L² diffusive expansion for neutron transport equation

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Abstract

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

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

converges to the interior solution U₀(x):

(0.2)$-\Delta_{x} U_{0}=0,\left.\quad U_{0}\right|_{\partial \Omega}=U_{0, \infty}^{B},$

in which $U_{0, \infty}^{B}$ is obtained by solving the Milne problem for the celebrated boundary layer correction U^B₀. The function g represents the inflow data, and n is the unit outward normal to the smooth bounded domain Ω. Surprisingly, we found [1,2] that the expected

(0.3)$\left\|u^{\varepsilon}-U_{0}-U_{0}^{B}\right\|_{L^{\infty}} \lesssim \varepsilon$

is invalid due to the grazing singularity of $U_{0}^{B}$. 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 [1–6] by constructing new boundary layers with favorable ε-geometric corrections. However, this approach is inapplicable in non-convex domains. In this paper, we settle this open question affirmatively in the L² sense. The convergence

(0.4)$\left\|u^{\varepsilon}-U_{0}\right\|_{L^{2}} \lesssim \varepsilon^{\frac{1}{2}}$

holds for general smooth domains, including non-convex ones. We achieve this by discovering a novel and optimal L² 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 $U_{0}^{B}$ 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 $U_{0}^{B}$ determines U₀, despite its absence in the estimate.

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