Strong Convergence towards homogeneous cooling states for dissipative Maxwell models

TL;DR

This paper proves strong convergence of solutions of the inelastic Maxwell model towards the homogeneous cooling state, using regularity propagation and Fisher information control, with implications for the inelasticity parameter.

Contribution

It establishes strong convergence in Sobolev and L^1 norms towards the homogeneous cooling state for the inelastic Maxwell model, extending previous weak convergence results.

Findings

Uniform regularity propagation in time

Strong convergence in Sobolev and L^1 norms

Bound on L^1 distance vanishing as inelasticity decreases

Abstract

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogenous cooling state present in the literature implies the strong convergence in Sobolev norms and in the $L^1$ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the $L^1$ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.