Regularity of stationary solutions to the linearized Boltzmann equations

TL;DR

This paper proves that stationary solutions to the linearized Boltzmann equations in convex domains are Hölder continuous away from the boundary, given regular incoming data, by transferring velocity regularity to space.

Contribution

It establishes the Hölder continuity of stationary solutions in convex domains for gases with cutoff potentials, extending regularity results to boundary-adjacent regions.

Findings

Stationary solutions are Hölder continuous with order 1/2^- away from the boundary.

Regularity of incoming data is preserved in the solution.

Method transfers velocity regularity to spatial regularity through transport and collision processes.

Abstract

We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are H\"{o}lder continuous with order $\frac1{2}^-$ away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.