Existence and conservation laws for the Boltzmann-Fermi equation in a general domain

TL;DR

This paper proves the existence of solutions for the Boltzmann-Fermi-Dirac equation in bounded domains with boundary reflections and demonstrates that these solutions conserve mass, momentum, and energy locally.

Contribution

It introduces a new existence proof for the Boltzmann-Fermi-Dirac equation in bounded domains with specular reflection, utilizing characteristic lines and dispersion techniques.

Findings

Existence of solutions in bounded domains with specular reflection

Solutions satisfy local conservation laws of mass, momentum, and energy

Applicable to integrable collision kernels

Abstract

We prove an existence theorem for the Boltzmann-Fermi-Dirac equation for integrable collision kernels in possibly bounded domains with specular reflection at the boundaries, using the characteristic lines of the free transport. We then obtain that the solution satisfies the local conservations of mass, momentum and kinetic energy thanks to a dispersion technique.