On the Cauchy Problem for the Homogeneous Boltzmann–Nordheim Equation for Bosons: Local Existence, Uniqueness and Creation of Moments

Abstract

The Boltzmann–Nordheim equation is a modification of the Boltzmann equation, based on physical considerations, that describes the dynamics of the distribution of particles in a quantum gas composed of bosons or fermions. In this work we investigate the Cauchy theory of the spatially homogeneous Boltzmann–Nordheim equation for bosons, in dimension $$d\geqslant 3$$ . We show existence and uniqueness locally in time for any initial data in $$L^\infty (1+\left| v\right| ^s)$$ with finite mass and energy, for a suitable s, as well as the instantaneous creation of moments of all order.