On the rate of convergence to equilibrium for the linear boltzmann equation with soft potentials

TL;DR

This paper establishes explicit algebraic and stretched exponential rates of convergence to equilibrium for the linear Boltzmann equation with soft potentials using an adapted entropy method, with potential extensions to non-cutoff cases.

Contribution

It introduces a novel approach involving functional inequalities relating entropy to its production, applicable to equations with mixed linear and non-linear terms.

Findings

Derived explicit algebraic convergence rates.

Obtained stretched exponential convergence results.

Discussed properties and conjectured rates for the non-cutoff case.

Abstract

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad's angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cutoff case and conjecture what we believe to be the right rate of convergence in that case.