Hölder regularity of solutions of the steady Boltzmann equation with soft potentials

We consider the Hölder regularity of solutions to the steady Boltzmann equation with in-flow boundary condition in bounded and strictly convex domains $\Omega \subset R^3$ 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 Hölder continuous, and its order depends not only on the regularity of the incoming boundary data, but also on the potential power γ. The result for modulated soft potential case $-2<\gamma <0$ is similar to hard potential case $(0\le \gamma <1)$ since we have $C^1$ velocity regularity from collision part. However, we observe that for very soft potential case $(-3<\gamma \le -2)$, the regularity in velocity obtained by the collision part is lower (Hölder only), but the boundary regularity still can transfer to solution (in both space and velocity) by transport and collision part under the restriction of γ.