Kinetic equations with Maxwell boundary conditions

Stéphane Mischler

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 5, page 719-760
  • ISSN: 0012-9593

Abstract

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We prove global stability results of DiPerna-Lionsrenormalized 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.

How to cite

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Mischler, Stéphane. "Kinetic equations with Maxwell boundary conditions." Annales scientifiques de l'École Normale Supérieure 43.5 (2010): 719-760. <http://eudml.org/doc/272103>.

@article{Mischler2010,
abstract = {We prove global stability results of DiPerna-Lionsrenormalized 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.},
author = {Mischler, Stéphane},
journal = {Annales scientifiques de l'École Normale Supérieure},
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},
language = {eng},
number = {5},
pages = {719-760},
publisher = {Société mathématique de France},
title = {Kinetic equations with Maxwell boundary conditions},
url = {http://eudml.org/doc/272103},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Mischler, Stéphane
TI - Kinetic equations with Maxwell boundary conditions
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 5
SP - 719
EP - 760
AB - We prove global stability results of DiPerna-Lionsrenormalized 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/272103
ER -

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