Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation

TL;DR

This paper proves exponential decay to equilibrium for a linear inhomogeneous Boltzmann equation with a confining potential, using hypoelliptic methods adapted to an equation lacking regularization.

Contribution

It introduces a novel application of hypoelliptic techniques to establish spectral gap and decay rates for a relaxation Boltzmann equation without diffusion.

Findings

Proves exponential decay to the Maxwellian equilibrium.

Provides explicit decay rate.

Extends hypoelliptic methods to non-regularizing equations.

Abstract

We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker-Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties.