Benoît Perthame

Kinetic formulation of the isentropic gas dynamics and

We consider the 2×2 hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called thep-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce newL∞ estimates using moments lemma and proveL∞−w* stability for γ≥3.

p

We construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws. In particular we prove that it is well-posed uniformly in ε — the microscopic scale. We also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions — analogous to Boltzmanns microscopicH-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, we prove by both — BV compactness arguments in the multidimensional case and by compensated compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a “continuum” limit, as it converges strongly with ε↓0 to the unique entropy solution of the corresponding conservation law.

-systems

Abstract Under appropriate assumptions on the collision kernel we prove the existence of global solutions of the Enskog equation with elastic or inelastic collisions. We consider also this equation with spin, that is, the case when the angular velocities of the colliding particles are taken into account. In this case we also prove global existence results.

A kinetic equation with kinetic entropy functions for scalar conservation laws

We study the structure of solutions for the evolution equation of gaseous stars. The concept of “gentle” solution is introduced to establish the local solvability of the Cauchy problem, and non-existence of non-trivial global “gentle” solutions is proved.

On the modified Enskog equation for elastic and inelastic collisions. Models with spin

We present a BGK-type collision model which approximates, by a Chapman-Enskog expansion, the compressible Navier-Stokes equations with a Prandtl number that can be chosen arbitrarily between 0 and 1. This model has the basic properties of the Boltzmann equation, including theH-theorem, but contains an extra parameter in comparison with the standard BGK model. This parameter is introduced multiplying the collision operator by a nonlinear functional of the distribution function. It is adjusted to the Prandtl number.

Sur les solution à symétrie sphérique de l’equation d’Euler-Poisson pour l’evolution d’etoiles gazeuses

We prove the blow-up in finite time for sherically symmetric solutions to the Euler-Poisson system with repulsive forces. We show that a “gentle” solution with initially bounded support can exist only for a finite time if its initial energy is large (in comparison with its mass). The method is to compare estimates on the inertial moment and on the characteristic curves.

A BGK model for small Prandtl number in the Navier-Stokes approximation

AbstractThe Radiative Transfer Equation is the nonlinear transport equation(RTE)n

Bounded speed of propagation for solutions to radiative transfer equations

partial _t f + frac{1}{varepsilon }v cdot nabla _x f + frac{1}{{varepsilon ^2 }}sigma (tilde f)(f - tilde f) = 0,