The Boltzmann Equation for Bose-Einstein Particles: Regularity and Condensation

Abstract

We study regularity and finite time condensation of distributional solutions of the space-homogeneous and velocity-isotropic Boltzmann equation for Bose-Einstein particles for the hard sphere model. Global in time existence of distributional solutions had been proven before. Here we prove that the equation is locally and can be globally (in time) well-posed for the class of distributional solutions having finite moment of the negative order -1/2, and solutions in this class with regular initial data are mild solutions in their regularity time-intervals. By observing a necessary condition on the initial data for the absence of condensation at some finite time, we also propose a sufficient condition on the initial data for the occurrence of condensation at all large time, and then using a positivity of a partial collision integral we prove further that the critical time of condensation can be strictly positive. © 2014 The Author(s).