Hypocoercivity for linear kinetic equations conserving mass

Dolbeault, Jean; Mouhot, Clément; Schmeiser, Christian

doi:10.1090/S0002-9947-2015-06012-7

Hypocoercivity for linear kinetic equations conserving mass
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Trans. Amer. Math. Soc. 367 (2015), 3807-3828 Request permission

Abstract:

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 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.

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