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

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.