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

Bernhard Kepka

Published 2021-03-19Version 1

We will 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 of the Boltzmann equation. Our goal is to prove that under smallness assumptions on the drift term the long-time asymptotics is given by self-similar solutions. We will 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 will prove that they have finite moments of arbitrary order by decreasing the norm of A, depending on that order. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. At the end we apply the results to study the asymptotics of homoenergetic solutions in particular cases. This extends previous results from the cutoff case to non cutoff Maxwell gases.