Radjesvarane Alexandre: 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).

Pascal Azerad: About a nonlocal model for morphodynamics

P. Azerad and A. Bouharguane, Universite Montpellier 2

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.

Piotr Biler: Barenblatt profiles for a nonlocal porous medium equation

Piotr Biler (Wrocław), Cyril Imbert (Paris IX), Grzegorz Karch (Wrocław), Régis Monneau (Paris-Est)

We study a generalization of the porous medium equation involving nonlocal terms. Explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present an argument to get the L^p decay of weak solutions of the Cauchy problem constructed by Caffarelli and Vázquez in the model case m=2.
Related equations appear in continuum mechanics to describe the evolution of dislocations in crystals.

Sébastien Boyaval: 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 numerical instabilities in most benchmark geometries. We suggest an analysis of some discretizations of the Oldroyd-B equations (our prototypical rheological model) that satisfy a discrete energy estimate similar to that for smooth solutions of the continuous equations, using a Lyapunov functional controlling the long-time asymptotics.

Piotr Gwiazda: 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.

Erika Hausenblas: 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 equidistant 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 convergence is given.

It is a joint work with Thomas Dunst and Andreas Prohl.

Matthias Hieber: Asymptotic properties and stability problems related to the Ekman Spiral

In this talk we consider Ekman boundary layers and in particular the Ekman spiral which is a stationary solution of the Navier-Stokes equations in the rotational setting. Based on the constructing a suitable weak solution, we discuss asymptotic and stability properties of the Ekman spiral and prove in particular asymptotic stability of the Ekman spiral for small Reynolds numbers.

Cyril Imbert: 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.

Jérôme Coville: Existence and uniqueness of positive solution of heterogeneous reaction

In this talk, I will discuss the asymptotic behaviour of positive solution of some heterogeneous integro-differential equation that have been recently introduced to model some pest invasion. The asymptotic behaviour is analysed through the properties (existence, uniqueness) of the stationary 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.

Grzegorz Karch: Infinite energy solutions to homogeneous Boltzmann equation

The goal of this talk is to present an approach to the homogeneous Boltzmann 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 Boltzmann 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 distributions.

Trygve Karper: 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 algorithms are based on evolving the solution by alternately applying the action of transport and diffusion. The main result is that both Gudnov and Strang splitting converges with the expected orders provided the initial 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.

François Murat: 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 solution. Such a definition (the "solution by transposition") was introduced by G. Stampacchia in 1973.
However, this definition is essentially restricted to the linear case. In this lecture, I will present the notion of "renormalized solution", which can be extended in a natural way
to the case of a second order monotone operator in divergence form posed on W₀^1,p(Ω)

Definition: u is a renormalized solution of the problem if
u : Ω -> R is measurable and a.e. finite
T_n(u) ∈ H¹₀(Ω) for every n > 0
(1/n) ∫_Ω |D T_n(u)|² -> 0 as n -> +∞
-div (h(u) A(x) Du) + h'(u) A(x) Du Du = h(u) f in D' (Ω) for every h ∈ C¹_c(R)

This definition allows one to prove that the problem has a renormalized solution, that this renormalized solution is unique and that it depends continuously on f, i.e. that in this framework the problem is well posed in the sense of Hadamard.

Alexander Ostermann: Exponential integrators

Exponential integrators are intended for the numerical solution of stiff differential equations. More precisely, they are designed for problems where the solution of the linearisation contains fast decaying (or highly oscillatory) 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.

Andreas Prohl: Numerics of the stochastic incompressible Navier-Stokes equation

We study finite element based space-time discretizations of the incompressible 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 boundary data.
This is joint work with E. Carelli (U Tübingen) and Z. Brzezniak (U York).

Michael Růžička: 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.

Russel Schwab: 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 optimal 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.

Guy Vallet: 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.

Julien Vovelle: 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 multiplicative noise.

Aneta Wróblewska: Generalised Stokes system in Orlicz spaces

We will investigate generalised Stokes system:
∂_tu-divS(t,x,Du)+ ∇ p = f in (0,T)xΩ,
divu =0 in (0,T)xΩ,
u(0,x)=u₀ in Ω,
u(t,x)=0 on (0,T)x∂Ω,
where Ω⊂Rⁿ 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.

Rico Zacher: 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, Hölder continuity and Harnack estimates).