KINETIC EQUATIONS WITH MAXWELL BOUNDARY CONDITION

Stéphane Mischler (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We prove global stability results of {\sl DiPerna-Lions} renormalized solutions to the initial boundary value problem for kinetic equations. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which include the so-called Maxwell boundary condition, and we prove that it is realized (it is not relaxed!). The techniques are illustrated with the Fokker-Planck-Boltzmann equation and with the Vlasov-Poisson-Fokker-Planck system, but can be readily extended to the Boltzmann equation and to the Vlasov-Poisson system when linear and diffuse boundary condition are imposed. The proof uses 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 Darroès-Guiraud information in a crucial way.

Domaines: Equations aux dérivées partielles [math.AP]

Stéphane Mischler. KINETIC EQUATIONS WITH MAXWELL BOUNDARY CONDITION. 2001. ⟨hal-00346628v1⟩