Stability of measure solutions to a generalized Boltzmann equation with collisions of a random number of particles

TL;DR

This paper investigates the stability of measure solutions to a generalized Boltzmann equation involving random particle collisions, using fixed point analysis and stability methods for both stationary and dynamic models.

Contribution

It introduces a measure-based framework for a generalized Boltzmann equation with random collisions and analyzes the stability of its solutions using novel mathematical techniques.

Findings

Existence of fixed points for the stationary model.

Asymptotic stability of the dynamic Boltzmann-type equation.

Application of Zolotarev seminorm and Kantorovich-Rubinstein methods.

Abstract

In the paper we study a measure version of the evolutionary nonlinear Boltzmann-type equation in which we admit a random number of collisions of particles. We consider first a stationary model and use two methods to find its fixed points: the first based on Zolotarev seminorm and the second on Kantorovich-Rubinstein maximum principle. Then a dynamic version of Boltzmann-type equation is considered and its asymptotical stability is shown.