H\"older regularity of solutions of the steady Boltzmann equation with soft potentials

TL;DR

This paper proves that solutions to the steady Boltzmann equation with soft potentials in convex domains are uniquely bounded and H"older continuous, with regularity depending on boundary data and potential strength.

Contribution

It establishes the H"older regularity of solutions for soft potentials, extending previous results to a broader range of potential parameters and boundary conditions.

Findings

Solutions are unique and bounded in $L^{ abla}$ norm.

Solutions exhibit H"older continuity depending on boundary data and potential.

Regularity in velocity decreases for very soft potentials, but boundary regularity transfers to solutions.

Abstract

We consider the H\"older regularity of solutions to the steady Boltzmann equation with in-flow boundary condition in bounded and strictly convex domains $\Omega\subset\mathbb{R}^{3}$ for gases with cutoff soft potential $(-3<\gamma<0)$. We prove that there is a unique solution with a bounded $L^{\infty}$ norm in space and velocity. This solution is H\"older continuous, and it's order depends not only on the regularity of the incoming boundary data, but also on the potential power $\gamma$. The result for modulated soft potential case $-2<\gamma<0$ is similar to hard potential case $(0\leq\gamma<1)$ since we have $C^{1}$ velocity regularity from collision part. However, we observe that for very soft potential case $(-3<\gamma\leq -2)$, the regularity in velocity obtained by the collision part is lower (H\"older only), but the boundary regularity still can transfer to solution (in both space and velocity) by transport and collision part under the restriction of $\gamma$.