Exponential trend to equilibrium for the inelastic Boltzmann equationdriven by a particle bath

TL;DR

This paper proves that solutions to the inelastic Boltzmann equation with a thermal bath converge exponentially fast to equilibrium, especially in the weakly inelastic regime, with explicit convergence rates based on spectral analysis.

Contribution

It establishes exponential convergence to equilibrium for the inelastic Boltzmann equation with a particle bath, using spectral and entropy methods, particularly near elastic collisions.

Findings

Exponential convergence towards equilibrium in the weakly inelastic regime.

Explicit convergence rate depending on the spectral gap of the elastic operator.

Convergence proven using spectral analysis and entropy estimates.

Abstract

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient $\alpha \in (0,1)$) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which $\alpha$ is close to $1$, and the rate of convergence is explicit and depends solely on the spectral gap of the \emph{elastic} linearised collision operator.