On a Boltzmann equation for Compton scattering, from non relativistic electrons at low density

TL;DR

This paper develops and analyzes a Boltzmann equation model for photon-electron interactions via Compton scattering at low electron densities, proving existence of solutions and exploring asymptotic behavior.

Contribution

It introduces a truncated, well-justified form of the Boltzmann equation for Compton scattering and proves the existence of weak solutions for both the full and simplified models.

Findings

Existence of weak solutions for the full Boltzmann equation.

Existence of solutions for the simplified low-temperature model.

Description of long-time behavior of solutions.

Abstract

A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon's energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.