On the Initial Boundary-Value Problem in the Kinetic Theory of Hard Particles II: Non-uniqueness

TL;DR

This paper demonstrates the non-uniqueness of weak solutions in the kinetic theory of hard particles, showing that multiple solutions can correspond to the same initial data, which challenges traditional assumptions of uniqueness.

Contribution

It establishes the existence of uncountably many weak solutions for the initial boundary-value problem in the kinetic theory of non-spherical particles, using advanced mathematical techniques.

Findings

Uncountably many weak solutions exist for given initial data.

Non-uniqueness is linked to solutions of a constrained Monge-Ampère equation.

Implications for the kinetic theory of hard particle systems are discussed.

Abstract

We prove that to each initial datum in a set of positive measure in phase space, there exist uncountably-many associated weak solutions of Newton's equations of motion which govern the dynamics of two non-spherical sets with real-analytic boundaries subject to the conservation of linear momentum, angular momentum and kinetic energy. We prove this result by first exhibiting non-uniqueness of classical solution to a constrained Monge-Amp\`ere equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we discuss the relevance of this observation to the kinetic theory of hard particle systems.