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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2010

Kinetic equations with Maxwell boundary conditions

Résumé

We prove global stability results of {\sl 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.
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Dates et versions

hal-00346628 , version 1 (11-12-2008)
hal-00346628 , version 2 (04-05-2009)
hal-00346628 , version 3 (29-01-2010)

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Citer

Stéphane Mischler. Kinetic equations with Maxwell boundary conditions. Annales Scientifiques de l'École Normale Supérieure, 2010, 43 (5), pp.719-760. ⟨hal-00346628v3⟩
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