Self-similar asymptotics for a modified Maxwell-Boltzmann equation in systems subject to deformations

TL;DR

This paper investigates the long-term behavior of solutions to a generalized Maxwell-Boltzmann equation with a linear deformation term, showing that solutions tend to a self-similar form when the deformation is small.

Contribution

It establishes the self-similar asymptotics for solutions under small deformation matrices and characterizes the finiteness of higher order moments.

Findings

Solutions tend to a self-similar form for small deformation matrices.

Higher order moments of the self-similar profile are finite under additional smallness conditions.

The results apply to solutions with finite second moments and non-negativity.

Abstract

In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of the paper is formulated in Theorem 2.1. Informally stated, this Theorem says that, for sufficiently small norm of A, any non-negative solution with finite second moment tends to a self-similar solution of relatively simple form for large values of time. This is what we call "the self-similar asymptotics". We also prove that the higher order moments of the self-similar profile are finite under further smallness condition on the matrix A.