An Alexander-like theorem for a particle model with inelastic collisions

TL;DR

This paper develops a mathematical framework for inelastic hard sphere systems, proving measure-preserving properties and establishing global well-posedness of trajectories despite energy loss at collisions.

Contribution

It introduces the Transport-Collision-Transport (TCT) dynamics for inelastic spheres and proves an Alexander-like theorem ensuring well-posedness of the system.

Findings

Scattering mapping preserves Lebesgue measure in velocity space.

Global well-posedness holds for almost all initial conditions.

The model extends classical theories to inelastic collision systems.

Abstract

We consider a finite system of hard spheres that collide inelastically according to a particular model, losing a fixed amount of kinetic energy at each collision. We develop the theory of the Transport-Collision-Transport (TCT) dynamics, which allows to study precisely the evolution of the Lebesgue measure in the phase space under the action of the flows of particle systems that can interact via instantaneous binary collision. We show that the scattering mapping associated to the inelastic hard sphere system we introduce preserves locally the Lebesgue measure in the velocity space, in spite of the fact that a positive amount of kinetic energy is lost at each inelastic collision. We prove the analog of Alexander's theorem for our model, which allows us to deduce the global well-posedness of the trajectories, for almost every initial datum.