Conditional $L^{\infty }$ estimates for the non-cutoff Boltzmann equation in a bounded domain

Abstract

We consider weak solutions of the inhomogeneous non-cutoff Boltzmann equation in a bounded domain with any of the usual physical boundary conditions: in-flow, bounce-back, specular-reflection and diffuse-reflection. When the mass, energy and entropy densities are bounded above, and the mass density is bounded away from vacuum, we obtain an estimate of the $L^{\infty }$ norm of the solution depending on the macroscopic bounds on these hydrodynamic quantities only. It is a regularization effect in the sense that the initial data is not required to be bounded.