$ L^2 $ diffusive expansion for neutron transport equation

One of the most classical and fundamental mathematical problems in kinetic theory is to study the diffusive limit of the neutron transport equation. As $ {\varepsilon}{\rightarrow}0 $, the phase space density $ u^{{\varepsilon}}({x}, {w}) $

$\begin{align} {w}\cdot\nabla_x u^{{\varepsilon}}+{\varepsilon}^{-1}\left(u^{{\varepsilon}}-\overline{u^{{\varepsilon}}}\right) = 0, \quad u^{{\varepsilon}}\big|_{{x}\in{\partial}\Omega, {w}\cdot{n}<0} = g, \quad \overline{u^{{\varepsilon}}}({x}): = \frac{1}{4\pi}\int_{{{\mathbb{S}}}^2}u^{{\varepsilon}}({x}, {w}){\mathrm{d}}{{w}}, \ \ \ \ \ \ \ \ {\rm{(0.1)}}\end{align} $

converges to the interior solution $ {U}_0({x}) $:

$\begin{align} -{\Delta_x}{U}_0 = 0, \quad {U}_0\big|_{{\partial}\Omega} = U^B_{0, \infty}, \ \ \ \ \ \ \ \ {\rm{(0.2)}}\end{align}$

in which $ U^B_{0, \infty} $ is obtained by solving the Milne problem for the celebrated boundary layer correction $ U^B_0 $. The function $ g $ represents the inflow data, and $ {n} $ is the unit outward normal to the smooth bounded domain $ \Omega $. Surprisingly, we found ^[1,2] that the expected $ L^{\infty} $ expansion

$\begin{align} \left\Vert{u^{{\varepsilon}}-{U}_0-U^B_0}\right\Vert_{L^{\infty}}{\lesssim}{\varepsilon} \ \ \ \ \ \ \ \ {\rm{(0.3)}}\end{align} $

is invalid due to the grazing singularity of $ U^B_0 $. As a result, the corresponding well-known mathematical theory breaks down, and the diffusive limit has remained an outstanding question. A satisfactory theory was developed for convex domains ^{[1,2,3,4,5,6]} by constructing new boundary layers with favorable $ {\varepsilon} $-geometric corrections. However, this approach is inapplicable in non-convex domains. In this paper, we settle this open question affirmatively in the $ L^2 $ sense. The convergence

$\begin{align} \left\Vert{u^{{\varepsilon}}-{U}_0}\right\Vert_{L^2}{\lesssim}{\varepsilon}^{\frac{1}{2}}\ \ \ \ \ \ \ \ {\rm{(0.4)}} \end{align}$

holds for general smooth domains, including non-convex ones. We achieve this by discovering a novel and optimal $ L^2 $ expansion theory that reveals a surprising $ {\varepsilon}^{\frac{1}{2}} $ gain for the average of the remainder, and by choosing a test function with a new cancellation via conservation of mass. We also introduce a cutoff boundary layer $ U^B_0 $ and investigate its delicate regularity estimates to control the source terms of the remainder equation with the help of Hardy's inequality. Notably, our new cutoff boundary layer $ U^B_0 $ determines $ {U}_0 $, despite its absence in the estimate.