The Global Well-Posedness of the Relativistic Boltzmann Equation with Diffuse Reflection Boundary Condition in Bounded Domains

TL;DR

This paper proves the global existence and stability of solutions to the relativistic Boltzmann equation with diffuse reflection boundary conditions in bounded domains, including uniform estimates independent of the speed of light.

Contribution

It establishes the first mathematical theory for global solutions of the relativistic Boltzmann equation in bounded domains with diffuse reflection boundary conditions.

Findings

Constructed a unique non-negative stationary solution.

Proved exponential stability of the stationary solution.

Obtained uniform $L^{ abla}$$^{ ext{infinity}}$ bounds independent of the speed of light.

Abstract

The relativistic Boltzmann equation in bounded domains has been widely used in physics and engineering, for example, Tokamak devices in fusion reactors.In spite of its importance, there has, to the best of our knowledge, been no mathematical theory on the global existence of solutions to the relativistic Boltzmann equation in bounded domains. In the present paper, assuming that the motion of single-species relativistic particles in a bounded domain is governed by the relativistic Boltzmann equation with diffuse reflection boundary conditions of non-isothermal wall temperature of small variations around a positive constant, and regarding the speed of light $\mathfrak{c}$ as a large parameter, we first construct a unique non-negative stationary solution $F_{*}$, and further establish the dynamical stability of such stationary solution with exponential time decay rate. We point out that the $L^{\infty}$-bound of perturbations for both steady and non-steady solutions are independent of the speed of light $\mathfrak{c}$, and such uniform in $\mathfrak{c}$ estimates will be useful in the study of Newtonian limit in the future.