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.

中文翻译：

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变形系统中修正 Maxwell-Boltzmann 方程的自相似渐近性

在本文中，我们研究了一类广义的 Maxwell-Boltzmann 方程，除了通常的碰撞项之外，它还包含一个由矩阵 A 描述的线性变形项。 例如，此类方程来自对 Boltzmann 的同能解的分析自 1950 年代以来，许多作者都考虑过方程。我们的主要目标是在矩阵 A 很小的假设下研究解的大时间渐近性。论文的主要结果在定理 2.1 中表述。非正式地说，这个定理说，对于 A 的足够小的范数，任何具有有限二阶矩的非负解趋于对于大的时间值具有相对简单形式的自相似解。这就是我们所说的“自相似渐近”。