Longtime behavior for homoenergetic solutions in the collision dominated regime for hard potentials

TL;DR

This paper analyzes the long-term behavior of homoenergetic solutions to the Boltzmann equation with hard potentials, showing convergence to an infinitely hot Maxwellian distribution under certain conditions.

Contribution

It establishes the stability and asymptotic behavior of homoenergetic solutions with high initial temperature for the Boltzmann equation with hard potentials.

Findings

Solutions with high initial temperature remain close to and converge to a Maxwellian with infinite temperature.

Provides explicit asymptotic formulas for the temperature evolution.

Results hold for both cutoff and non-cutoff collision kernels.

Abstract

In this paper, we consider a particular class of solutions to the Boltzmann equation which are referred to as homoenergetic solutions. They describe the dynamics of a dilute gas due to collisions and the action of either a shear, a dilation or a combination of both. We prove that solutions with initially high temperature remain close and converge to a Maxwellian distribution with temperature going to infinity. Furthermore, we give precise asymptotic formulas for the temperature. This local stability result is a consequence of a dominant shear and the homogeneity $ \gamma>0 $ of the collision operator with respect to relative velocities. The proof relies on an ansatz which is motivated by a Hilbert-type expansion. We consider both non-cutoff and cutoff kernels.