On the spatially homogeneous Boltzmann equation

Abstract

We consider the question of existence and uniqueness of solutions to the spatially homogeneous Boltzmann equation. The main result is that to any initial data with finite mass and energy, there exists a unique solution for which the same two quantities are conserved. We also prove that any solution which satisfies certain bounds on moments of order s < 2 must necessarily also have bounded energy.A second part of the paper is devoted to the time discretization of the Boltzmann equation, the main results being estimates of the rate of convergence for the explicit and implicit Euler schemes.Two auxiliary results are of independent interest: a sharpened form of the so called Povzner inequality, and a regularity result for an iterated gain term.