Regularity of Stationary Boltzmann Equation in Convex Domains

Abstract

The higher regularity estimate has been a challenging question regarding the Boltzmann equation in bounded domains. Indeed, it is well-known to have “the non-existence of a second order derivative at the boundary” in Guo et al. (Invent Math 207:115–290, 2017) even for symmetric convex domains such as a disk or sphere. In this paper, we answer this question in the affirmative by constructing the \(C^{1,\beta }\) solutions away from the grazing boundary, for any \(\beta <1\), to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely.