We consider the High field limit of the Boltzmann equation of semiconductors

The limiting equation is a nonlinear conservation law. Thanks to a Hilbert expansion method and to the L¹ contractivity of both the kinetic and the limiting equations, we show the convergence of the kinetic solutions towards the solution of the limiting conservation whenever the latter is smooth.

Then, we derive a series of entropies for the kinetic equation, which "converge" to the convex entropies of the limiting conservation law. By using these entropies, we prove the convergence as e tends to zero, of the kinetic solutions f_e towards the unique entropy solution, even when shocks do appear. The same entropy allows to construct kinetic profiles for entropic shocks. Diffusive correction can be handled by a linearized version of these entropies.

This is a work in collaboration with Hédia Chaker (Tunis) and Christian Schmeiser (Vienna).

The long-time asymptotics for the doubly nonlinear diffusion equations r_t = div (|Ñr^m|^p-2 Ñr^m) in Rⁿ, is studied for p > 1 and m > [(n-p)/(n(p-1))]. The non-negative solutions of the equation are shown to behave asymptotically, as t®¥, like Barenblatt-type solutions, and a polynomial decay is established for the convergence with respect to the L¹-norm. The rate of decay is proved to be optimal when m ³ [(n-p+1)/(n(p-1))]. The method used is based on optimal transportation arguments when m ³ [(n-p+1)/(n(p-1))], and on a linear analysis when [(n-p)/(n(p-1))] < m < [(n-p+1)/(n(p-1))].

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball W of radius R in Rⁿ, n ³ 2, so that the following inequalities hold for all u Î C₀^¥ (W):

This is joint work with Amir Moradifam.

We are interested in asymptotic problems for kinetic equations where the small parameter is intended to describe both the (high) strength and the (fast) period of oscillations of the force field the particles are subject to.

In this problem the force field is derived from random principles, which in turn imply some dissipation mechanism and lead to effective equations involving diffusion with respect to the velocity variable.

We present in this talk some recent results about the application of L² hypocoercive methods to the explicit exponential trend to the equilibrium for some simple kinetic models (Fokker-Planck equation and linear relaxation) in bounded domains.

Starting from a Vlasov-type kinetic model we derive nonlocal macroscopic models generalizing the models of conservation type first introduced by Aw-Rascle and Zhang. We discuss reasonable examples of braking and acceleration forces as functions of density and relative speed. Removing the nonlocality by Taylor expansions produces nonlinear PDEs for both braking and acceleration scenarios. A traveling wave ansatz produces stop-and-go waves in good qualitative agreement with practical observations.

The behaviour of a large number of particles interacting with a singular potential is studied. In the repulsive case and for almost all initial configurations, it is possible to show a stability result for the flow, quantitative and uniform in the number of particles. In particular, the dynamics remains close to any regularized flow, and therefore to the dynamics predicted by the Vlasov equation at the limit.

One outstanding open question of fundamental interest in fluid mechanics concerns the existence of long-time smooth solutions for the 3D Euler equations. It is argued in the literature that the surface quasi-geostrophic (SQG) equations bear some resemblance with the 3D Euler vorticity equations and as such they provide an interesting model for studying long time behaviour of solutions in a context similar to Euler equations. As an alternative to previous interesting approaches by others, we propose here an inviscid regularization for the SQG equations to facilitate the analysis. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter tends to zero, for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy.

Therefore, the new regularization provides an attractive numerical procedure for finite time blow up testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. Some numerical tests will be presented as well.

Joint work with Edriss Titi.

We study a coupled system of kinetic and fluid equations modeling fluid-particles interactions arising in sprays, aerosols or sedimentation problems. More precisely, we consider a Vlasov-Fokker-Planck equation coupled to compressible Navier-Stokes equation via a drag force. We establish the existence of solutions and we rigorously derive the asymptotic regime corresponding to a strong drag force and a strong Brownian motion.

We present a simple conservation law model describing the population level swarming of organisms. The organism's speed of motion results from the combination of a nonlocal, long range, attraction and short range repulsion. The kernel of the long range attraction term is not symmetric and can be thought of as modeling the response of higher organisms to visual cues where the organisms sense others primarily in the direction in which they are moving. The resulting model has compact travelling waves (swarms) whose speed increases with the swarm "mass" up to a largest swarm. There are also front and periodic solutions. We discuss a possible kinetic model that leads to our conservation law and some extensions of our macroscopic model.

This is joint work with Xu Yang.

I will consider the three dimensional relativistic gravitational Vlasov-Poisson system

This is a joint program with Mohammed Lemou (IMT, Toulouse) and Florian Mehats (IRMAR, Rennes).