On the Initial Boundary-Value Problem in the Kinetic Theory of Hard Particles I: Non-existence

TL;DR

This paper investigates the initial boundary-value problem in the kinetic theory of non-spherical particles, demonstrating non-existence of solutions under certain conditions and highlighting the necessity of rolling in rigid body dynamics.

Contribution

It is the first to show that, for certain boundary conditions, no local solutions exist, emphasizing the importance of rolling in energy-conserving rigid body dynamics.

Findings

Non-existence of solutions for specific initial conditions.

Necessity of rolling in the dynamics of convex rigid bodies.

First demonstration of non-existence in this context.

Abstract

In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of associated boundary conditions, we identify initial conditions for which there do not exist local-in-time weak solutions of Newton's equations of motion. To our knowledge, this is the first time the necessity of rolling in the energy-conserving dynamics of strictly-convex rigid bodies has been demonstrated. This study was, in part, motivated by a recent observation of Palffy-Muhoray, Virga, Wilkinson and Zheng on the interpenetration of strictly-convex rigid bodies.