Free Molecular Flow with Boundary Effect

We study the dissipative effect of the boundary condition in the kinetic theory. We focus our study on the simplest situation of the free molecular flow with diffuse reflection boundary condition and constant boundary temperature, T_*. The geometry is also chosen to be the simplest ones, a bounded symmetric domain in ℝ^d: an interval for d = 1, a disk for d = 2, and a ball for d = 3. It is shown that the solution converges to the global Maxwellian with the given boundary temperature T_*. We obtain the optimal convergence rates of (t + 1)^-d. The stochastic formulation of Shih-Hsien Yu is refined and generalized for our analysis.