On the self-similar asymptotics for generalized non-linear kinetic Maxwell models

TL;DR

This paper develops a general theory for Maxwell models in nonlinear kinetic equations, proving existence and convergence of self-similar solutions across various applications including physics and economics.

Contribution

It introduces a broad framework for Maxwell models with polynomial nonlinearities, establishing conditions for solution behavior and self-similarity in multiple dimensions.

Findings

Existence of self-similar solutions for generalized Maxwell models.

Convergence of scaled solutions to self-similar states over time.

Application of theory to Boltzmann equations with elastic and inelastic interactions.

Abstract

Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economy, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties which one of them can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. The properties of these self-similar solutions, leading to non classical equilibrium stable states, are studied in detail. We apply the results to three different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this paper.