Global stability of vacuum for the relativistic Vlasov-Maxwell-Boltzmann system

TL;DR

This paper proves the global stability of vacuum solutions for the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system with small initial data, using advanced mathematical techniques that are uniform in the speed of light.

Contribution

It introduces a novel chain rule for the relativistic Boltzmann collision operator compatible with vector field methods, enabling uniform bounds in the speed of light.

Findings

Established global existence and nonlinear stability of vacuum

Derived a chain rule for the relativistic Boltzmann collision operator

Achieved bounds uniform in the speed of light

Abstract

We consider the three-dimensional relativistic Vlasov-Maxwell-Boltzmann system, where the speed of light $c$ is an arbitrary constant no less than 1, and we establish global existence and nonlinear stability of the vacuum for small initial data, with bounds that are uniform in $c$. The analysis is based on the vector field method combined with the Glassey-Strauss decomposition of the electromagnetic field, and does not require any compact support assumption on the initial data. A key ingredient of the proof is the derivation of a chain rule for the relativistic Boltzmann collision operator that is compatible with the commutation properties of the vector fields. These tools allow us to control the coupled kinetic and electromagnetic equations and to obtain global stability near vacuum.