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 $\varphi (\mathbf{x})$ can be approximately written as the delta function plus a certain attractive potential such that the Fourier transform $\hat{\varphi }$ of $\varphi$ behaves like $0\le \hat{\varphi }(\xi )\le const.|\xi {|}^{\eta }$ 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 $\hat{\varphi }(\xi )\ge const.|\xi {|}^{\eta }$ 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.

References in corpus (1)

• Not to normal order - Notes on the kinetic limit for weakly interacting quantum fluids