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 $d\geq 3$. We show existence and uniqueness locally in time for any initial data in $L^\infty\left(1+|v|^s\right)$ with finite mass and energy, for a suitable $s$, as well as the instantaneous creation of moments of all order.