$L^2$ Diffusive Expansion For Neutron Transport Equation

TL;DR

This paper introduces a new $L^2$ diffusive expansion theory for the neutron transport equation that overcomes previous limitations caused by grazing singularities, applicable to both convex and non-convex domains.

Contribution

It develops an optimal $L^2$ expansion approach that achieves an $rac{1}{2}$-power gain for the average of the remainder, extending mathematical understanding to non-convex domains.

Findings

Established an $L^2$ expansion with $rac{1}{2}$-power gain

Extended diffusive expansion theory to non-convex domains

Overcame previous barriers caused by grazing singularities

Abstract

Grazing set singularity leads to a surprising counter-example and breakdown of the classical mathematical theory for $L^{\infty}$ diffusive expansion of neutron transport equation with in-flow boundary condition in term of the Knudsen number $\varepsilon$, one of the most classical problems in the kinetic theory. Even though a satisfactory new theory has been established by constructing new boundary layers with favorable $\varepsilon$-geometric correction for convex domains, the severe grazing singularity from non-convex domains has prevented any positive mathematical progress. We develop a novel and optimal $L^2$ expansion theory for general domain (including non-convex domain) by discovering a surprising $\varepsilon^{\frac{1}{2}}$ gain for the average of remainder.