Recent Trends in Differential Equations: Analysis and Discretisation Methods

November 15-17, 2010 at

Organised by Etienne Emmrich (Bielefeld), Moritz Kaßmann (Bielefeld), and Petra Wittbold () Contact e-mail: [email protected] fax: +49.521.106-6498 http://www.math.uni-bielefeld.de/recent2010/

Etienne Emmrich and Moritz Kaßmann Universitat¨ Bielefeld Fakultat¨ fur¨ Mathematik Postfach 100131 33501 Bielefeld, e-mail: [email protected] [email protected] http: www.math.uni-bielefeld.de/˜emmrich www.math.uni-bielefeld.de/˜kassmann

Petra Wittbold Universitat¨ -Essen, Campus Essen Fakultat¨ fur¨ Mathematik 45117 Essen, Germany e-mail: [email protected] http: 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 and nonlocal phenomena by partial differential and integro-differential equa- tions. The aim is to discuss the use of various methods and concepts such as fractional derivatives, truncation techniques, appropriate function spaces, and stochastic perturbations.

We wish you all an inspiring workshop and a pleasant stay in Bielefeld.

Etienne Emmrich Moritz Kaßmann Petra 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 the middle 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 rooms in the university are labelled like V3-106. This means the room is in part V, 3rd floor and has number 106. For a plan of the building, see the last page.

Restaurants at the university

Westend, opening hours: 11am – 4pm Located 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 – 2pm You must buy a receipt at the service point in front of the mensa. Four menus to choose every day. Salad bar on the east end, here you can also pay in cash. Self-service.

Univarza–Restaurant, opening hours: 10am – 12pm Serves full meals, but also pizza, salad, or just coffee or tea. Besides the restaurant, there is also a snack bar.

Cafeteria, opening hours: 8am – 6pm Between the main entrance and the Mensa. Serves full meals, sandwiches and 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.

1 Computers and internet

There is a document in your conference folder with an accountname and password. Please use it to log-in in U5-139 or any other computer in the department of mathematics. There is also a WLAN-account, which you can 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 are from the first floor of the main hall. To go to the mathematics part use the entrances 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 each station. 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 machines inside 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 cross the street to get to the underground. Take the tram Stadtbahn Linie 4 to “Lohmannshof” and get off at “Universitat”.¨

Get together

There will be a “Get together” in “Wernings Weinstube” on Sunday evening starting 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 1 33602 Bielefeld http://www.wernings-weinstube.de

Conference dinner

You are invited to attend the conference dinner on Monday evening at 7pm in the restaurant “Bernstein”. We kindly ask for a contribution of 35 Euro (20 Euro for PhD students, no fee for students).

From the university: take the tram Stadtbahn Linie 4 to the city center and get off at “Jahnplatz”. Turn into the pedestrian area between “Sport Scheck” and “”. Turn left directly after “Sport Scheck” and get into 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 2 33602 Bielefeld http://www.bernstein-live.de

Reimbursement

Reasonable travel expenses will be covered for invited participants. For the reimbursement, we need a copy of your travel documents and your international banc account. Please, have both with you then. For German residents, we probably also need your tax number (due to regulations by the 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 the street to get to the underground. Take any of the trams Stadtbahn Linie 1 to “”, Stadtbahn Linie 2 to “Sieker”, Stadtbahn Linie 3 to “Stieghorst” or Stadtbahn Linie 4 to “Rathaus” and get off at “Rathaus”. Turn left (in the direction of train) and cross the street.

From the university: take the tram Stadtbahn Linie 4 to the city center and get off at “Rathaus”. Turn left (in the direction of train) and cross the street.

Niederwall 31-35 33602 Bielefeld http://www.arcadia-hotellerie.com/tulip inn bielefeld

4 5 2 Program

Monday and Wednesday the workshop takes place in the main building of the university in room V2-105. Tuesday it takes place in room V3-201.

November 14th, 2010 (Sunday)

-18:00 Arrival 18:00-22:30 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 Registration 09:45-10:00 Opening 10:00-10:40 Imbert 10:40-11:20 Gwiazda 11:20-12:00 Break (V3-106) 12:00-12:40 Biler 12:40-14:00 Lunch 14:00-14:40 Ru˚ziˇ ckaˇ 14:40-15:20 Boyaval 15:20-16:00 Break (V3-106) 16:00-16:40 Wroblewska´ 16:40-17:20 Hieber 19:00 Conference Dinner

6 November 16th, 2010 (Tuesday)

10:00-10:40 Karch 10:40-11:20 Alexandre 11:20-12:00 Break (V3-106) 12:00-12:40 Coville 12:40-14:00 Lunch 14:00-14:40 Ostermann 14:40-15:20 Karper 15:20-16:00 Break (V3-106) 16:00-16:40 Vovelle 16:40-17:20 Hausenblas 17:20-18:00 Prohl

November 17th, 2010 (Wednesday)

10:00-10:40 Vallet 10:40-11:20 Azerad 11:20-12:00 Break (V3-106) 12:00-12:40 Murat 12:40-14:00 Lunch 14:00-14:40 Zacher 14:40-15:20 Schwab 15:20-16:00 Break (V3-106)

7 3 Abstracts

(ordered alphabetically)

Radjesvarane Alexandre Ecole Navale/ENSAM, France

Boltzmann equation: some qualitative properties

We report on some recent works on Boltzmann equation, with non cut off 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 Tong Yang (Hong Kong).

8 Pascal Azerad Universite´ de Montpellier II, France

About a nonlocal model for morphodynamics P. Azerad and A. Bouharguane

We will introduce a PDE describing the morphodynamics of sand dunes sheared by a fluid flow. This model involves a non local term which can be seen as a fractional differential operator. The equation is able to describe both erosion and accretion phenomena. We will present mathematical results about the well posedness, violation of maximum principle, existence and instability of travelling waves. We will also give some insight about the numerical discretization of the non local term.

9 Piotr Biler Uniwersytet Wrocławski,

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

α−1 m−1 ∂tu − ∇ · u∇ |u| = 0, d with α ∈ (0, 2), m > 1, and a nonlocal gradient operator in R 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

1 α   m−1 − d  2 2 − 2  2 u(t, x) = kt d+α R − |x| t d+α + with R > 0 and suitable k > 0. We also present an argument to get the Lp decay of weak solutions of the Cauchy problem constructed by Caffarelli and Vazquez´ in the model case m = 2. Related equations appear in continuum mechanics to describe the evolu- tion of dislocations in crystals.

10 Sebastien´ Boyaval Ecole des Ponts ParisTech, CERMICS, France

Energy-dissipative discretizations of the Oldroyd-B system

The numerical simulation of Non-Newtonian flows using the Oldroyd-B system of equations e.g. is difficult. Although there is no global existence theory, 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 of some discretizations of the Oldroyd-B equations (our prototypical rheologi- cal model) that satisfy a discrete energy estimate similar to that for smooth solutions of the continuous equations, using a Lyapunov functional con- trolling the long-time asymptotics.

11 Jerome Coville INRA – Unite´ de Biostatistiques et Processus Spatiaux (UR546)

Existence and uniqueness of positive solution of heterogeneous reaction dispersion equation

In this talk, I will discuss the asymptotic behaviour of positive solution of some heterogeneous integro-differential equation that have been re- cently introduced to model some pest invasion. The asymptotic behaviour is analysed through the properties (existence, uniqueness) of the station- ary solutions of the equation. We present a simple criteria of existence and uniqueness of positive solution stationary solution similar to the one known for the classical reaction diffusion equation.

12 Piotr Gwiazda Uniwersytetu Warszawskiego, Poland

Split-up algorithm in the metric space for the equations of structured population dynamics

The talk is based on the joint research with Jose Carillo, Rinaldo Colombo, Anna Marciniak-Czochra and Agnieszka Ulikowska. As the example of the structured population equations we mean the equation of so-called age-structured model (transport equation in a half space with non-local boundary conditions) or size structured model (transport equation with an integral term in space on the right hand side), see for more details B. Perthame “Transport equations in mathematical biology” 2007. From the biological reason there is a need for using initial data in the space of Radon measures. Using the Lipschitz-bounded distance (flat metric) we prove Lipschitz dependence of the solutions to linear and nonlinear system w.r.t. initial data and coefficients of equations. Significant simplifications of the calculations is done by using the split-up algorithm, dealing separately with a semigroup of transport and a semigroup of an integral kernel operator.

13 Erika Hausenblas Montanuniversitat¨ Leoben, Austria

The numerical approximation of spdes driven by Levy process

Usually, in contrary to a Brownian motion, it is difficult to simulate a Levy walk. There exists algorithm, but most of the algorithm uses not equidis- tant time steps. In particular the time step is an exponentiall distributed random variable. In low dimension this fact is no drawback, but in high dimension this leads to difficulties. However, if one want to simulate the solution of an spdes driven by Levy process one has to deal with high dimensional Levy processes. In the talk an 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 Hieber TU , Germany

Asymptotic properties and stability problems related 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 equations in the rotational setting. Based on the constructing a suitable weak so- lution, we discuss asymptotic and stability properties of the Ekman spiral and prove in particular asymptotic stability of the Ekman spiral for small Reynolds numbers.

15 Cyril Imbert Universite´ Paris Dauphine, France

A higher order non-local equation appearing in crack dynamics

This is a joint work with Antoine Mellet (University of Maryland). When modeling the propagation of an hydraulic fracture in a rock, one has to deal with a non-local version of the well known thin film equation. The analysis of such an equation implies difficulties and the construction of non-negative weak solutions is delicate. We will explain these difficuties, present existence results and discuss open problems.

16 Grzegorz Karch Uniwersytet 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 unique solutions to the initial value problem in a space (of probability measures) defined via the Fourier transform. In that space, the second moment of a measure is not assumed to be finite, so infinite energy solutions are not a 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 Karper Norwegian 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-geostrophic equation modeling strongly rotating atmospheric flow. The numerical al- gorithms are based on evolving the solution by alternately applying the action of transport and diffusion. The main result is that both Godunov and Strang splitting converges with the expected orders provided the ini- tial data is sufficiently regular. The analysis can be generalized to a large class of well-posed active scalar equations including the Topaz-Bertozzi aggregation equation and the quasi-geostrophic equation with dispersion.

This is joint work with Helge Holden and Kenneth Karlsen.

18 Franc¸ois Murat Laboratoire Jacques-Louis Lions, Univ. Paris VI, France

Renormalized solutions of second order elliptic equations with 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 and when 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 is measurable and a.e. finite

1 Tn(u) ∈ H0 (Ω) for every n > 0 Z 1 2 |DTn(u)| → 0 as n → +∞ n Ω 0 0 1 − div(h(u)A(x)Du)+h (u)A(x)DuDu = h(u)f in D (Ω) for every h ∈ Cc (R) This definition allows one to prove that the problem has a renormalized solution, 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 the sense of Hadamard. This definition and the result of well-posedness can be extended in the natural way to the case of a second order monotone 1,p operator in divergence form posed on W0 (Ω).

19 Alexander Ostermann Universitat¨ Innsbruck, Austria

Exponential integrators

Exponential integrators are intended for the numerical solution of stiff dif- ferential equations. More precisely, they are designed for problems where the 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 such integrators. Similarities and differences to standard integrators (implicite Runge–Kutta methods and multistep methods) will be addressed.

20 Andreas Prohl Universitat¨ 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, iterates construct martingale solutions for vanishing discretization parameters. In the 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 R ˚uˇzickaˇ Universitat¨ Freiburg, Germany

Analysis of non-Newtonian fluid flows

In the talk we present recent results on the existence of weak solutions for the equations describing the motion of generalized Newtonian fluids and electrorheological fluids.

22 Russell Schwab Carnegie Mellon University, USA

Periodic homogenization for nonlinear integro-differential equations

We will discuss the homogenization for viscosity solutions of a general class of nonlinear integro-differential equations which includes, e.g. those arising as the Bellman-Isaacs equations from differential games and opti- mal control with pure jump processes. The appropriate notion of corrector equation and the use of an obstacle problem in the determination of the effective equation will be presented.

23 Guy Vallet Universite´ 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 problem of Barenblatt’s type. We will first present models leading to such type of equation. Then, we will give a result of existence of a solution and derive some applications to the equation of Barenblatt.

24 Julien Vovelle Universite´ Claude Bernard Lyon 1, France

Stochastic perturbation of first-order non-linear scalar conservation law

In this joint work with A. Debussche, we solve the Cauchy Problem for a multi-dimensional, first-order non-linear scalar conservation law with mul- tiplicative noise.

25 Aneta Wroblewska´ University 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, n where Ω ⊂ R open bounded set with Lipschitz boundary. We assume that the stress tensor S is monotone and coercivity conditions are given by convex function. To prove existence of weak solution to our equations we will show Korn-Sobolev inequality for Orlicz spaces and the fact that closures of smooth compactly supported functions w.r.t. modular and weak star topology of symmetric gradient coincides.

26 Rico Zacher Universitat¨ , Germany

De Giorgi-Nash-Moser estimates for time fractional diffusion equations

We discuss several recent results on a priori estimates for weak solutions to linear and quasilinear fractional diffusion equations of time order less than one (boundedness, Holder¨ continuity and Harnack estimates).

27 4 List of Participants

First name Last name City Radjesvarane Alexandre Brest Pascal Azerad Montpellier Piotr Biler Wrocław Robin Beier Bielefeld Wolf-Jurgen¨ Beyn Bielefeld Sebastien´ Boyaval Marne la Vallee´ Mario Bukal Wien Jerome Coville Avignon Bartlomiej Dyda Bielefeld Matthias Eisenmann Etienne Emmrich Bielefeld Matthieu Felsinger Bielefeld Barbara Gentz Bielefeld Piotr Gwiazda Warsaw Christopher Hartleb Bielefeld Erika Hausenblas Leoben Leonie Herden Matthias Hieber Darmstadt Cyril Imbert Paris Grzegorz Karch Wrocław Trygve Karper Trondheim Moritz Kaßmann Bielefeld Mariko Alexa Mathuis Bielefeld Ante Mimica Bielefeld Franc¸ois Murat Paris Alexander Ostermann Innsbruck Andreas Prohl Tubingen¨ Dimitri Puhst Berlin Michael Ru˚ziˇ ckaˇ Freiburg Russell Schwab Pittsburgh David Siˇ skaˇ Bielefeld Guy Vallet Pau Julien Vovelle Lyon Petra Wittbold Essen Aneta Wroblewska´ Warsaw Rico Zacher Halle Alexandra Zimmermann Essen Notes Notes Notes Notes

Program

Monday (V2-105) Tuesday (V3-201) Wednesday(V2-105) 09:15-09:45 Registration 09:45-10:00 Opening 10:00-10:40 Imbert Karch Vallet 10:40-11:20 Gwiazda Alexandre Azerad 11:20-12:00 Break (V3-106) Break (V3-106) Break (V3-106) 12:00-12:40 Biler Coville Murat 12:40-14:00 Lunch Lunch Lunch 14:00-14:40 Ru˚ziˇ ckaˇ Ostermann Zacher 14:40-15:20 Boyaval Karper Schwab 15:20-16:00 Break (V3-106) Break (V3-106) Break (V3-106) 16:00-16:40 Wroblewska´ Vovelle 16:40-17:20 Hieber Hausenblas 17:20-18:00 Prohl 19:00h Conference Dinner