Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules

TL;DR

This paper proves the existence, uniqueness, and stability of self-similar solutions for a modified non-cutoff Maxwell Boltzmann equation with a drift term, extending previous cutoff results.

Contribution

It extends the analysis of homoenergetic solutions to non-cutoff Maxwell gases, establishing properties of self-similar solutions under small drift terms.

Findings

Existence and uniqueness of self-similar solutions for small drift matrices.

Solutions have finite moments of all orders if the drift is sufficiently small.

Solutions are smooth due to the non-cutoff collision operator.

Abstract

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term that is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Our goal is to prove that, under smallness assumptions on the drift term, the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p > 2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases.