Particle dynamics with elastic collision at the boundary: existence and partial uniqueness of solutions

TL;DR

This paper studies the motion of particles in a bounded domain with elastic boundary collisions, establishing existence of solutions and partial uniqueness, while demonstrating that full uniqueness does not always hold.

Contribution

It introduces a precise solution concept for particle dynamics with elastic boundary collisions and proves existence and partial uniqueness of solutions, addressing a gap in the mathematical understanding.

Findings

Existence of solutions for the particle system with elastic boundary collisions.

Partial uniqueness of solutions under certain conditions.

Counterexample showing full uniqueness does not always hold.

Abstract

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $\Omega$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.