INTRODUCTION

Consider the Boltzmann equation, that famous kinetic model of rarefied gases driven by binary collisons. Since a linearization would remove some of the interesting behaviour, we shall here retain the truly non-linear collision mechanism.

A fairly common characteristic of non-linear evolution equations is that the behaviour close to equilibrium is smooth and regular with nice asymptotic convergence, whereas initial values further away may result in wilder, possibly non-smooth behaviour. For space-dependent Boltzmann gases in full space or bounded containers, various contraction mapping estimates can be used to prove the existence of unique, smooth solutions converging to an equilibrium (Maxwellian) value with time, if the initial values are close enough to equilibrium. Such methods break down when the initial values are further away from equilibrium, and so do natural compactness arguments.

For another approach, let us first recall the physical background. From the point of view of physics, what happens at distances or within volumes below, say, the scale of elementary particle phenomena, is an artefact of the model with little experimental relevance. From this perspective, the question whether the model starts from an underlying set of rationally, really, or infinitesimally spaced points, should be decided purely on mathematical grounds.