Boundary Layer of Transport Equation with In-Flow Boundary

Consider the steady neutron transport equation in 2D convex domains with in-flow boundary condition. In this paper, we establish the diffusive limit while the boundary layers are present. Our contribution relies on a delicate decomposition of boundary data to separate the regular and singular boundary layers, novel weighted $W^{1,\mathrm{\infty }}$ estimates for the Milne problem with geometric correction in convex domains, as well as an $L^{2m}-L^{\mathrm{\infty }}$ framework which yields stronger remainder estimates.