Diffusive Limit of Transport Equation in 3D Convex Domains

Abstract

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 Laplace’s equation, and the solution of unsteady problem converges to that of heat equation. This is the most difficult case of a long-term project on asymptotic analysis of kinetic equations in bounded domains. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. The upshot of this paper is a novel boundary layer decomposition argument in 3D and $L^2-L^{2m}-L^{\infty }$ bootstrapping method for time-dependent problem.

Funding

National Science Foundation(1853002)