Decay of the Boltzmann Equation with the Specular Boundary Condition in Non-convex Cylindrical Domains

The basic question about the existence, uniqueness, and stability of the Boltzmann equation in general non-convex domains with the specular reflection boundary condition has been widely open. In this paper, we consider cylindrical domains whose cross section is generally non-convex analytic bounded planar domain. We establish a global well-posedness and asymptotic stability of the Boltzmann equation with the specular reflection boundary condition. Our method consists of the delicate construction of \({\epsilon}\)-tubular neighborhoods of billiard trajectories which bounce infinitely many times or hit the boundary tangentially at some moment, and sharp estimates of the size of such neighborhoods.