Recent Trends in Differential Equations:
Analysis and Discretisation Methods
November 15-17, 2010 at Bielefeld University
Organised by Etienne Emmrich (Bielefeld),Moritz Kaßmann (Bielefeld),and Petra Wittbold (Essen)
Contact
e-mail: [email protected]: +49.521.106-6498http://www.math.uni-bielefeld.de/recent2010/
Etienne Emmrich and Moritz KaßmannUniversitat BielefeldFakultat fur MathematikPostfach 10013133501 Bielefeld, Germanye-mail:[email protected]@math.uni-bielefeld.dehttp:www.math.uni-bielefeld.de/˜emmrichwww.math.uni-bielefeld.de/˜kassmann
Petra WittboldUniversitat Duisburg-Essen, Campus EssenFakultat fur Mathematik45117 Essen, Germanye-mail:[email protected]:www.uni-due.de/˜mat201/wittbold/wittbold.html
This workshop is supported by Collaborative Research Center 701 at Biele-feld University, which is funded by DFG (German Research Foundation).
The workshop focusses on mathematical approaches to nonlinear andnonlocal phenomena by partial differential and integro-differential equa-tions. The aim is to discuss the use of various methods and conceptssuch as fractional derivatives, truncation techniques, appropriate functionspaces, and stochastic perturbations.
We wish you all an inspiring workshop and a pleasant stay in Bielefeld.
Etienne EmmrichMoritz KaßmannPetra Wittbold
Contents
1 Useful Information 1
2 Program 6
3 Abstracts 8
4 List of Participants 28
1 Useful Information
The university building
The university is located in one large building. There is one main hall in themiddle connecting all parts of the building. These are labelled by letters.The main hall is on level 0 and it houses shops (books, stationaries, gro-cery), a post office, and several restaurants and coffee shops. All roomsin the university are labelled like V3-106. This means the room is in partV, 3rd floor and has number 106. For a plan of the building, see the lastpage.
Restaurants at the university
Westend, opening hours: 11am – 4pmLocated next to the swimming pool at one end of the main hall.Serves full meals but also cake and salad. Self-service.
Mensa, opening hours: 11:30am – 2pmYou must buy a receipt at the service point in front of the mensa. Fourmenus to choose every day. Salad bar on the east end, here you can alsopay in cash. Self-service.
Univarza–Restaurant, opening hours: 10am – 12pmServes full meals, but also pizza, salad, or just coffee or tea.Besides the restaurant, there is also a snack bar.
Cafeteria, opening hours: 8am – 6pmBetween the main entrance and the Mensa. Serves full meals, sandwichesand light meals as well as cakes, salad, or just coffee or tea.
There are also coffee shops on the first floor of the main hall.
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Computers and internet
There is a document in your conference folder with an accountname andpassword. Please use it to log-in in U5-139 or any other computer in thedepartment of mathematics. There is also a WLAN-account, which youcan use to connect your own computer to the internet.
Library
The university’s library is situated on the first floor, around the whole build-ing. Math books and journals are in part V1. Entrances to the library arefrom the first floor of the main hall. To go to the mathematics part use theentrances in part U or V.
Opening hours:Entrance U1: Sun 9am – 10pm, Mon – Wed 8am – 1 am.Entrance V1: Mon – Wed 9am – 4pm.
Public transport. How to find the university
The tram Stadtbahn Linie 4 connects the university to the city center.Tickets for one or four trips can be obtained from the machines at eachstation. You need tickets “Preisstufe 1”, single ticket 2.10 Euro, four tick-ets 6.80 Euro. Make sure to validate your ticket at the stamp machinesinside the tram when boarding.
From the hotel: cross the street and take the tram Stadtbahn Linie 4 to“Lohmannshof” and get off at “Universitat”.
2
From the main station: take the exit in direction to the city and crossthe street to get to the underground. Take the tram Stadtbahn Linie 4to “Lohmannshof” and get off at “Universitat”.
Get together
There will be a “Get together” in “Wernings Weinstube” on Sunday eveningstarting at 6pm. We will be there until 22:30pm.
From the hotel: Cross the street into the pedestrian area “Rathausstraße”.“Wernings Weinstube” will appear on the right-hand side at “Alter Markt”.
Alter Markt 133602 Bielefeldhttp://www.wernings-weinstube.de
Conference dinner
You are invited to attend the conference dinner on Monday evening at 7pmin the restaurant “Bernstein”.We kindly ask for a contribution of 35 Euro (20 Euro for PhD students, nofee for students).
From the university: take the tram Stadtbahn Linie 4 to the city centerand get off at “Jahnplatz”. Turn into the pedestrian area between “SportScheck” and “Bertelsmann”. Turn left directly after “Sport Scheck” and getinto the entrance with the elevator.
From the hotel: Cross the street into the pedestrian area “Rathausstraße”.Turn right at “Alter Markt” and follow the road. The elevator, which is the
3
entrance of “Bernstein” will appear on the right-hand side.
Niederwall 233602 Bielefeldhttp://www.bernstein-live.de
Reimbursement
Reasonable travel expenses will be covered for invited participants. Forthe reimbursement, we need a copy of your travel documents and yourinternational banc account. Please, have both with you then. For Germanresidents, we probably also need your tax number (due to regulations bythe university). The hotel will be paid directly by us.
How to find “Arcadia Hotel Bielefeld”(previously “Tulip Inn”)
From the main station: take the exit in direction to the city and cross thestreet to get to the underground. Take any of the trams Stadtbahn Linie 1to “Senne”, Stadtbahn Linie 2 to “Sieker”, Stadtbahn Linie 3 to “Stieghorst”or Stadtbahn Linie 4 to “Rathaus” and get off at “Rathaus”. Turn left (in thedirection of train) and cross the street.
From the university: take the tram Stadtbahn Linie 4 to the city centerand get off at “Rathaus”. Turn left (in the direction of train) and cross thestreet.
Niederwall 31-3533602 Bielefeldhttp://www.arcadia-hotellerie.com/tulip inn bielefeld
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5
2 Program
Monday and Wednesday the workshop takes place in the main building ofthe university in room V2-105. Tuesday it takes place in room V3-201.
November 14th, 2010 (Sunday)
-18:00 Arrival18:00-22:30 Get together in the restaurant Wernings Weinstube,
which is at Alter Markt. For those arriving early on Sun-day, we recommend to visit the Kunsthalle Bielefeld.
November 15th, 2010 (Monday)
09:15-09:45 Registration09:45-10:00 Opening10:00-10:40 Imbert10:40-11:20 Gwiazda11:20-12:00 Break (V3-106)12:00-12:40 Biler12:40-14:00 Lunch14:00-14:40 Ruzicka14:40-15:20 Boyaval15:20-16:00 Break (V3-106)16:00-16:40 Wroblewska16:40-17:20 Hieber
19:00 Conference Dinner
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November 16th, 2010 (Tuesday)
10:00-10:40 Karch10:40-11:20 Alexandre11:20-12:00 Break (V3-106)12:00-12:40 Coville12:40-14:00 Lunch14:00-14:40 Ostermann14:40-15:20 Karper15:20-16:00 Break (V3-106)16:00-16:40 Vovelle16:40-17:20 Hausenblas17:20-18:00 Prohl
November 17th, 2010 (Wednesday)
10:00-10:40 Vallet10:40-11:20 Azerad11:20-12:00 Break (V3-106)12:00-12:40 Murat12:40-14:00 Lunch14:00-14:40 Zacher14:40-15:20 Schwab15:20-16:00 Break (V3-106)
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3 Abstracts
(ordered alphabetically)
Radjesvarane AlexandreEcole Navale/ENSAM, France
Boltzmann equation: some qualitative properties
We report on some recent works on Boltzmann equation, with non cutoff cross sections, and in particular existence, regularization properties ...Some of these works were done in collaboration with Yoshinori Morimoto(Kyoto), Seiji Ukai (Kyoto), Chao-Jiang Xu (Wuhan and Rouen) and TongYang (Hong Kong).
8
Pascal AzeradUniversite de Montpellier II, France
About a nonlocal model for morphodynamicsP. Azerad and A. Bouharguane
We will introduce a PDE describing the morphodynamics of sand dunessheared by a fluid flow.This model involves a non local term which can be seen as a fractionaldifferential operator.The equation is able to describe both erosion and accretion phenomena.We will present mathematical results about the well posedness, violationof maximum principle, existence and instability of travelling waves. We willalso give some insight about the numerical discretization of the non localterm.
9
Piotr BilerUniwersytet Wrocławski, Poland
Barenblatt profiles for a nonlocal porous medium equation
Piotr Biler (Wrocław), Cyril Imbert (Paris IX), Grzegorz Karch (Wrocław),Regis Monneau (Paris-Est)
We study a generalization of the porous medium equation involving nonlo-cal terms.More precisely, our equation is
∂tu−∇ ·(u∇α−1
(|u|m−1
))= 0,
with α ∈ (0, 2), m > 1, and a nonlocal gradient operator in Rd defined by∇βu = F−1(iξ|ξ|β−1Fu).Explicit self-similar solutions with compact support generalizing the Baren-blatt solutions are constructed. They are of the form
u(t, x) =
(kt−
dd+α
(R2 − |x|2t−
2d+α
)α2
+
) 1m−1
with R > 0 and suitable k > 0.We also present an argument to get the Lp decay of weak solutions of theCauchy problem constructed by Caffarelli and Vazquez in the model casem = 2.Related equations appear in continuum mechanics to describe the evolu-tion of dislocations in crystals.
10
Sebastien BoyavalEcole des Ponts ParisTech, CERMICS, France
Energy-dissipative discretizations of the Oldroyd-B system
The numerical simulation of Non-Newtonian flows using the Oldroyd-Bsystem of equations e.g. is difficult. Although there is no global existencetheory, even for the simplest homogeneous Dirichlet problem without in-and out-flows, many sensible discretizations exist; but they report numeri-cal instabilities in most benchmark geometries. We suggest an analysis ofsome discretizations of the Oldroyd-B equations (our prototypical rheologi-cal model) that satisfy a discrete energy estimate similar to that for smoothsolutions of the continuous equations, using a Lyapunov functional con-trolling the long-time asymptotics.
11
Jerome CovilleINRA – Unite de Biostatistiques et Processus Spatiaux (UR546)
Existence and uniqueness of positive solution of heterogeneousreaction dispersion equation
In this talk, I will discuss the asymptotic behaviour of positive solutionof some heterogeneous integro-differential equation that have been re-cently introduced to model some pest invasion. The asymptotic behaviouris analysed through the properties (existence, uniqueness) of the station-ary solutions of the equation. We present a simple criteria of existenceand uniqueness of positive solution stationary solution similar to the oneknown for the classical reaction diffusion equation.
12
Piotr GwiazdaUniwersytetu Warszawskiego, Poland
Split-up algorithm in the metric space for the equations of structuredpopulation dynamics
The talk is based on the joint research with Jose Carillo, Rinaldo Colombo,Anna Marciniak-Czochra and Agnieszka Ulikowska. As the example ofthe structured population equations we mean the equation of so-calledage-structured model (transport equation in a half space with non-localboundary conditions) or size structured model (transport equation with anintegral term in space on the right hand side), see for more details B.Perthame “Transport equations in mathematical biology” 2007. From thebiological reason there is a need for using initial data in the space of Radonmeasures. Using the Lipschitz-bounded distance (flat metric) we proveLipschitz dependence of the solutions to linear and nonlinear system w.r.t.initial data and coefficients of equations. Significant simplifications of thecalculations is done by using the split-up algorithm, dealing separately witha semigroup of transport and a semigroup of an integral kernel operator.
13
Erika HausenblasMontanuniversitat Leoben, Austria
The numerical approximation of spdes driven by Levy process
Usually, in contrary to a Brownian motion, it is difficult to simulate a Levywalk. There exists algorithm, but most of the algorithm uses not equidis-tant time steps. In particular the time step is an exponentiall distributedrandom variable. In low dimension this fact is no drawback, but in highdimension this leads to difficulties.However, if one want to simulate the solution of an spdes driven by Levyprocess one has to deal with high dimensional Levy processes. In the talkan alogorithm to simulate a Levy walk is proposed and the rate of conver-gence is given.
It is a joint work with Thomas Dunst and Andreas Prohl.
14
Matthias HieberTU Darmstadt, Germany
Asymptotic properties and stability problemsrelated to the Ekman spiral
In this talk we consider Ekman boundary layers and in particular the Ek-man spiral which is a stationary solution of the Navier-Stokes equationsin the rotational setting. Based on the constructing a suitable weak so-lution, we discuss asymptotic and stability properties of the Ekman spiraland prove in particular asymptotic stability of the Ekman spiral for smallReynolds numbers.
15
Cyril ImbertUniversite Paris Dauphine, France
A higher order non-local equation appearing in crack dynamics
This is a joint work with Antoine Mellet (University of Maryland). Whenmodeling the propagation of an hydraulic fracture in a rock, one has todeal with a non-local version of the well known thin film equation. Theanalysis of such an equation implies difficulties and the construction ofnon-negative weak solutions is delicate. We will explain these difficuties,present existence results and discuss open problems.
16
Grzegorz KarchUniwersytet Wrocławski, Poland
Infinite energy solutions to homogeneous Boltzmann equation
The goal of this talk is to present an approach to the homogeneous Boltz-mann equation for the Maxwellian gas, which allows us to construct uniquesolutions to the initial value problem in a space (of probability measures)defined via the Fourier transform. In that space, the second moment of ameasure is not assumed to be finite, so infinite energy solutions are nota priori excluded. It is well-known that finite energy solutions of the Boltz-mann equation converge towards a stationary solution called Maxwellian.In our study of the large time asymptotics of infinite energy solutions,we discover new self-similar profiles which resemble alpha-stable distri-butions.
17
Trygve KarperNorwegian University of Science and Technology, Norway
Operator splitting for the dissipative quasi-geostrophic equation
In this talk I will discuss operator splitting for the surface quasi-geostrophicequation modeling strongly rotating atmospheric flow. The numerical al-gorithms are based on evolving the solution by alternately applying theaction of transport and diffusion. The main result is that both Godunovand Strang splitting converges with the expected orders provided the ini-tial data is sufficiently regular. The analysis can be generalized to a largeclass of well-posed active scalar equations including the Topaz-Bertozziaggregation equation and the quasi-geostrophic equation with dispersion.
This is joint work with Helge Holden and Kenneth Karlsen.
18
Francois MuratLaboratoire Jacques-Louis Lions, Univ. Paris VI, France
Renormalized solutions of second order elliptic equationswith right-hand side in L1
In this lecture, I will consider the problem: find u such that
−div(A(x)Du) = f in Ω
u = 0 on ∂Ω
when the matrix A is coercive with measurable bounded coefficients andwhen f belongs to L1(Ω).The main difficulty of the problem is to define a convenient notion of solu-tion. I will introduce the notion of renormalized solution, i.e.
Definition: u is a renormalized solution of the problem if u : Ω → R ismeasurable and a.e. finite
Tn(u) ∈ H10 (Ω) for every n > 0
1
n
∫Ω|DTn(u)|2 → 0 as n→ +∞
−div(h(u)A(x)Du)+h′(u)A(x)DuDu = h(u)f in D′(Ω) for every h ∈ C1c (R)
This definition allows one to prove that the problem has a renormalizedsolution, that this renormalized solution is unique and that it depends con-tinuously on f, i.e. that in this framework the problem is well posed in thesense of Hadamard. This definition and the result of well-posedness canbe extended in the natural way to the case of a second order monotoneoperator in divergence form posed on W 1,p
0 (Ω).
19
Alexander OstermannUniversitat Innsbruck, Austria
Exponential integrators
Exponential integrators are intended for the numerical solution of stiff dif-ferential equations. More precisely, they are designed for problems wherethe solution of the linearisation contains fast decaying (or highly oscilla-tory) components.In my talk I will focus on the construction and numerical analysis of suchintegrators. Similarities and differences to standard integrators (impliciteRunge–Kutta methods and multistep methods) will be addressed.
20
Andreas ProhlUniversitat Tubingen, Germany
Numerics of the stochastic incompressible Navier-Stokes equation
We study finite element based space-time discretizations of the incom-pressible Navier-Stokes equations with noise. In three dimensions, iteratesconstruct martingale solutions for vanishing discretization parameters. Inthe two dimensional case, iterates converge to the unique strong solution.Rates of convergence will be obtained in the 2D case with periodic bound-ary data.
This is joint work with E. Carelli (U Tubingen) and Z. Brzezniak (U York).
21
Michael RuzickaUniversitat Freiburg, Germany
Analysis of non-Newtonian fluid flows
In the talk we present recent results on the existence of weak solutionsfor the equations describing the motion of generalized Newtonian fluidsand electrorheological fluids.
22
Russell SchwabCarnegie Mellon University, USA
Periodic homogenization for nonlinear integro-differential equations
We will discuss the homogenization for viscosity solutions of a generalclass of nonlinear integro-differential equations which includes, e.g. thosearising as the Bellman-Isaacs equations from differential games and opti-mal control with pure jump processes. The appropriate notion of correctorequation and the use of an obstacle problem in the determination of theeffective equation will be presented.
23
Guy ValletUniversite de Pau et des pays de L’Adour, France
On pseudoparabolic problems of Barenblatt’s type
In this talk, we will be interested in a nonlinear pseudoparabolic problemof Barenblatt’s type. We will first present models leading to such type ofequation. Then, we will give a result of existence of a solution and derivesome applications to the equation of Barenblatt.
24
Julien VovelleUniversite Claude Bernard Lyon 1, France
Stochastic perturbation of first-order non-linearscalar conservation law
In this joint work with A. Debussche, we solve the Cauchy Problem for amulti-dimensional, first-order non-linear scalar conservation law with mul-tiplicative noise.
25
Aneta WroblewskaUniversity of Warsaw, Poland
Generalised Stokes system in Orlicz spaces
We will investigate generalised Stokes system:
∂tu− div S(t, x,Du) +∇p = f in (0, T )× Ω,
div u = 0 in (0, T )× Ω,
u(0, x) = u0 in Ω,
u(t, x) = 0 on (0, T )× ∂Ω,
where Ω ⊂ Rn open bounded set with Lipschitz boundary. We assumethat the stress tensor S is monotone and coercivity conditions are givenby convex function. To prove existence of weak solution to our equationswe will show Korn-Sobolev inequality for Orlicz spaces and the fact thatclosures of smooth compactly supported functions w.r.t. modular and weakstar topology of symmetric gradient coincides.
26
Rico ZacherUniversitat Halle, Germany
De Giorgi-Nash-Moser estimates for time fractionaldiffusion equations
We discuss several recent results on a priori estimates for weak solutionsto linear and quasilinear fractional diffusion equations of time order lessthan one (boundedness, Holder continuity and Harnack estimates).
27
4 List of ParticipantsFirst name Last name CityRadjesvarane Alexandre BrestPascal Azerad MontpellierPiotr Biler WrocławRobin Beier BielefeldWolf-Jurgen Beyn BielefeldSebastien Boyaval Marne la ValleeMario Bukal WienJerome Coville AvignonBartlomiej Dyda BielefeldMatthias Eisenmann BerlinEtienne Emmrich BielefeldMatthieu Felsinger BielefeldBarbara Gentz BielefeldPiotr Gwiazda WarsawChristopher Hartleb BielefeldErika Hausenblas LeobenLeonie Herden AachenMatthias Hieber DarmstadtCyril Imbert ParisGrzegorz Karch WrocławTrygve Karper TrondheimMoritz Kaßmann BielefeldMariko Alexa Mathuis BielefeldAnte Mimica BielefeldFrancois Murat ParisAlexander Ostermann InnsbruckAndreas Prohl TubingenDimitri Puhst BerlinMichael Ruzicka FreiburgRussell Schwab PittsburghDavid Siska BielefeldGuy Vallet PauJulien Vovelle LyonPetra Wittbold EssenAneta Wroblewska WarsawRico Zacher HalleAlexandra Zimmermann Essen
Notes
Notes
Notes
Notes
Program
Monday
(V2-105)
Tuesday(V
3-201)W
ednesday(V2-105)
09:15-09:45R
egistration09:45-10:00
Opening
10:00-10:40Im
bertK
archVallet
10:40-11:20G
wiazda
Alexandre
Azerad
11:20-12:00B
reak(V
3-106)B
reak(V
3-106)B
reak(V
3-106)12:00-12:40
Biler
Coville
Murat
12:40-14:00Lunch
LunchLunch
14:00-14:40R
uzickaO
stermann
Zacher14:40-15:20
Boyaval
Karper
Schw
ab15:20-16:00
Break
(V3-106)
Break
(V3-106)
Break
(V3-106)
16:00-16:40W
roblewska
Vovelle16:40-17:20
Hieber
Hausenblas
17:20-18:00P
rohl19:00hC
onferenceD
inner
