Asymptotic Analysis of Transport Equation in Bounded Domains

Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number $\epsilon\rightarrow 0^+$. Using Hilbert expansion, we rigorously justify that the solution of steady problem converges to that of the Laplace's equation, and the solution of unsteady problem converges to that of the heat equation. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. This problem can be formulated in many different settings, and the above one is probably the most physically significant and most mathematically challenging. We have to utilize almost all methods and techniques we developed in a series of papers in the past decade, and bring novel ideas to treat the new complications. The difficulty mainly comes from three sources: 3D domain, boundary layer regularity, and time dependence. To fully solve this problem, we introduce several techniques: (1) boundary layer with geometric correction; (2) remainder estimates with $L^2-L^{2m}-L^{\infty}$ framework; boundary layer decomposition.