On the spatially homogeneous Boltzmann equation for Bose-Einstein particles with balanced potentials

Abstract

The paper is concerned with the spatially homogeneous isotropic Boltzmann equation for Bose-Einstein particles with quantum collision kernel where the interaction potential $\phi({\bf x})$ can be approximately written as the delta function plus a certain attractive potential such that the Fourier transform $\widehat{\phi}$ of $\phi$ behaves like $0 \le \widehat{\phi}(\xi) \le {\rm const.} |\xi|^{\eta}$ for $|\xi|<<1$ for some constant $\eta\ge 1$. We prove that in this case, there is no condensation in finite time for all temperatures and all solutions, and thus it is completely different from the case $\widehat{\phi}(\xi) \ge {\rm const.}|\xi|^{\eta}$ for $|\xi|<<1$ with $0\le \eta<1/4$ as considered in \cite{Cai-Lu}. For a class of initial data that have some nice integrability near the origin, we also get some regularity, stability and $L^{\infty}$ estimate.