The Mixed Convex-Concave effect on the regularity of a Boltzmann solution

The geometric properties of domains are well-known to be crucial factors influencing the regularity of Boltzmann boundary problems. In the presence of non-convex physical boundaries, highly singular behaviors are investigated including the propagation of discontinuities \cite{Kim11} and Hölder regularity \cite{CD2022}. In this paper, we focus on studying the $C^{0,\frac{1}{4}-}$ Hölder regularity of the Boltzmann equation within concentric cylinders where both convex and concave features are present on the boundaries. Our findings extend the previous results of \cite{CD2022}, which primarily addressed concavity while considering the exterior of convex objects.