Regularity of the Boltzmann equation in convex domains

A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical C^1 solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct W^{1,p} solutions for 1 < p < 2 and weighted W^{1,p} solutions for 2 ≤ p ≤∞as well.