31
University of Novi Sad Faculty of Sciences Department of Mathematics and Informatics Problems on Kinetic theory and PDE’s Book of Abstracts Novi Sad, 25–27 September 2014
University of Novi SadFaculty of Sciences
Department of Mathematics and Informatics
Problems on
Kinetic theory and PDE’sBook of Abstracts
Novi Sad, 25–27 September 2014
Organizers
• Department of Mathematics and Informatics, Faculty ofSciences, Novi Sad
• Department of Mechanics, Faculty of Technical Sciences,Novi Sad
• Laboratoire MAP5, Universite Paris Descartes & CNRS,Paris
Organizing Committee
• Marko Nedeljkov, Department of Mathematics and Infor-matics, Novi Sad
• Berenice Grec, Laboratoire MAP5, Universite Paris Descartes,Paris
• Milana Pavic, Department of Mathematics and Informat-ics, Novi Sad
• Srboljub Simic, Department of Mechanics, Novi Sad
Supported by
• Ministry of Education, Science and Technological Devel-opment of the Republic of Serbia
• Provincial Secretariat for Science and Technological De-velopment, Province of Vojvodina
• Centre National de la Recherche Scientifique (CNRS), France
• Bilateral project (France-Serbia) CNRS/MSTD No.25794– Fluid and kinetic models for gaseous mixtures
• PICS France-Serbia CNRS No. 06370 – Kinetic and macro-scopic modelling of gaseous mixtures
• Project ON174024 – Methods of functional and harmonicanalysis and PDEs with singularities
• Project ON174016 - Mechanics of nonlinear and dissipa-tive systems - contemporary models, analysis and appli-cations
Program
Thursday, 25 September
Morning session – DMI, Amphitheater 5
10.15-10.25 Opening
10.25-11.05 Francesco Salvarani: Transport phenomena in evo-lutionary domains
11.05-11.20 Coffee break
11.25-12.05 Jelena Aleksic: Strong traces of ultra-parabolic equa-tion via averaged traces of ultra-parabolic transport equa-tion
Afternoon session – University of Novi Sad, Central building
15.00-17.00 Milana Pavic: Mathematical modelling and anal-ysis of polyatomic gases and mixtures in the context ofkinetic theory of gases and fluid mechanics
17.00-18.00 Cocktail
20.00 Dinner
Friday, 26 September
Morning session – DMI, Amphitheater 5
09.00-09.40 Tommaso Ruggeri: Molecular Extended Thermo-dynamics of Rarefied Polyatomic Gases and Wave Veloc-ities for Increasing Number of Moments
09.40-10.20 Maria Groppi: Kinetic relaxation models for re-acting gas mixtures
10.20-10.50 Coffee break
10.50-11.30 Laurent Desvillettes: The incompressible Navier-Stokes limit of the Boltzmann equation for mixtures ofgases
11.30-12.10 Klemens Fellner: Convergence to Equilibrium fora Coagulation-Fragmentation Model with Degenerate Spa-tial Diffusion
12.30-14.00 Lunch break
Afternoon session – DMI, Amphitheater 5
14.00-14.40 Marzia Bisi: Recent advances in kinetic theory formixtures of polyatomic gases
14.40-15.20 Laurent Boudin: Kinetic modelling for respira-tory aerosols, numerical treatment
15.20-15.50 Coffee break
15.50-16.30 Valeria Ricci: About the validation of models formulticomponent systems
16.30-17.10 Dusan Zorica: Generalizations of the classical waveequation within the theory of fractional calculus
Saturday, 27 September
Morning session – DMI, Amphitheater 5
09.00-09.40 Stephane Brull: Derivation of BGK models forgas mixtures
09.40-10.20 Damir Madjarevic: Shock structure for macro-scopic multi-temperature model of binary mixtures: com-parison with kinetic models
10.20-10.50 Coffee break
10.50-11.30 Marko Nedeljkov: Shadow waves
11.30-12.10 Berenice Grec: A diffusion limit for gaseous mix-tures
12.10-12.30 Closing
Abstracts
Strong traces of ultra-parabolic equation viaaveraged traces of ultra-parabolic transport
equation
Jelena ALEKSICDepartment of Mathematics and Informatics
University of Novi Sad, Serbia
Our aim is to prove the existence of strong traces for entropysolutions to ultraparabolic equations in heterogeneous media,
ut + divx f(x, u) =
k∑i,j=1
∂2xixjbij(t, x, u), k ≤ d.
We obtain this as consequence of the following result: Namely,we prove that if ”traceability conditions” are fulfilled then aweak solution h ∈ L∞(IR+ × IRd × IR) to the ultra-parabolictransport equation
∂th+ divx (F (t, x, λ)h) =
k∑i,j=1
∂2xixj(bij(t, x, λ)h) + ∂λγ(t, x, λ),
is such that for every ρ ∈ C1c (IR), the velocity averaged quantity∫
IRh(t, x, λ) ρ(λ)dλ admits the strong L1
loc(IRd)-limit as t→ 0,
i.e. there exist h0(x, λ) ∈ L1loc(IR
d× IR) and set E ⊂ IR+ of fullmeasure such that for every ρ ∈ C1
c (IR),
L1loc(IR
d)− limt→0, t∈E
∫IR
h(t, x, λ)ρ(λ)dλ =
∫IR
h0(x, λ)ρ(λ)dλ.
Recent advances in kinetic theory for mixtures ofpolyatomic gases
Marzia BISIDepartment of Mathematics and Computer Sciences
University of Parma, Italy
It is well known that gas mixtures involved in physical ap-plications are usually composed also of polyatomic species, forinstance in simple dissociation and recombination problems orin the evolution of powders in the atmosphere. In kinetic ap-proaches, each polyatomic gas is endowed with a (discrete orcontinuous) internal energy variable, to mimic non-translationaldegrees of freedom. The mathematical properties of the Boltz-mann operator allowing energy transfer in each interaction arestill under investigation even in absence of chemical reactions(implying also transfer of mass). In this talk we present somerecent generalizations to the polyatomic frame of models of BGKtype and of hydrodynamic limits well established for monatomicgas mixtures. Since Boltzmann-like equations are quite awk-ward to deal with, a consistent BGK relaxation model is pro-posed for inert or reactive polyatomic gases, determining in aunique way parameters of the Maxwellian attractors in terms ofspecies number densities, mass velocities and temperatures; cor-rect collision invariants and Maxwellian equilibria are recovered,
as well as fulfillment of Boltzmann H-theorem. Then, we inves-tigate the asymptotic limit of the Boltzmann equations leadingto the incompressible Navier-Stokes system; analogies and dif-ferences with respect to monatomic mixtures are presented, andcontributions due to inelastic scattering are explicitly computed.
Kinetic modelling for respiratory aerosols,numerical treatment
Laurent BOUDINLaboratoire Jacques-Louis Lions
Universite Pierre et Marie Curie, Paris, France
In this talk, we first deal with the modelling and the dis-cretization of an aerosol evolving in the air, in the respirationframework, within a domain which can be fixed or moving. Themodel consists in strongly coupling a Vlasov-type equation forthe aerosol with the incompressible Navier-Stokes equations forthe air, through a drag term between aerosol and air. We alsodiscuss some basic numerical properties of the numerical codewhich was developped, and focus on the influence of the aerosolon the air flow through the drag term, which implies a stabilitycondition.
This set of works was partially funded by the ANR-08-JCJC-013-01 and ANR-10-BLAN-1119 projects of the French researchagency.
Derivation of BGK models for gas mixtures
Stephane BRULLInstitut de Mathematiques de Bordeaux
Universite Bordeaux, France
This paper is devoted to the construction of a BGK operatorfor gas mixtures. The construction is based as in introduced insome previous works on the introduction of relaxation coeficientsand a principle of minimization of the entropy under constraintsof moments. These free parameters are com pared with the freeparameters introduced in the Thermodynamics of IrreversibleProcesses approach of the Navier-Stokes system. At the endthe BGK model is proved to satisfy Fick and Newton laws.
The incompressible Navier-Stokes limit of theBoltzmann equation for mixtures of gases
Laurent DESVILLETTESCentre de Mathematiques et Leurs Applications
Ecole normale superieure de Cachan, France
We present a work in common with Marzia Bisi, in which weextend the now classical paper by Claude Bardos, Francois Golseand Dave Levermore on the formal limit from the Boltzmannequation (for one monoatomic gas) towards the incompressibleNavier-Stokes equation. Our starting point is the Boltzmannequation for a mixture of monoatomic gases, and we develop aformal asymptotics which leads to a system of incompressibleNavier-Stokes equations for a mixture which includes the diffu-sion between different species (Fick’s law), the viscosity terms,and the diffusion of temperature (Fourier’s law). The paraboliccharacter of the obtained equations is observed, and a compar-ison with the asymptotics from compressible to incompressibleequations is described.
Convergence to Equilibrium for aCoagulation-Fragmentation Model with
Degenerate Spatial Diffusion
Klemens FELLNERInstitute for Mathematics and Scientific Computing
University of Graz, Austria
We prove explicit convergence to equilibrium for a coagulation-fragmentation model with spatial diffusion. In particular, westudy the continuous-in-size Smoluchowski’s equation with con-stant coefficients. The main difficulties arise from consideringrealistic diffusion coefficients, which degenerate for large sizeclusters. The main techniques include a-priori estimates basedon the dissipation of an entropy functional, entropy entropy-dissipation estimates, moment bounds and duality methods.
A diffusion limit for gaseous mixtures
Berenice GRECLaboratoire MAP5
Universite Paris Descartes, France
Many works dealing with the derivation of macroscopic equa-tions starting from kinetic theory considered a mono-species,monatomic and ideal gas. However, common physical situa-tions can be far more intricate, e.g. multi-species mixtures. Inthe case of mixtures, it is of interest to derive the macroscopicequations -even formally- from kinetic models, in order to linkdifferent modelling levels and identify the range of validity ofthe equations.
On the other hand, the time evolution of diffusive phenom-ena for mixtures at a macroscopic level is well described bythe Maxwell-Stefan equations. The mathematical study of theMaxwell-Stefan system is, however, very recent and solid resultson the subject appeared only in the last years.
In this talk, we present the Maxwell-Stefan equations andsome of their properties, and we establish formally the rela-tionship between the kinetic description of a mixture and thediffusion phenomena governed by the Maxwell-Stefan equations.
This is a joint work with Laurent Boudin and FrancescoSalvarani.
Kinetic relaxation models for reacting gasmixtures
Maria GROPPIDepartment of Mathematics and Computer Sciences
University of Parma, Italy
Recent relaxation time-approximation models of BGK typefor the kinetic description of chemically reacting gas mixturesare briefly reviewed [1, 5]. In spite of their simplicity, their capa-bility in retaining the most significant mathematical and phys-ical properties of the Boltzmann-type kinetic equations madethem useful and tractable tools of investigation of chemical re-actions in rarefied gas dynamics. For the numerical approxima-tion of these BGK-type models, high order numerical methods,based on a semi-lagrangian formulation, have been studied andimplemented [4].
As well known, the main drawback of the BGK approachis an uncorrect prediction of transport coefficients in the con-tinuum limit. To overcome this problem, ellipsoidal (ES) BGKmodels for inert mixtures have been investigated [2,3]. Mov-ing towards this direction, in this talk we present an ES-BGKmodel for a slowly reacting binary gas mixture, which is able tocorrectly reproduce, in the hydrodynamic limit, Fick’s law fordiffusion velocities and Newton’s law for the viscous stress.
References
[1] M. Bisi, M. Groppi, G. Spiga, Kinetic Bhatnagar-Gross-Krook model for fast reactive mixtures and its hydrodynamiclimit, Phys. Rev. E 81 (2010) 036327 (pp. 1–9).
[2] S. Brull, V. Pavan, J. Schneider, Derivation of a BGK modelfor mixtures, Eur. J. Mech. B–Fluids 33 (2012) 74–86.
[3] M. Groppi, S. Monica, G. Spiga, A kinetic ellipsoidal BGKmodel for a binary gas mixture, Europhys. Lett. 96 (2011)64002 (pp. 1–6).
[4] M. Groppi, G. Russo, G. Stracquadanio, High order semila-grangian methods for the BGK equation, preprint July 2014,submitted.
[5] M. Groppi, G. Spiga, A Bhatnagar-Gross-Krook type ap-proach for chemically reacting gas mixtures, Phys. Fluids16 (2004) 4273–4284.
Shock structure for macroscopicmulti-temperature model of binary mixtures:
comparison with kinetic models
Damir MADJAREVICDepartment of Mechanics
University of Novi Sad, Serbia
The present study deals with the shock wave profiles in themacroscopic multi-temperature (MT) model of binary gaseousmixtures. For that purpose we have adopted the hyperbolicmodel developed within the framework of extended thermody-namics, with diffusion as only dissipative mechanism. Diffusiv-ity and relaxation times are taken from kinetic theory for themixture of monatomic gases. Recently, this model proved togive good agreement with experimental data in the case of he-lium and argon mixture. However, systematic study ought berestricted to shock structures propagating at speeds lower thanthe highest characteristic speed of the system. In the presentstudy we include extra dissipation to eliminate restriction on theshock speed. This allows as to compare our results with moresophisticated kinetic solutions which were computed for hypo-thetic mixtures of gases. Numerical implementation of the MTmodel is considerably simpler than the one for Boltzmann equa-tions for mixtures or the direct simulation Monte Carlo method(DSMC).
Shadow waves
Marko NEDELJKOVDepartment of Mathematics and Informatics
University of Novi Sad, Serbia
Approximate solutions to conservation law systems calledShadow Waves (SDWs) have been used in lot of situations whenclassical solutions do not exist. More precisely, they appears insituations when one expect delta function to appear in solution(”infinite concentration” of some variable). SDWs are repre-sented by piecewise constant functions for a fixed time so theyare well adopted to the following important issues: - one caneasily check an entropy condition when (semi-)convex entropy-entropy flux pair exists, and - wave interactions can be rela-tively easy investigated (simple use of the ideas from Wave FrontTracking algorithm)
The idea of this talk is to show some strong points of SDWuse, but also show some limitations (”blow up” of SDW-solutions).Finally, we will present some preliminary results about multi-dimensional cases obtained with the collaboration with MichaelOberguggenberger, Lucas Neumann, and Manas Sahoo.
Mathematical modelling and analysis ofpolyatomic gases and mixtures in the context of
kinetic theory of gases and fluid mechanics
Milana PAVICDepartment of Mathematics and Informatics
University of Novi Sad, Serbia
This talk is dedicated to the problems arising in the mathe-matical modelling of polyatomic gases, and mixtures of monatomicand polyatomic gases, in the context of the kinetic theory ofgases and fluid mechanics. The kinetic theory of gases (Boltz-mann equation and its variants) is a very active field of appliedmathematics. At the same time, continuum theories of physicshave quite similar aims and very often treat the same problemsas kinetic theory, although from a different point of view. Theissues related to their mutual relationships are rather involvedand call for the application of mathematical techniques, as wellas elaborate physical explanations of modeling problems.
Considering polyatomic gases, our aim is to derive a macro-scopic model for 14 moments starting from kinetic theory. Atthe microscopic level, one single parameter is introduced andit becomes an additional argument of the distribution func-tion that enables to recover the proper equation of state at themacroscopic level. We first propose two independent hierarchiesof the moment equations for polyatomic gases, which allow to
obtain conservation laws for mass density, momentum and to-tal energy of a gas. Such hierarchies are usually truncated atsome order. A method which provides an appropriate solutionto the closure problem when one performs such a truncationis the maximization of entropy method. We formulate a varia-tional problem for polyatomic gases, and give the solution forany number of moments. We explore in detail the physical caseof 14 moments, in which the appropriate approximative distribu-tion function yields the closed system, that is further comparedwith the model arising from extended thermodynamics. In par-ticular, we compute production terms, and obtain the explicitexpressions for relaxation times in terms of two parameters thatcan be fitted in order to obtain a correct value of the Prandtlnumber and/or temperature dependence of viscosity.
When dealing with mixtures of polyatomic gases, the hydro-dynamic approximation in which collisions between moleculesof the same component of a mixture are much more frequentthan collisions between the molecules of different components isstudied. It leads to the so-called maxwellization of a distribu-tion function: the distribution function of each species convergestowards a Maxwellian distribution function, each with its ownbulk velocity and temperature. With the help of this specifieddistribution function, balance laws for mass density, momentumand energy can be obtained for each component of the mixture,that can be compared with the multitemperature models formixtures of Eulerian fluids coming out of extended thermody-namics. In particular, if we restrict the attention to processeswhich occur in the neighborhood of the average velocity andtemperature of the mixture, the phenomenological coefficients
of extended thermodynamics can be determined from the sourceterms provided by the kinetic theory.
Regarding mixtures of monatomic gases, we discuss the dif-fusion asymptotics of the Boltzmann equations. It amounts toscale the macroscopic arguments of the distribution function -time and space position - with the help of a small parameter in-terpreted as the mean free path. This asymptotics correspondsto a slow dynamics in space and an even slower one in time.The Hilbert expansion of each distribution function yields twoequations. The first equation allows to state that the mixture isclose to equilibrium. The second equation is a linear functionalequation in the velocity variable. We prove the existence of asolution to this equation. On the one hand, when molecularmasses are equal, the techniques introduced by Grad in orderto prove the compactness of one part of the kernel can be ex-tended to the multispecies case. On the other hand, we proposea new approach based on a change of variables in velocities forthe same issue, which only holds when molecular masses aredifferent.
About the validation of models formulticomponent systems
Valeria RICCIDipartimento di Metodi e Modelli Matematici
Universita di Palermo, Italy
We shall discuss the derivation of systems of partial differ-ential equations, where hydrodynamic equations are coupled tomean field (Vlasov type) kinetic equations, as the asymptoticlimit of suitable models on a smaller scale. This kind of PDEsystems can be considered as simply modelling multicomponentflows in mixtures containing a dispersed phase, such as spraysor aerosols. In this talk we shall give an overview of resultsof various type obtained in cooperation with E. Bernard, L.Desvillettes, F. Golse.
Molecular Extended Thermodynamics of RarefiedPolyatomic Gases and Wave Velocities for
Increasing Number of Moments
Tommaso RUGGERIDepartment of MathematicsUniversity of Bologna, Italy
Molecular Extended Thermodynamics of rarefied polyatomicgases is characterized by two hierarchies of equations for mo-ments of a suitable distribution function in which the internaldegrees of freedom of a molecule is taken into account [1]. Onthe basis of physical relevance the truncation orders of the twohierarchies are proven to be not independent on each other, andthe closure procedures based on the maximum entropy princi-ple (MEP) and on the entropy principle (EP) are proven to beequivalent [2],[3].
The characteristic velocities of the emerging symmetric hy-perbolic system of differential equations are compared to thoseobtained for monatomic gases and the lower bound estimatefor the maximum equilibrium characteristic velocity establishedfor monatomic gases (characterized by only one hierarchy formoments with truncation order of moments N) by Boillat andRuggeri [4]
λE,max(N)
c0≥
√6
5
(N − 1
2
),
(c0 =
√5
3
k
mT
)
is proven to hold also for rarefied polyatomic gases indepen-dently from the degrees of freedom of a molecule [2].
References
[1] M. Pavic, T. Ruggeri and S. Simic, Physica A 392, 1302-1317 (2013).
[2] T. Arima, A. Mentrelli and T. Ruggeri, Annals of Physics,345 111-140 (2014).
[3] T. Arima, A. Mentrelli and T. Ruggeri, Rend. Lincei Mat.Appl. 25 275291 (2014).
[4] G. Boillat and T. Ruggeri, Cont. Mech. Thermodyn. 9,205–212 (1997).
Transport phenomena in evolutionary domains
Francesco SALVARANIDipartimento di matematica F. Casorati
Universita degli Studi di Pavia, Italia
We study the transport equation in a time-dependent ves-sel with absorbing boundary, in any space dimension. We firstprove existence and uniqueness, and subsequently we considerthe problem of the time-asymptotic convergence to equilibrium.We show that the convergence towards equilibrium heavily de-pends on the initial data and on the evolution law of the vessel.Subsequently, we describe a numerical strategy to simulate theproblem, based on a particle method implemented on general-purpose graphics processing units (GPGPU). We observe thatthe parallelization procedure on GPGPU allows for a markedimprovement of the performances when compared with the stan-dard approach on CPU.
Generalizations of the classical wave equationwithin the theory of fractional calculus
Dusan ZORICAMathematical Institute
Serbian Academy of Sciences and Arts, Belgrade, Serbia
dusan [email protected]
The classical wave equation, containing integer order par-tial derivatives with respect to spatial and time coordinate,can be written in the form of system of equations consistingof: equation of motion of deformable body, constitutive equa-tion (Hooke’s law) and strain. The equation of motion, beingthe consequence of the Second Newton’s law, is not general-ized. Constitutive equation describes the response of the mate-rial to the applied force and it is generalized within the theoryof fractional calculus, so that a class of linear constitutive equa-tions describing viscoelastic materials is obtained. Constitutiveequations containing fractional derivatives with respect to timeproved to successfully model hereditary properties of viscoelas-tic materials. Spatial non-locality can be modeled not only byintroducing derivatives (of integer, or fractional order) in theconstitutive equation, but also generalizing strain to a non-localstrain measure provided that it satisfies certain physical require-ment.
Laplace and Fourier transform methods are used in orderto solve system of partial differential equations of integer and
fractional order. The solution is obtained as the convolutionof initial conditions and solution kernel. Illustrative numericalexamples are presented as well.
Results are obtained in collaboration with Prof. TeodorAtanackovic, Prof. Stevan Pilipovic, dr Sanja Konjik, dr LjubicaOparnica and dr Marko Janev. The joint work of Prof. TeodorAtanackovic and Prof. Bogoljub Stankovic is also addressed.
