Kinetic Wedge Layers and Diffusive Limit of Neutron Transport in Polygonal Domains

Abstract

We establish the diffusive limit of the stationary neutron transport equation with velocity-dependent inflow data in polygonal domains. On a bounded convex polygon we assemble a composite approximation from an interior harmonic field, flat side layers, and kinetic wedge layers, and we prove that the solution converges to this composite in $L^{\infty}$ at an explicit algebraic rate, and, uniformly on compact subsets of the interior, to the harmonic field itself. We also give a complete formulation and well-posedness theory for the kinetic wedge layer, including its algebraic decay. The proof combines a characteristic stability estimate, a weighted Mellin mapping theorem, a two-depth construction and matching scheme, and a shifted superharmonic barrier for the wedge corrector. As a secondary result, we prove convergence at the square-root rate in $L^{2}$ on any bounded simple polygon, including those with reentrant vertices. That argument needs only an endpoint-truncated side layer and a two-test cancellation, and no kinetic wedge layer at all.