Frédéric Poupaud

HOMOGENIZATION LIMITS AND WIGNER TRANSFORMS

We present a theory for carrying out homogenization limits for quadratic functions (called “energy densities”) of solutions of initial value problems (IVPs) with anti-self-adjoint (spatial) pseudo-differential operators (PDOs). The approach is based on the introduction of phase space Wigner (matrix) measures that are calculated by solving kinetic equations involving the spectral properties of the PDO. The weak limits of the energy densities are then obtained by taking moments of the Wigner measure. The very general theory is illustrated by typical examples like (semi)classical limits of Schrodinger equations (with or without a periodic potential), the homogenization limit of the acoustic equation in a periodic medium, and the classical limit of the Dirac equation.

A Wigner‐function approach to (semi)classical limits: Electrons in a periodic potential

A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band‐structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

Classical and quantum transport in random media

We study in this article the transport of particles in time-dependent random media, in the so-called weak coupling limit. We show the convergence of a Liouville equation to a Fokker–Planck equation. We also obtain the semi-classical limit of Schrodinger equations. This limit is described by a linear Boltzmann equation. In both cases, the ratio between a typical time scale and the scale of the media determines whether the limit diffusion and the collision process are elastic or not.  2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.

APPROXIMATION BY HOMOGENIZATION AND DIFFUSION OF KINETIC EQUATIONS

This work is devoted to the approximation by the diffusion of kinetic equations. The approximation is justified by homogenization and in the limit of vanishing mean free path. Our analysis includes the grey transfer model but also discrete velocity model. The technics of proofs are based on compensated compactness and double scale limit.

HOMOGENIZATION OF TRANSPORT EQUATIONS: WEAK MEAN FIELD APPROXIMATION ∗

We are interested, with respect to the small parameter

Semi-classical limits in a crystal with exterior potentials and effective mass theorems

\epsilon

The Beker--Döring System and Its Lifshitz--Slyozov Limit

, in the behavior of solutions

Berenger absorbing boundary condition with time finite-volume scheme for triangular meshes

\rho^\epsilon

Derivation of a hydrodynamic system hierarchy for semiconductors from the Boltzmann equation

of the conservative advection-diffusion equation

EXISTENCE OF SOLUTIONS OF A KINETIC EQUATION MODELING COMETARY FLOWS

\partial_t\rho^\epsilon + \nabla_x\cdot(\rho^\epsilon u^\epsilon)=\eta\Delta_x\rho^\epsilon