Diffusive Limit of the Unsteady Neutron Transport Equation in Bounded Domains

The justification of hydrodynamic limits in non-convex domains has long been an open problem due to the singularity at the grazing set. In this paper, we investigate the unsteady neutron transport equation in a general bounded domain with the in-flow, diffuse-reflection, or specular-reflection boundary condition. Using a novel kernel estimate, we demonstrate the optimal $L^2$ diffusive limit in the presence of both initial and boundary layers. Previously, this result was only proved for convex domains when the time variable is involved. Our approach is highly robust, making it applicable to all basic types of physical boundary conditions.