L 2 Hypocoercivity methods for kinetic Fokker-Planck equations with factorised Gibbs states

Emeric Bouin (1) , Jean Dolbeault (1) , Luca Ziviani (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This contribution deals with L2 hypocoercivity methods for kinetic Fokker-Planck equations with integrable local equilibria and a factorisation property that relates the Fokker-Planck and the transport operators. Rates of convergence in presence of a global equilibrium, or decay rates otherwise, are estimated either by the corresponding rates in the diffusion limit, or by the rates of convergence to local equilibria, under moment conditions. On the basis of the underlying functional inequalities, we establish a classification of decay and convergence rates for large times, which includes for instance sub-exponential local equilibria and subexponential potentials.

Keywords: Hypocoercivity; linear kinetic equations; entropy -entropy production inequalities; Fokker-Planck operator; transport operator; diffusion limit; Nash inequality; Caffarelli-Kohn-Nirenberg inequality; Hardy-Poincaré inequality; weighted Poincaré inequality; Poincaré inequality

Domains: Mathematics [math]

Emeric Bouin, Jean Dolbeault, Luca Ziviani. L 2 Hypocoercivity methods for kinetic Fokker-Planck equations with factorised Gibbs states. 2023. ⟨hal-04079910v1⟩