Damping of kinetic transport equation with diffuse boundary condition

Abstract

We prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as $t^{-3}$ when the domain is any general strictly convex subset of ${\mathbb{R}}^3$ with the smooth boundary of the diffuse boundary condition. We prove the theorem by establishing a novel $L^1$-$L^{\infty }$ framework via stochastic cycles.

15 pages