Hypocoercivity for linear kinetic equations conserving mass

Jean Dolbeault (1) , Clément Mouhot (2, 3) , Christian Schmeiser (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 DAMTP - Department of Applied Mathematics and Theoretical Physics
4 Fakultät für Mathematik [Wien]

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.

Mots clés: spectral gap; confinement; Fokker-Planck; nonlinear diffusion; kinetic equations; hypocoercivity; Boltzmann; BGK; relaxation; diffusion limit; Poincaré inequality; Hardy-Poincaré inequality

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Jean Dolbeault, Clément Mouhot, Christian Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Transactions of the American Mathematical Society, 2015, 367, pp.3807-3828. ⟨10.1090/S0002-9947-2015-06012-7⟩. ⟨hal-00482286⟩