On the Boltzmann-Fermi-Dirac Equation for Hard Potential: Global Existence and Uniqueness, Gaussian Lower Bound, and Moment Estimates

TL;DR

This paper extends classical Boltzmann equation results to the Fermi-Dirac setting, establishing global existence, uniqueness, bounds, and moment estimates for solutions with hard potentials and angular cutoff.

Contribution

It provides the first comprehensive analysis of the Boltzmann-Fermi-Dirac equation, including existence, uniqueness, Gaussian bounds, and moment propagation for hard potentials.

Findings

Existence and uniqueness of global solutions

Gaussian lower bounds for non-saturated equilibria

Propagation of polynomial and exponential moments

Abstract

In this paper, we study the global existence and uniqueness, Gaussian lower bound, and moment estimates in the spatially homogeneous Boltzmann equation for Fermi-Dirac particles for hard potential ($0\leq \gamma\leq 2$) with angular cutoff $b$. Our results extend classical results to the Boltzmann-Fermi-Dirac setting. In detail, (1) we show existence, uniqueness, and $L^1_2$ stability of global-in-time solutions of the Boltzmann-Fermi-Dirac equation. (2) Assuming the solution is not a saturated equilibrium, we prove creation of a Gaussian lower bound for the solution. (3) We prove creation and propagation of $L^1$ polynomial and exponential moments of the solution under additional assumptions on the angular kernel $b$ and $0<\gamma\leq 2$. (4) Finally, we show propagation of $L^\infty$ Gaussian and polynomial upper bounds when $b$ is constant and $0<\gamma\leq 1$.