Existence of solutions to Dirichlet boundary value problems of the stationary relativistic Boltzmann equation

TL;DR

This paper establishes the existence of solutions to the steady-state relativistic Boltzmann equation with Dirichlet boundary conditions, analyzing how the Mach number influences solvability and extending prior non-relativistic results.

Contribution

It provides the first rigorous analysis of Dirichlet boundary value problems for the relativistic Boltzmann equation, highlighting the role of Mach number and boundary data proximity.

Findings

Unique solutions exist for Mach number less than -1 with data close to Maxwellian.

Solutions exist for Mach number greater than -1 under small boundary data conditions.

The approach combines macro-micro decomposition with artificial damping and velocity-weighted energy estimates.

Abstract

In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the outgoing particles at the boundary and a relativistic global Maxwellian with nonzero macroscopic velocities at the far field. We first explicitly address the sound speed for the relativistic Maxwellian in the far field, according to the eigenvalues of an operator based on macro-micro decomposition. Then we demonstrate that the solvability of the problem varies with the Mach number $\mathcal{M}^\infty$. If $\mathcal{M}^\infty<-1$, a unique solution exists connecting the Dirichlet data and the far field Maxwellian when the boundary data is sufficiently close to the Maxwellian. If $\mathcal{M}^\infty>-1$, such a solution exists only if the outgoing boundary data is small and satisfies certain solvability conditions. The proof is based on the macro-micro decomposition of solutions combined with an artificial damping term. A singular in velocity (at $p_1=0$ and $|p|\gg 1$) and spatially exponential decay weight is chosen to carry out the energy estimates. The result extends the previous work [Ukai, Yang, Yu, Comm. Math. Phys. 236, 373-393, 2003] to the relativistic problem.