The Non-cutoff Boltzmann Equation in Bounded Domains

The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a challenging open problem. In this paper, we study the stability and long-time dynamics of the Boltzmann equation near a global Maxwellian without angular cutoff assumption in a general $C^3$ bounded domain $\Omega$ (including convex and non-convex cases) with physical boundary conditions: inflow boundary and Maxwell-reflection boundary with accommodation coefficient $\al\in(0,1)$. We obtain the global-in-time existence, which has an exponential decay rate towards the global Maxwellian for both hard and soft potentials. The crucial methods are the forward-backward extension of the boundary problem to the whole space by Vlasov-type equations, a level-function trace lemma, an improved velocity averaging lemma with less regularity but without cutoff in velocity, and an extra damping provided by the advection operator, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ energy method.