Generalized kinetic Maxwell type models of granular gases

TL;DR

This paper develops a comprehensive mathematical framework for generalized Maxwell models of granular gases, analyzing existence, convergence, and stability of self-similar solutions with power-law tails in various dimensions.

Contribution

It introduces a broad class of Maxwell-type kinetic models with arbitrary nonlinearities, establishing existence, asymptotic behavior, and stability of self-similar solutions in a unified way.

Findings

Existence of self-similar solutions for generalized Maxwell models.

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

Identification of stable power-law tail equilibria.

Abstract

We consider generalizations of kinetic granular gas models given by Boltzmann equations of Maxwell type. These type of models for non-linear elastic or inelastic interactions, have many applications in physics, dynamics of granular gases, economy, etc. We present the problem and develop its form in the space of characteristic functions, i.e. Fourier transforms of probability measures, from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. We find a whole class of generalized Maxwell models that satisfy properties that characterize the existence and asymptotic of dynamically scaled or self-similar solutions, often referred as {\em homogeneous cooling states}. Of particular interest is a concept interpreted as an operator generalization of usual Lipschitz conditions which allows to describe the behavior of solutions to the corresponding initial value problem. In particular, we present, in the most general case, existence of self similar solutions and study, in the sense of probability measures, the convergence of dynamically scaled solutions associated with the Cauchy problem to those self-similar solutions, as time goes to infinity. In addition we show that the properties of these self-similar solutions lead to non classical equilibrium stable states exhibiting power tails. These results apply to 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 presentation.