Diffusive Limit of the Boltzmann Equation in Bounded Domains

Abstract: The rigorous justification of the hydrodynamic limits of kinetic equations in bounded domains has been actively investigated in recent years. In spite of the progress for the diffuse-reflection boundary case, the more challenging in-flow boundary case, in which the leading-order boundary layer effect is non-negligible, still remains open. In this work, we consider the stationary and evolutionary Boltzmann equation with the in-flow boundary in general (convex or non-convex) bounded domains, and demonstrate their incompressible Navier-Stokes-Fourier (INSF) limits in $L^2$. Our method relies on a novel and surprising gain of $\varepsilon^{\frac{1}{2}}$ in the kernel estimate, which is rooted from a key cancellation of delicately chosen test functions and conservation laws. We also introduce the boundary layer with grazing-set cutoff and investigate its BV regularity estimates to control the source terms of the remainder equation with the help of Hardy's inequality.