Validity and regularization of classical half-space equations

TL;DR

This paper investigates the classical half-space equation (CHS) in 2D, revealing its limitations in boundary layer representation, but also demonstrating its validity in providing correct boundary conditions for the interior Laplace equation through regularization.

Contribution

The paper introduces a regularization technique for CHS and proves its effectiveness in ensuring correct boundary conditions for the Laplace equation despite boundary layer misrepresentation.

Findings

CHS does not accurately capture boundary layer behavior on the 2D disk.

First-order regularization of CHS corrects boundary condition issues.

CHS remains valid for interior boundary condition recovery despite boundary layer inaccuracies.

Abstract

Recent result [Wu and Guo, Comm. Math. Phys., 2015] has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this paper we develop a regularization technique for CHS to any arbitrary order and use its first-order regularization to show that in the case of the 2D unit disk, although CHS misrepresents the boundary layer behaviour, it does give the correct boundary condition for the interior macroscopic (Laplace) equation. Therefore CHS is still a valid equation to recover the correct boundary condition for the interior Laplace equation over the 2D unit disk.