Uniqueness and regularity of steady states of the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath

TL;DR

This paper investigates the uniqueness and regularity of steady states in the Boltzmann equation for viscoelastic hard-spheres driven by a thermal bath, introducing novel methods that do not rely on prior time-dependent analysis or entropy inequalities.

Contribution

It presents a new approach to study regularity without prior time-dependent knowledge and proves uniqueness in the small thermalization regime via the quasi-elastic limit.

Findings

Established regularity of steady states without prior time-dependent analysis.

Proved uniqueness of steady states in the small thermalization regime.

Developed a novel approach avoiding entropy functional inequalities.

Abstract

We study the uniqueness and regularity of the steady states of the diffusively driven Boltzmann equation in the physically relevant case where the restitution coefficient depends on the impact velocity including, in particular, the case of viscoelastic hard-spheres. We adopt a strategy which is novel in several aspects, in particular, the study of regularity does not requires a priori knowledge of the time-dependent problem. Furthermore, the uniqueness result is obtained in the small thermalization regime by studying the so-called quasi-elastic limit for the problem. An important new aspect lies in the fact that no entropy functional inequality is needed in the limiting process.