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Hypocoercivity for linear kinetic equations conserving mass. (English) Zbl 1342.82115

Authors’ 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.

MSC:

82C40 Kinetic theory of gases in time-dependent statistical mechanics
35H10 Hypoelliptic equations
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
35F10 Initial value problems for linear first-order PDEs
35Q83 Vlasov equations
35Q84 Fokker-Planck equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q20 Boltzmann equations
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References:

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