Existence and uniqueness of solutions to the quantum Boltzmann equation for soft potentials

TL;DR

This paper proves the global existence and uniqueness of solutions to a modified quantum Boltzmann equation with soft potentials, uniformly in the quantum parameter, bridging quantum and classical cases.

Contribution

It establishes uniform estimates for solutions to the quantum Boltzmann equation across the quantum parameter range, including the classical limit.

Findings

Solutions exist globally with small defect mass, energy, and entropy.

Estimates are uniform in the quantum parameter $oldsymbol{\delta}$.

Classical Boltzmann solutions are recovered as $oldsymbol{\delta o 0}$.

Abstract

In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter $\delta$ that can decrease from $\delta=1$ for the Fermi-Dirac particles to $\delta=0$ for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space $L^{\infty}_{T}L^{\infty}_{v,x}\cap L^{\infty}_{T}L^{\infty}_{x}L^1_v$ with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound $F(t,x,v)\leq \frac{1}{\delta}$. The key point is that the obtained estimates are uniform in the quantum parameter $0< \delta\leq1$. In particular, as $\delta\to 0$ we can recover the results on the classical Boltzmann equation around global Maxwellians for which solutions may have arbitrarily large oscillations.