Asymptotic Analysis of Transport Equation in Annulus

TL;DR

This paper investigates the diffusive limit of a steady neutron transport equation in an annulus, providing a counterexample to classical approximation results through a novel boundary layer expansion with geometric correction.

Contribution

It introduces a new boundary layer expansion with geometric correction that challenges the classical approximation theorem for neutron transport in an annulus.

Findings

Classical theorem on solution approximation is invalid in certain cases.

Constructs a counterexample demonstrating limitations of existing theory.

Proposes a modified boundary layer expansion with geometric correction.

Abstract

We consider the diffusive limit of a steady neutron transport equation with one-speed velocity in a two-dimensional annulus. A classical theorem states that the solution can be approximated in $L^{\infty}$ by the leading order interior solution plus the corresponding Knudsen layers in the diffusive limit. In this paper, we construct a counterexample of this result via a different boundary layer expansion with geometric correction.