Boundary Layer of Transport Equation with In-Flow Boundary

TL;DR

This paper analyzes the diffusive limit of the steady neutron transport equation in convex domains, addressing boundary layers through advanced decomposition and estimates, advancing understanding of boundary layer behavior in transport problems.

Contribution

It introduces a novel boundary data decomposition and weighted estimates for the Milne problem, improving the analysis of boundary layers in neutron transport equations.

Findings

Established diffusive limit with boundary layers present

Developed weighted $W^{1,inity}$ estimates for the Milne problem

Provided stronger remainder estimates using an $L^{2m}-L^{inity}$ framework

Abstract

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 $W^{1,\infty}$ estimates for the Milne problem with geometric correction in convex domains, as well as an $L^{2m}-L^{\infty}$ framework which yields stronger remainder estimates.