Kinetic equations with Maxwell boundary conditions

Mischler, Stéphane

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 5, pp. 719-760.

Nous montrons la stabilité des solutions renormalisées au sens de DiPerna-Lions pour des équations cinétiques avec conditions initiale et aux limites. La condition aux limites (qui peut être non linéaire) est partiellement diffuse et est réalisée (c'est-à-dire qu'elle n'est pas relaxée). Les techniques que nous introduisons sont illustrées sur l'équation de Fokker-Planck-Boltzmann et le système de Vlasov-Poisson-Fokker-Planck ainsi que pour des conditions aux limites linéaires sur l'équation de Boltzmann et le système de Vlasov-Poisson. Les démonstrations utilisent des théorèmes de trace du type de ceux introduits par l'auteur pour les équations de Vlasov, des résultats d'analyse fonctionnelle sur les convergences faible-faible (la convergence renormalisée et la convergence au sens du biting lemma), ainsi que l'information de Darrozès-Guiraud d'une manière essentielle.

We prove global stability results of DiPerna-Lions renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darrozès-Guiraud information in a crucial way.

Classification : 76P05, 82B40, 82C40, 82D05
Keywords: Vlasov-Poisson, Boltzmann and Fokker-Planck equations, Maxwell or diffuse reflection, nonlinear gas-surface reflection laws, Darrozès-Guiraud information, trace theorems, renormalized convergence, biting lemma, Dunford-Pettis lemma
Mot clés : Équations de Vlasov-Poisson, Boltzmann et Fokker-Planck, réflexion de Maxwell ou diffuse, réflexion non linéaire, information de Darrozès-Guiraud, théorèmes de trace, convergence renormalisée, convergence au sens de Chacon (biting lemma), lemme de Dunford-Pettis

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