The Boltzmann equation for a multi-species inelastic mixture

TL;DR

This paper develops and analyzes a multi-species Boltzmann equation model for granular gases, providing mathematical inequalities, existence theory, and demonstrating that the system exhibits a generalized Haff's Law over time.

Contribution

It introduces a novel multi-species kinetic model for inelastic granular gases, including mathematical analysis and large-time behavior results.

Findings

Established Povzner-type inequalities for the model.

Proved existence and uniqueness in Orlicz spaces.

Showed the system obeys a mixture analogue of Haff's Law.

Abstract

A granular gas is a collection of macroscopic particles that interact through energy-dissipating collisions, also known as inelastic collisions. This inelasticity is characterized by a collision mechanics in which mass and momentum are conserved and kinetic energy is dissipated. Such a system can be described by a kinetic equation of the Boltzmann type. Nevertheless, due to the macroscopic aspect of the particles, any realistic description of a granular gas should be written as a mixture model composed of M different species, each with its own mass. We propose in this work such a granular multi-species model and analyse it, providing Povzner-type inequalities, and a Cauchy theory in general Orlicz spaces. We also analyse its large time behavior, showing that it exhibits a mixture analogue of the seminal Haff's Law.