Time averages for kinetic Fokker-Planck equations

We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equa-tions 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 Poincare ' 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 vari-ous regimes corresponding to subexponential, exponential and superexponen-tial (including Maxwellian) local equilibria. As a consequence, we also derive hypocoercivity estimates, which are compared to similar results obtained by other techniques.