On the convergence to equilibrium for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles

TL;DR

This paper proves strong and time-averaged convergence to equilibrium for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles, including Coulomb interactions, using entropy and moment methods.

Contribution

It introduces a novel approach leveraging positive kernel dominance to establish convergence without analyzing cubic collision integrals.

Findings

Proves strong convergence to equilibrium for Fermi-Dirac particles.

Includes Coulomb potential with weaker angular cutoff.

Utilizes entropy dissipation and moment estimates.

Abstract

In this paper we prove the strong and time-averaged strong convergence to equilibrium for solutions (with general initial data) of the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. The assumption on the collision kernel includes the Coulomb potential with a weaker angular cutoff. The proof is based on moment estimates, entropy dissipation inequalities, regularity of the collision gain operator, and a new observation that many collision kernels are larger than or equal to some completely positive kernels, which enables us to avoid dealing with the convergence problem of the cubic collision integrals.