Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media

TL;DR

This paper studies the behavior of a granular gas modeled by the Boltzmann equation with diffusive forcing, proving uniqueness and stability of the steady state in the regime of small inelasticity.

Contribution

It establishes the uniqueness and stability of the steady state for the inelastic Boltzmann equation with diffusive forcing when inelasticity is small.

Findings

Uniqueness of the stationary solution for small inelasticity.

Linear and nonlinear stability results for the steady state.

Analysis within the physical regime of small inelasticity.

Abstract

We consider a space-homogeneous gas of {\it inelastic hard spheres}, with a {\it diffusive term} representing a random background forcing (in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$ for the inelasticity). In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_* \in [0,1)$) we prove uniqueness of the stationary solution for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum, and we give various results on the linear stability and nonlinear stability of this stationary solution.