Geometric Correction in Diffusive Limit of Neutron Transport Equation in 2D Convex Domains

TL;DR

This paper establishes the diffusive limit of the neutron transport equation in 2D convex domains, introducing new estimates for the Milne problem with geometric correction to improve the mathematical understanding of neutron behavior.

Contribution

It provides the first proof of diffusive limit in 2D convex domains using novel $W^{1,ty}$ estimates and an $L^{2m}-L^{ty}$ framework for the Milne problem with geometric correction.

Findings

Established diffusive limit in 2D convex domains.

Developed novel $W^{1,ty}$ estimates for the Milne problem.

Introduced an $L^{2m}-L^{ty}$ framework for stronger remainder estimates.

Abstract

Consider the steady neutron transport equation with diffusive boundary condition. In [Wu and Guo(2015) Comm. Math. Phys.] and [Wu and Yang and Guo(2016) Preprint], it was discovered that geometric correction is necessary for the Milne problem of Knudsen-layer construction in a disk or annulus. In this paper, we establish diffusive limit for a 2D convex domain. Our contribution relies on novel $W^{1,\infty}$ estimates for the Milne problem with geometric correction in the presence of a convex domain, as well as an $L^{2m}-L^{\infty}$ framework which yields stronger remainder estimates.