$L^2$ Diffusive Expansion For Neutron Transport Equation

Grazing set singularity leads to a surprising counter-example and breakdown of the classical mathematical theory for $L^{\mathrm{\infty }}$ diffusive expansion of neutron transport equation with in-flow boundary condition in term of the Knudsen number $\epsilon$, 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 $\epsilon$-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 ${\epsilon }^{\frac{1}{2}}$ gain for the average of remainder.