Long time asymptotics for homoenergetic solutions of the Boltzmann equation. Collision-dominated case

TL;DR

This paper analyzes the long-time behavior of homoenergetic solutions to the Boltzmann equation in the collision-dominated regime, showing that solutions tend toward a time-dependent Maxwellian distribution with evolving temperature.

Contribution

It extends previous work by focusing on the collision-dominated case, providing a formal analysis of the asymptotic behavior of solutions with a Maxwellian limit.

Findings

Solutions tend to a Maxwellian distribution over time

The temperature of the Maxwellian evolves according to the collision dynamics

The hyperbolic term becomes negligible in the long-time limit

Abstract

In this paper we present a formal analysis of the long-time asymptotics of a particular class of solutions of the Boltzmann equation, known as homoenergetic solutions, which have the form $f\left( x,v,t\right)=g\left( v-L\left( t\right) x,t\right)$ where $L\left( t\right) =A\left(I+tA\right) ^{-1}$ with the matrix $A$ describing a shear flow or a dilatation or a combination of both. We began this study in \cite{JNV1}. Homoenergetic solutions satisfy an integro-differential equation which contains, in addition to the classical Boltzmann collision operator, a linear hyperbolic term. In \cite{JNV1} it has been proved rigorously the existence of self-similar solutions which describe the change of the average energy of the particles of the system in the case in which there is a balance between the hyperbolic and the collision term. In this paper we focus in homoenergetic solutions for which the collision term is much larger than the hyperbolic term (collision-dominated behavior). In this case the long time asymptotics for the distribution of velocities is given by a time dependent Maxwellian distribution with changing temperature.