Hypocoercivity and sub-exponential local equilibria

TL;DR

This paper extends hypocoercivity techniques to linear kinetic equations with sub-exponential local equilibria, deriving decay rates that align with the heat equation in the diffusion limit, using weighted inequalities.

Contribution

It introduces a novel approach applying hypocoercivity to equations with sub-exponential equilibria without extra regularity, broadening the scope of decay rate analysis.

Findings

Global decay rates match the heat equation in the diffusion limit

Weighted Poincaré inequalities effectively describe convergence without extra regularity

Method applies to Fokker-Planck and scattering collision operators

Abstract

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincar\'e inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincar\'e inequalities compared to the more classical weak Poincar\'e inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.