Hypocoercivity for linear kinetic equations conserving mass

TL;DR

This paper introduces a new method to prove hypocoercivity for a broad class of linear kinetic equations with a single conservation law, demonstrating exponential convergence to equilibrium under various models and conditions.

Contribution

The paper develops a novel approach to establish hypocoercivity for linear kinetic equations with one conservation law, applicable to diverse models including Vlasov-Fokker-Planck and linear Boltzmann equations.

Findings

Proves exponential convergence to equilibrium in weighted L^2 norms.

Applicable to models with unbounded confining potentials.

Unified method covering diffusive and scattering kinetic equations.

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 like the linear Boltzmann equation 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.