Songting (SS 30), 110Luo, Ting (SS 107), 359Luo, Xudan (SS 94), 314Luque, Alejandro (SS 26), 95Luque, Alejandro (SS 89), 302Lushnikov, Pavel (SS 19), 76Lushnikov, Pavel (SS 45), 161Luzzatto-Fegiz, Paolo (SS 70), 239Lv, Xiang (SS 16), 64Lye, Kjetil Olsen (SS 56), 187Lyons, Je↵rey (SS 57), 192

Lyons, Je↵rey (SS 63), 217Lyons, Tony (SS 93), 310

Ma, Huanfei (SS 86), 294Ma, Pei (SS 104), 351Ma, To Fu (SS 42), 148Ma, To Fu (SS 60), 205Ma, Tongyi (SS 50), 173Ma, Wen-Xiu (SS 57), 192Ma, Wen-Xiu (SS 97), 326Madeira, Gustavo (SS 60), 205Maderna, Ezequiel (SS 26), 95Madzvamuse, Anotida (SS 16), 65Maharaj, Sunil (SS 57), 192Mahmood, Irfan (SS 109), 366Mahmoudi, Fethi (SS 54), 185Mahomed, Fazal (SS 57), 192Mahomed, Komal (SS 97), 326Mainini, Edoardo (SS 68), 234Makanda, Gilbert (SS 11), 45Makhmali, Omid (SS 111), 372Makino, Kyoko (SS 81), 276Makki, Ahmad (SS 3), 21Malik, Muhammad Yousaf (SS 97), 326Malik, Numann (SPC), 426Malik, Pradeep (CS 4), 413Malik, Tufail (SS 97), 326Mamonov, Alexander (SS 62), 213Man, Chiang Yik (CS 1), 398Man, Chiang Yik (SS 94), 315Manakova, Natalia (SS 87), 295Manasevich, Raul (SS 58), 197Mantzavinos, Dionyssios (SS 38), 138Mao, Zhiping (SS 6), 34Maraj, Ehnber (SS 97), 326Marano, Salvatore Angelo (SS 71), 243Marche, Fabien (SS 84), 285Marciniak-Czochra, Anna (SS 2), 15Marciniak-Czochra, Anna (SS 78), 264Marcus, Adam (SS 59), 201Mariconda, Carlo (SS 72), 246Marigonda, Antonio (SS 72), 246Marin-Rubio, Pedro (SS 3), 21Marin-Rubio, Pedro (SS 64), 221Marion, Martine (SS 3), 21Marion, Martine (SS 61), 209Marras, Monica (SS 13), 51Marras, Monica (SS 32), 116Martcheva, Maia (SS 110), 368Martcheva, Maia (SS 96), 321Martin, Joerg (SS 41), 144Martin, Pau (SS 26), 96Martin, Pau (SS 31), 113Martins, Paulo (CS 3), 409Marzuola, Jeremy (SS 33), 121Masahiko, Shimojo (SS 12), 48Maspero, Alberto (SS 69), 238Massaccesi, Annalisa (SS 28), 103Massetti, Jessica Elisa (SS 26), 96Matsue, Kaname (SS 81), 276


Matsue, Kaname (SS 89), 302Matsuzawa, Hiroshi (SS 12), 48Matsuzawa, Hiroshi (SS 21), 82Matzavinos, Anastasios (SS 101), 339Maurelli, Mario (SS 41), 145Mavinga, Nsoki (SS 29), 106Mayer, John (SS 44), 156Mayer, John (SS 80), 271Maza, Susanna (CS 1), 396Mazzocco, Pauline (SS 40), 140Mckibben, Mark (SS 74), 251Mckinley, Scott (SS 101), 339McOwen, Robert (SS 117), 379Meddaugh, Jonathan (SS 80), 271Mederski, Jaroslaw (SS 104), 351Medjo, Theodore Tachim (SS 43), 152Medkova, Dagmar (SS 99), 334Meir, Amnon (SS 16), 65Melo, Cesar Hernandez (SS 60), 204Mena, Hermann (CS 4), 413Merker, Jochen (SS 16), 65Merker, Jochen (SS 65), 224Meskhi, Alexander (SS 52), 179Messaoudi, Salim (SS 43), 151Messias, Marcelo (CS 1), 396Messias, Marcelo (PS), 424Meszaros, Alpar (SS 118), 382Metzger, Stefan (SS 3), 21Metzler, Ralf (SS 53), 182Mihalia, Cornelia (SS 17), 70Mikhailov, Sergey (SS 99), 335Miller, Judith (SS 5), 31Milliken, Evan (SS 51), 176Milner, Fabio (SS 20), 78Min, Chohong (SS 73), 249Ming, Ju (SS 30), 110Minhos, Feliz (SS 63), 217Miranville, Alain (SS 2), 15Miranville, Alain (SS 43), 151Miroshnikov, Alexey (SS 111), 373Mitake, Hiroyoshi (SS 28), 104Mitrea, Dorina (SS 91), 304Mitrea, Dorina (SS 99), 335Mitrea, Marius (SS 91), 304Mitrea, Marius (SS 99), 335Miyagaki, Olimpio (SS 54), 185Miyaji, Tomoyuki (CS 6), 420Miyoshi, Hironari (SS 21), 82Mizukami, Masaaki (SPC), 426Mizukami, Masaaki (SS 78), 264Molati, Motlatsi (SS 57), 192Monobe, Harunori (SS 12), 48Montalto, Riccardo (SS 69), 238Montefusco, Francesco (SS 85), 288Moon, Byungsoo (SS 32), 116Mooney, Connor (SS 17), 70Moore, Brian (SS 76), 258Moore, Brian (SS 97), 327Mora, Jose Galaz (SS 84), 284Morales-Rodrigo, Cristian (SS 13), 52

Morandotti, Marco (SS 8), 37Moreles, Miguel (SS 48), 166Mori, Tatsuki (SS 21), 82Morita, Yoshihisa (SS 21), 82Morris, Quinn (SS 29), 106Moschini, Luisa (SS 122), 392Mosco, Umberto (SS 122), 392Mothibi, Dimpho (SS 57), 192Motta, Monica (SS 72), 246Mousa, Abdelrahim (SS 22), 85Mousa, Abdelrahim (SS 24), 90Mozgunov, Evgeny (SS 97), 327Muatjetjeja, Ben (SS 57), 192Muha, Boris (SS 35), 127Muite, Benson (SS 76), 258Muneepeerakul, Rachata (SS 96), 322Munoz, Claudio (SS 33), 121Munoz, Claudio (SS 69), 238Muntean, Adrian (SS 78), 265Munyakazi, Justin B. (SS 97), 327Murai, Minoru (SS 21), 82Murakawa, Hideki (SS 2), 15Murase, Yusuke (SS 78), 265Murray, Maxime (SS 89), 302Murray, Rua (SS 103), 346Murzabekova, Gulden (SS 43), 151Mydlarczyk, Wojciech (CS 1), 397

Naeem, Imran (SS 97), 327Nagayama, Masaharu (SS 21), 82Nagy, John (SS 20), 79Nakamura, Ken-Ichi (SS 2), 15Nakamura, Tohru (SS 32), 116Nakayashiki, Ryota (SS 3), 22Naoki, Honda (SS 59), 201Nara, Mitsunori (SS 12), 48Nara, Mitsunori (SS 2), 15Narayanan, Ranga (SS 53), 182Natali, Fabio (SS 33), 121Natali, Fabio (SS 60), 205Naudot, Vincent (SS 89), 302Naz, Rehana (SS 97), 327Neamtu, Alexandra (SS 74), 251Neamtu, Alexandra (SS 75), 255Necasova, Sarka (SS 61), 209Nedeljkov, Marko (CS 2), 402Nepomnyashchy, Alexander (SS 53), 182Neugebauer, Je↵rey (SS 63), 217Neumayer, Robin (SS 119), 386Neumayer, Robin (SS 17), 70Neustupa, Jiri (SS 61), 209Nevai, Andrew (SS 5), 31Newby, Jay (SS 19), 76Newhall, Katie (SS 19), 76Newhall, Katie (SS 45), 161Ngonghala, Calistus (SS 47), 164Nguyen, Anh (SS 16), 63Nguyen, Phuoc Tai (SS 58), 197Nguyen, Phuong (SS 9), 40Nguyen, Thanh Nam (SS 2), 15


Nguyen, Truyen (SS 108), 362Nguyen, Truyen (SS 58), 197Nguyen, Xuan (SS 12), 48Ni, Wei-Ming (Plenary), 4Ninomiya, Hirokazu (SS 12), 48Nitica, Viorel (SS 44), 156Nitsch, Carlo (SS 92), 307Nitta, Takashi (SS 100), 336Niu, Ben (SS 49), 168Niu, Yi (SS 32), 116Nkashama, M. (SS 29), 106Nobili, Camilla (SS 56), 187Nodari, Simona Rota (SS 33), 122Noja, Diego (SS 33), 121Noutchie, Suares Clovis Oukouomi (SS 57), 193Novick-Cohen, Amy (SS 2), 16Novick-Cohen, Amy (SS 3), 22Novikov, Vladimir (SS 38), 138Nowak, Magdalena (SS 80), 271Nteumagne, Bienvenue Feugang (SS 97), 324Nurbekyan, Levon (SS 17), 70Nurtazina, Karlygash (SS 43), 151Ny, Jerome (SS 118), 382

Oberlack, Martin (SS 57), 193Ohtsuka, Takeshi (SS 121), 390Okabe, Takahiro (SS 61), 209Oliveira, Bruno (SS 22), 85, 86Oliveira, Lina (SS 94), 314Olivera, Christian (SS 75), 255Olshanii, Maxim (SS 94), 314Olson, Eric (SS 75), 255Ong, Kiah Wah (SS 95), 317Onitsuka, Masakazu (CS 1), 397Oprocha, Piotr (SS 80), 271Orhan, Ozlem (SS 57), 193Orlando, Gianluca (SS 68), 234Osher, Stan (Plenary), 5Oshita, Yoshihito (SS 12), 49Ott, Katharine (SS 91), 304Ott, William (SS 103), 346Ott, William (SS 44), 157Ou, Chunhua (SS 34), 124Ou, Yvonne (SS 108), 363Ouyang, Cheng (SS 75), 255Ouyang, Tiancheng (SS 31), 113Ovcharova, Nina (SS 113), 376Ovsyannikov, Ivan (SS 97), 328Ozturk, Eylem (SS 43), 151

Padberg-Gehle, Kathrin (SS 103), 346Padhi, Seshadev (SS 63), 218Padial, Juan Francisco (SS 16), 65Pal, Samares (SS 110), 369Pal, Samares (SS 5), 31Pan, Hongjing (SS 11), 45Pan, Meng (SS 85), 288Pan, Shuxia (SS 50), 173Pang, Peter (SS 11), 45Panigrahi, Saroj (SS 63), 218

Pardo, Rosa (SS 29), 107Pardo, Rosa (SS 60), 206Park, Anna (PS), 424Patel, Mainak (SS 19), 76Patidar, Kailash C. (SS 57), 193Patrizi, Stefania (SS 68), 234Patterson, Denis (SS 74), 251Pava, Jaime Angulo (SS 33), 119Pava, Jaime Angulo (SS 60), 204Pei, Long (SS 14), 56Peitz, Sebastian (SS 103), 346Peletier, Mark (SS 8), 37Peng, Yuanyuan (SS 111), 373Pennybacker, Matt (SS 19), 77Pennybacker, Matt (SS 2), 16Pereira, Marcone (SS 60), 206Perera, Kanishka (SS 71), 243Perera, Kanishka (SS 92), 307Perez, Luis Franco (SS 31), 113Perez-Chavela, Ernesto (SS 31), 114Peszynska, Malgorzata (SS 108), 363Petcu, Madalina (SS 3), 22Petronilho, Gerson (SS 38), 139Pfei↵er, Laurent (SS 118), 383Pham, Du (SS 9), 40Pham, Kim (SS 68), 235Piao, Daxiong (CS 1), 397Piccoli, Benedetto (SS 18), 72Piccoli, Benedetto (SS 28), 104Pierre, Morgan (SS 3), 22Pierre, Morgan (SS 4), 26Pimentel, Edgard (SS 118), 383Pimentel, Juliana (SS 60), 206Pinaud, Olivier (SS 75), 255Pinto, Alberto (SS 22), 86Plaza, Ramon (SS 64), 221Plechac, Petr (SS 82), 280Poh, Ai Ling Amy (SS 97), 323Pokorny, Milan (SS 61), 209Polacik, Peter (SS 2), 16Polacik, Peter (SS 60), 206Poll, Daniel (SS 101), 339Ponsiglione, Marcello (SS 27), 100Ponsiglione, Marcello (SS 8), 37Porchia, Donald (SS 110), 369Portet, Stephanie (SS 40), 140Porzio, Maria Michaela (SS 32), 116Powathil, Gibin (SS 59), 202Pozar, Norbert (SS 28), 104Procesi, Michela (SS 26), 96Promislow, Keith (SS 2), 16Promislow, Keith (SS 23), 88Prosper, Olivia (SS 51), 176Protas, Bartosz (SS 70), 240Ptashnyk, Mariya (SS 2), 16Pujo-Menjouet, Laurent (SS 34), 124Pujo-Menjouet, Laurent (SS 40), 140Pusateri, Fabio (SS 26), 96Pusuluri, Krishna (SS 120), 388Pyzza, Pamela (SS 19), 77


Qi, Jiangang (SS 93), 310Qi, Yuanwei (SS 11), 45Qi, Yuanwei (SS 2), 16Qiao, Zhijun (SS 109), 366Qiao, Zhijun (SS 38), 139Qiao, Zhonghua (SS 23), 88Quintero, Jose (SS 58), 197

Rabinovich, Vladimir (SS 99), 335Radu, Petronela (SS 35), 127Radu, Petronela (SS 43), 151Ragusa, Maria Alessandra (SS 52), 179Raines, Brian (SS 80), 271Ramanan, Kavita (SS 24), 91Rammaha, Mohammad (SS 35), 127Ran, Zheng (SS 79), 268Ran, Zheng (SS 97), 328Rasheed, Amer (SS 3), 22Rasheed, Amer (SS 97), 328Rautmann, Reimund (SS 61), 210Razafimandimby, Paul (SS 75), 256Recio, Elena (SS 109), 366Recio, Elena (SS 57), 193Reichelt, Sina (SS 2), 17Reinhardt, Christian (SS 81), 276Reitmann, Volker (SS 103), 347Remonato, Filippo (SS 4), 26Reynolds, Angela (SS 85), 289Rezaei, Human (SS 40), 141Ricca, Renzo (SS 70), 240Rihan, Fathalla (CS 3), 410Rihan, Fathalla (SS 77), 261Rinaldi, Matteo (SPC), 427Rinaldi, Matteo (SS 68), 235Rinaldi, Matteo (SS 8), 37Rindler, Filip (SS 8), 38Rivero, Felipe (SS 60), 206Roberts, Eric (PS), 424Robertson-Tessi, Mark (SS 66), 227Robinson, Stephen (SS 113), 376Robinson, Stephen (SS 29), 107Rodrigues, Miguel (SS 33), 122Rodrigues, Tatiana (SS 44), 157Rodriguez, Marisabel (SS 64), 221Rodriguez-Lopez, Salvador (SS 119), 386Rogers, Stuart (SS 111), 373Roldan, Pablo (SS 26), 96Rong, Libin (SS 110), 369Rossmanith, James (SS 30), 110Rousseau, Antoine (SS 4), 26Rousseau, Antoine (SS 84), 285Rowell, Jonathan (CS 3), 410Roychowdhury, Mrinal (SS 44), 157Roychowdhury, Mrinal (SS 80), 272Rozanova-Pierrat, Anna (SS 122), 392Rockner, Michael (SS 24), 91Rockner, Michael (SS 74), 251Rost, Gergely (SS 77), 261Ruan, Shigui (SS 110), 369Ruan, Weihua (SS 12), 49

Ruediger, Barbara (SS 41), 145Ru�no, Paulo (SS 64), 222Rupp, Florian (SS 25), 93Rutherford, Blake (SS 103), 347Rutter, Erica (SS 20), 79Ryden, David (SS 80), 272

Saal, Juergen (SS 61), 210Sabin, Julien (SS 69), 238Sadiq, Kamran (SS 62), 214Sagadeeva, Minzilia (SS 87), 295Sagara, Nobusumi (SS 72), 247Sakajo, Takashi (SS 70), 240Salceanu, Paul (SS 34), 124Salgado, Abner (SS 23), 88Salgado, Luciana (SS 83), 283Sanchez-Gabites, Jaime J. (SS 80), 272Sanjurjo, Jose (SS 80), 272Sano, Megumi (SS 119), 386Santoprete, Manuele (SS 22), 86Santos, Carlos (SS 54), 185Sanz-Alonso, Daniel (SS 88), 299Sare, Hugo Fernandez (SS 42), 147Sasaki, Toru (SS 12), 49Sastre-Gomez, Silvia (SS 93), 310Sato, Naoki (SS 78), 265Sattler, Elizabeth (SS 44), 157Saucedo, Omar (SS 51), 176Saussol, Benoit (SS 103), 347Savatorova, Viktoria (CS 2), 403Savina, Tatiana (SS 53), 183Scapellato, Andrea (SS 52), 179Scheutzow, Michael (SS 41), 145Schiattarella, Roberta (SS 52), 180Schindler, Ian (SS 71), 243Schmalfuß, Bjorn (SS 41), 145Schmalfuß, Bjorn (SS 75), 256Schmidt, Deena (SS 101), 340Schmidt, Paul (SS 58), 197Schober, Constance (SS 106), 356Schugart, Richard (SS 85), 289Schurz, Henri (SS 25), 94Schurz, Henri (SS 74), 252Schwab, Russell (SS 17), 70Schwarz, Michael (SS 19), 77Sciammetta, Angela (PS), 424Sciammetta, Angela (SS 92), 307Scott, Jacob (SS 66), 228Sedjro, Marc (SS 56), 188Seibold, Benjamin (SS 18), 72Seidman, T (SS 78), 265Seidman, Thomas (SS 2), 17Senger, Steven (SS 91), 304Sengul, Taylan (SS 4), 27Seok, Jinmyoung (SS 95), 317Sequeira, Adelia (SS 61), 210Shabeer, Muhammad (CS 2), 403Sharma, Veena (CS 2), 403Sharma, Veena (CS 5), 417Sharpe, Je↵ (SS 110), 369


She, Zhen-Su (SS 79), 268Sheils, Natalie (SS 38), 139Shen, Jie (SS 23), 88Shen, Jie (SS 6), 34Shen, Lihua (SS 11), 45Shen, Wenxian (SS 15), 60Shen, Wenxian (SS 29), 107Shen, Zhongwei (SS 16), 66Shi, Guannan (CS 2), 404Shi, Junping (SS 16), 66Shi, Junping (SS 37), 133Shi, Kehan (CS 2), 403Shi, Shengzhu (SS 6), 35Shi, Xia (SS 85), 289Shibayama, Mitsuru (SS 10), 43Shieh, Tien-Tsan (SS 21), 83Shioji, Naoki (SS 71), 243Shipman, Patrick (SS 44), 157Shirakawa, Ken (SS 78), 265Shivaji, Ratnasingham (SS 29), 107Shomberg, Joseph (SS 14), 56Shomberg, Joseph (SS 97), 328Shrauner, Barbara (SS 57), 194Shtylla, Blerta (SS 101), 340Shu, Hongying (SS 34), 124Shu, Hongying (SS 86), 294Shuai, Zhisheng (SS 5), 31Shuai, Zhisheng (SS 51), 176Silantyev, Denis (SS 19), 77Silantyev, Denis (SS 45), 161Silva, Erica de Mello (SS 97), 324Silva, Marcio Jorge (SS 42), 147Silva, Ricardo (SS 60), 207Silwal, Sharad (SS 106), 356Sim, Inbo (SS 29), 107Sim, Inbo (SS 71), 243Sindi, Suzanne (SS 40), 141Singh, Anuraj (CS 3), 410Sivasankaran, Anoop (PS), 425Sjoedin, Tomas (SS 53), 183Skalak, Zdenek (SS 61), 210Smarrazzo, Flavia (SS 32), 116Smith, Hal (Plenary), 6Smith, Leslie (SS 4), 27Smith, Michel (SS 80), 272Sobajima, Motohiro (SS 78), 266Softova, Lyoubomira (SS 58), 198Soh, Celestin Wafo (SS 97), 329Sohrab, Siavash (SS 100), 336Solombrino, Francesco (SS 68), 235Somoza, Lucıa Lopez (SS 63), 217Son, Byungjae (SS 29), 108Son, Ju Hee (SS 95), 317Song, Changming (SS 32), 117Song, Fangying (SS 6), 35Song, Kyungwoo (SS 95), 317Song, Yongli (SS 37), 134Sovrano, Elisa (SS 80), 273Sovrano, Elisa (SS 92), 307Spinolo, Laura (SS 28), 104

Spirito, Stefano (SS 56), 188Sprinkle, Jonathan (SS 18), 72Stacho, Laszlo (SS 94), 314Stankewitz, Rich (SS 44), 157Stepien, Tracy (SS 37), 134Stern, Raphael (SS 18), 73Stimming, Hans Peter (SS 76), 258Stoica, Cristina (SS 26), 96Stoica, Cristina (SS 31), 114Stolarska, Magdalena (SS 59), 202Strani, Marta (SS 122), 392Strani, Marta (SS 69), 238Strohmer, Gerhard (SS 61), 210Stumpf, Eugen (SS 18), 73Sturman, Rob (SS 103), 347Su, Jianzhong (SS 85), 289Su, Ying (SS 49), 169Suazo, Erwin (SS 93), 310Subramaniam, Pushpavanam (SS 103), 348Sukhinin, Alexey (SS 45), 161Sumi, Hiroki (SS 44), 158Sun, De-Jun (SS 79), 269Sun, Huaqing (SS 93), 310Sun, Jiebao (SS 49), 169Sun, Xiaojuan (SS 85), 290Sun, Xingping (SS 91), 305Sun, Yi (SS 67), 230Sun, Yuhua (SS 104), 351Sundar, Padmanabhan (SS 75), 256Suzuki, Masahiro (SS 32), 117Suzuki, Takashi (SS 13), 52Suzuki, Toshiyuki (SS 78), 266Svanadze, Merab (CS 2), 404Sviridyuk, Georgy (SS 87), 296Svoboda, Zdenek (SS 77), 261Swanson, Jason (SS 75), 256Swierczewska-Gwiazda, Agnieszka (SS 61), 210Swigon, David (SS 85), 290Swigon, David (SS 97), 329Sy, Mouhamadou (SS 9), 40

Tacuri, Patricia (PS), 425Taddei, Valentina (SS 92), 307Taekyoung, Kim (PS), 425Takac, Peter (SS 16), 66Takac, Peter (SS 29), 108Takada, Ryo (SS 119), 387Takahashi, Futoshi (SS 119), 387Takahashi, Hiroshi (SS 97), 329Takeuchi, Shingo (SS 21), 83Takiguchi, Takashi (SS 62), 214Takimoto, Kazuhiro (SS 36), 130Tanaka, Satoshi (SS 58), 198Tanaka, Yoshitaro (SPC), 427Tang, Chao (SS 14), 56Tang, Min (SS 67), 230Tang, Wenbo (SS 103), 348Tang, Yong-Li (SS 21), 83Taniguchi, Masaharu (SS 12), 49Tanne, Erwan (SS 68), 235


Tantet, Alexis (SS 103), 348Tao, Molei (SS 82), 280Tao, Molei (SS 9), 40Tappe, Stefan (SS 74), 252Tarfulea, Andrei (SS 2), 17Tarfulea, Nicolae (SS 57), 194Tarfulea, Nicoleta (SS 34), 124Tawiah, Bismark Owusu (CS 3), 410Tchizawa, Kiyoyuki (SS 100), 336Teboh-Ewungkem, Miranda (SS 47), 164Tebou, Louis (SS 42), 148Tegegn, Tesfalem Abate (SS 57), 194Tellini, Andrea (SS 36), 130Tellini, Andrea (SS 58), 198Tello, J. Ignacio (SS 13), 52Tello, J. Ignacio (SS 16), 66Tello, Lourdes (SS 16), 66Tello, Lourdes (SS 36), 130Telyakovskiy, Aleksey (SS 105), 354Temam, Roger (SS 1), 10Temam, Roger (SS 9), 40Tembine, Hamidou (SS 118), 383Tenali, Gnana Bhaskar (SS 63), 218Teodorescu, Razvan (SS 53), 183Teplyaev, Alexander (SS 122), 392Teplyaev, Alexander (SS 41), 145Termuer, Chaolu (SS 109), 366Thie↵eault, Jean-Luc (SS 103), 348Thi↵eault, Jean-Luc (SS 70), 240Thomas, Marita (SS 2), 17Thomas, Marita (SS 68), 236Thompson, Ryan (SS 38), 139Thuswaldner, Jorg (SS 44), 158Tian, Gang (Plenary), 6Tian, Yun (SS 89), 302Tiba, Dan (SS 35), 127Tiglay, Feride (SS 38), 139Tine, Leon (SS 40), 141Tine, Leon Matar (SS 40), 141Ting, Yung (CS 4), 414Todorov, Michail (SS 106), 356F. Tojo, F. Adrian (SS 63), 215Tokman, Cecilia (SS 103), 345Tomoeda, Kyoko (CS 2), 404Tone, Florentina (SS 23), 89Tone, Florentina (SS 43), 152Tonon, Daniela (SS 118), 383Tonon, Daniela (SS 72), 247Toppin, Kelly (SS 85), 290Torok, Andrew (SS 103), 349Totz, Nathan (SS 117), 379Toundykov, Daniel (SS 25), 94Toundykov, Daniel (SS 43), 152Tournus, Magali (SS 40), 142Touzi, Nizar (SS 24), 91Tovstolis, Alexander (SS 52), 180Tolle, Jonas (SS 41), 146Tracina, Rita (SS 57), 194Tracina, Rita (SS 97), 329Trautwein, Christoph (SS 74), 252

Tribbia, Joseph (SS 1), 10Tripathi, Jai (CS 3), 410Trombetti, Cristina (SS 92), 308Trucu, Dumitru (SS 59), 202Tsang, Chiu Yin (SS 94), 315Tsutsui, Yohei (SS 119), 387Tsuzuki, Yutaka (SS 78), 266Tulu, Thomas (CS 3), 411Tuncer, Necibe (SS 49), 169Tuncer, Necibe (SS 51), 177Turhan, Nezihe (CS 4), 414Turi, Janos (SS 25), 94Turi, Janos (SS 77), 262Turinici, Gabriel (SS 118), 383

Uchida, Shun (SS 14), 56Umezu, Kenichiro (SS 36), 130Ushijima, Takeo (SS 2), 17Usman, Muhammad (SS 57), 194

Vaidya, Naveen (SS 34), 125Vaiter, Samuel (SS 98), 331van den Berg, Jan Bouwe (SS 81), 277van Gennip, Yves (SS 121), 390van Gennip, Yves (SS 97), 329van Gorder, Robert (SS 70), 240van Meurs, Patrick (SS 68), 236Vannitsem, Stephane (SS 1), 11Varnhorn, Werner (SS 61), 211Vas, Gabriella (SS 77), 262Vasquez, Fernando Guevara (SS 62), 213Vasquez, Paula (SS 67), 230Vaughan, Benjamin (SS 37), 134Velasco, M. Victoria (SS 94), 315Velho, Roberto (SS 118), 382Velho, Roberto Machado (PS), 423Veraar, Mark (SS 41), 146Veraar, Mark (SS 74), 252Verma, Ram Baran (CS 2), 404Vernescu, Bogdan (SS 122), 393Vernier-Piro, Stella (SS 13), 52Vernier-Piro, Stella (SS 32), 117Vernole, Paola (SS 122), 393Vidal-Lopez, Alejandro (SS 60), 207Vieiro, Arturo (SS 89), 302Viens, Frederi (SS 75), 256Viglialoro, Giuseppe (SS 122), 393Viglialoro, Giuseppe (SS 13), 52Villamizar, Vianey (SS 105), 354Vishwakarma, Sumit (SS 32), 117Vivaldi, Maria (SS 122), 393Vladimirova, Natalia (SS 45), 162Voigt, Axel (SS 70), 241Vosshall, Robert (SS 74), 252Vromans, Arthur (SS 78), 266

Wadade, Hidemitsu (SS 119), 387Wadade, Hidemitsu (SS 21), 83Wakasa, Tohru (SS 21), 83Walsh, Samuel (SS 117), 379


Wang, Chong (CS 2), 404Wang, Chuncheng (SS 49), 170Wang, Chuntian (SS 64), 222Wang, Chuntian (SS 9), 41Wang, Feng-Bin (SS 37), 134Wang, Feng-Bin (SS 49), 170Wang, Hong (SS 105), 354Wang, Hong (SS 6), 35Wang, Hongbin (SS 49), 169Wang, Huiju (SPC), 427Wang, Huiju (SS 104), 352Wang, Jin (SS 102), 342Wang, Jin (SS 96), 322Wang, Jinfeng (SS 49), 170Wang, Kun (SS 83), 283Wang, Lina (SS 104), 352Wang, Min (SS 63), 218Wang, Qi (SS 102), 343Wang, Qing (SS 85), 290Wang, Shanshan (CS 5), 417Wang, Shanshan (SS 105), 355Wang, Shouhong (SS 1), 11Wang, Wendong (SS 117), 380Wang, Xiang-Sheng (SS 49), 169Wang, Xiaohui (SS 85), 290Wang, Xiaoming (SS 23), 89Wang, Xiaoming (SS 4), 27Wang, Xinjing (SS 104), 351Wang, Xueying (SS 110), 369Wang, Yan (SS 49), 170Wang, Yi (SS 15), 61Wang, Yi (SS 16), 67Wang, Ying (SS 102), 343Wang, Ying (SS 107), 359Wang, Yuliang (SS 98), 332Wang, Yunjiao (SS 85), 291Wang, Yutong (SS 42), 148Wang, Zhen (SS 93), 311Wang, Zhi-Cheng (SS 2), 18Wang, Zhi-Cheng (SS 50), 173Wang, Zhian (SS 13), 52Wang, Zhian (SS 49), 170Wang, Zhongming (SS 30), 110Watanabe, Hiroshi (SS 78), 266Watanabe, Tatsuya (SS 14), 57Webster, Justin (SS 35), 128Weinmueller, Ewa (CS 1), 397Westdickenberg, Maria (SS 8), 38Wheeler, Mary (SS 108), 363Wheeler, Miles (SS 117), 380Wheeler, Miles (SS 69), 238Wieners, Christian (SS 68), 236Wilkerson, Mary (SS 44), 158Winkert, Patrick (SS 92), 308Wise, Steven (SS 3), 23Wise, Steven (SS 67), 230Wolkowicz, Gail (SS 37), 135Wong, Ngai-Ching (SS 94), 315Woolley, Thomas (CS 3), 411Woolley, Thomas (SS 97), 329

Work, Dan (SS 18), 73Wright, J. Douglas (SS 117), 380Wrzosek, Dariusz (SS 13), 53Wu, Boying (SS 49), 171Wu, Jianhong (SS 15), 61Wu, Jianhong (SS 77), 262Wu, Leyun (SS 104), 352Wu, Sainan (SS 49), 171Wu, Yixiang (SS 51), 177

Xiang, Tian (SS 13), 53Xiao, Pengcheng (SS 85), 291Xiao, Yanyu (SS 110), 370Xiao, Yanyu (SS 34), 125Xiaofei, Cao (SS 107), 359Xie, Dexuan (SS 67), 230Xie, Qiangjun (CS 4), 414Xie, Weiqing (CS 2), 405Xie, Xiaoxia (SS 16), 67Xie, Xilin (SS 79), 269Xie, Zhifu (SS 31), 114Xie, Zhifu (SS 36), 131Xing, Jianhua (SS 59), 202Xing, Ruixiang (SS 11), 45Xing, Yulong (SS 102), 343Xing, Yulong (SS 30), 111Xu, Benlong (CS 2), 405Xu, Runzhang (SS 32), 117Xu, Shihe (SS 93), 311Xu, Xin (SS 42), 148Xu, Zhiliang (SS 105), 355Xu, Zijian (CS 3), 411Xue, Chuan (SS 67), 231Xue, Chuan (SS 85), 291Xue, Jinxin (SS 26), 97

Yagasaki, Kazuyuki (SS 120), 388Yam, Phillip (SS 118), 383Yam, Phillip (SS 24), 91Yamada, Tetsuya (SS 36), 131Yamada, Yoshio (SS 36), 131Yamazaki, Kazuo (SS 117), 380Yamazaki, Noriaki (SS 78), 266Yan, Duokui (SS 31), 114Yan, Jue (SS 102), 343Yan, Jue (SS 30), 111Yang, Chi-Ru (SS 10), 43Yang, Fan (SS 44), 158Yang, Lihong (SS 97), 330Yang, Xiaofeng (SS 67), 231Yang, Xing (SS 97), 330Yang, Xinguang (SS 42), 148Yang, Yanbing (SS 32), 117Yang, Yang (SS 102), 343Yang, Yixin (SS 93), 311Yang, Yunyun (CS 5), 417Yang, Zhuoqin (SS 85), 291Yao, Guang (SS 59), 203Yao, Song (SS 24), 91Yao, Wenjuan (CS 5), 418


Yasar, Emrullah (SS 57), 194Yayla, Sema (SS 3), 23Ye, Xiaojing (SS 98), 332Ye, Zhuan (SS 94), 315Yi, Fengqi (SS 2), 18Yin, Rong (CS 2), 405Yin, Xiaolong (SS 108), 363Yin, Zhaoyang (SS 93), 311Yip, Nung Kwan (SS 11), 46Yokota, Tomomi (SS 13), 53Yokoyama, Keita (SS 100), 336Yoo, Minha (SS 95), 318Yoon, Changwook (SS 13), 53Yorke, James (SS 44), 158Yorke, James (SS 80), 273Yoshii, Kentarou (SS 78), 267Yoshimura, Hiroaki (SS 111), 373Yoshino, Noriaki (SS 21), 84Yoshino, Noriaki (SS 78), 267You, Yuncheng (SS 64), 222Young, Robin (SS 111), 374Younis, Muhammad (PS), 425Yu, Cheng (SS 117), 380Yu, Fajun (SS 93), 311Yu, Hui (SS 30), 111Yu, Meng (SS 98), 332Yu, Xiao (SS 34), 125Yu, Xinwei (SS 117), 380Yu, Yifeng (SS 11), 46Yu, Yifeng (SS 28), 104Yuan, Juan-Ming (SS 33), 122Yuan, Rong (SS 12), 49Yue, Pengtao (SS 23), 89Yuste, Santos (SS 53), 183Yvinec, Romain (SS 40), 142

Zada, Muhammad Ibrahim Badsha (CS 2), 399Zagrebina, Sophiya (SS 87), 296Zaidi, Ali Ashher (CS 2), 405Zaliapin, Ilya (SS 1), 11Zampieri, Gaetano (SS 57), 195Zamyshlyaeva, Alyona (SS 87), 296Zaslavski, Alexander (SS 72), 247Zaytseva, Sofya (SS 86), 294Zecca, Gabriella (SS 52), 180Zehra, I↵at (SS 97), 330Zelenko, Igor (SS 72), 247Zenkov, Dmitry (SS 111), 374Zeppieri, Caterina Ida (SS 27), 100Zhang, Aijun (SS 16), 67Zhang, Aijun (SS 49), 171Zhang, Binlin (SS 32), 118Zhang, Duan (SS 108), 364Zhang, Fan (SS 5), 32Zhang, Fu (SS 25), 94Zhang, Fubao (SS 107), 359Zhang, Guannan (SS 88), 299Zhang, Honghui (SS 85), 292Zhang, Jimin (SS 49), 171Zhang, Jinshun (SS 93), 311

Zhang, Ke (SS 26), 97Zhang, Lei (SS 67), 231Zhang, Lijun (SS 57), 195Zhang, Lu (SS 119), 387Zhang, Mingji (SS 34), 125Zhang, Qiang (SS 102), 343Zhang, Qingtian (SPC), 427Zhang, Qingtian (SS 28), 104Zhang, Shuhai (SS 79), 269Zhang, Tongli (SS 37), 135Zhang, Zhengce (SS 11), 46Zhao, Dong (SS 24), 92Zhao, Guangyu (SS 104), 352Zhao, Jia (SS 23), 89Zhao, Jia (SS 67), 231Zhao, Zhizhen (SS 88), 299Zheng, Zhong (SS 108), 364Zhong, Xinghui (SS 30), 111Zhou, Haomin (SS 24), 92Zhou, Qinghua (SS 93), 312Zhou, Shouming (SS 93), 312Zhou, Yuan (SS 109), 367Zhou, Zhenyu (CS 2), 405Zhu, Chao (SS 24), 92Zhu, Meijun (SS 104), 352Zhu, Min (SS 107), 359Zhu, Yi (SS 11), 46Zhu, Yujun (SS 83), 283Zielinski, Przemyslaw (SS 74), 252Zima, Miroslawa (SS 63), 218Zou, Xingfu (SS 15), 61Zou, Xingfu (SS 49), 171Zuo, Wenjie (SS 49), 171

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ABSTRACTS...Special Session 87 Direct, Inverse and Control Problems for Di erential Equations 295...

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