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

TL;DR

This paper investigates the spatially homogeneous Boltzmann equation for Bose-Einstein particles with specific interaction potentials, proving the absence of condensation in finite time and establishing regularity and stability results for certain initial data.

Contribution

It demonstrates that for potentials with Fourier transform behavior like $|\xi|^{eta}$ with $eta eq 1/4$, no finite-time condensation occurs, contrasting previous cases with $eta<1/4$.

Findings

No finite-time condensation for all temperatures and solutions.

Established regularity, stability, and $L^{ abla}$ estimates for certain initial data.

Differentiates behavior based on the Fourier transform decay rate of the potential.

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.