Time averages for kinetic Fokker-Planck equations

TL;DR

This paper develops explicit methods to estimate decay rates of time averages for solutions to kinetic Fokker-Planck equations with various local equilibria, extending hypocoercivity results across different regimes.

Contribution

It introduces an explicit, constructive approach using adapted inequalities to analyze decay rates for kinetic Fokker-Planck equations with diverse equilibria.

Findings

Derived decay rate estimates for solutions in different regimes.

Extended hypocoercivity results to subexponential, exponential, and superexponential equilibria.

Compared new estimates with existing techniques.

Abstract

We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the torus with Maxwellian or fat tail local equilibria. Results based on weak norms have recently been achieved by S. Armstrong and J.-C. Mourrat in case of Maxwellian local equilibria. Using adapted Poincar\'e and Lions-type inequalities, we develop an explicit and constructive method for estimating the decay rate of time averages of norms of the solutions, which covers various regimes corresponding to subexponential, exponential and superexponential (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.