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Hypocoercivity for linear kinetic equations conserving mass

Dolbeault, Jean; Mouhot, Clément; Schmeiser, Christian (2012), Hypocoercivity for linear kinetic equations conserving mass, Transactions of the American Mathematical Society

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00482286/fr/
Date
2012
Journal name
Transactions of the American Mathematical Society
Publisher
American Mathematical Society
Pages
21
Metadata
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Author(s)
Dolbeault, Jean cc
Mouhot, Clément
Schmeiser, Christian
Abstract (EN)
We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann equation or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.
Subjects / Keywords
hypocoercivity; Poincaré inequality; kinetic equations; Hardy-Poincaré inequality; spectral gap; confinement; Fokker-Planck; nonlinear diffusion; diffusion limit; relaxation; BGK; Boltzmann