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arXiv · 2101.00144

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

Shuzhe Cai
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Abstract

The paper is concerned with the spatially homogeneous isotropic Boltzmann equation for Bose-Einstein particles with quantum collision kernel where the interaction potential $ϕ({\bf x})$ can be approximately written as the delta function plus a certain attractive potential such that the Fourier transform $\widehatϕ$ of $ϕ$ behaves like $0 \le \widehatϕ(ξ) \le {\rm const.} |ξ|^η$ for $|ξ|<<1$ for some constant $η\ge 1$. 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 $\widehatϕ(ξ) \ge {\rm const.}|ξ|^η$ for $|ξ|<<1$ with $0\le η<1/4$ 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.

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Shuzhe Cai. 2021-01-01. On the spatially homogeneous Boltzmann equation for Bose-Einstein particles with balanced potentials. https://arxiv.org/abs/2101.00144

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