Homogenization and localization in locally periodic transport

In this paper, we study the homogenization and localization of a spectral transport equation posed in a locally periodic heterogeneous domain. This equation models the equilibrium of particles interacting with an underlying medium in the presence of a creation mechanism such as, for instance, neutrons in nuclear reactors. The physical coefficients of the domain are $\epsilon$-periodic functions modulated by a macroscopic variable, where $\epsilon$ is a small parameter. The mean free path of the particles is also of order $\epsilon$. We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point $x_0$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around $x_0$, as $\epsilon$ goes to $0$, at a new scale of the order of $\sqrt{\epsilon }$ which is superimposed with the usual $\epsilon$ oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The first one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the first eigenvector of a homogenized diffusion equation in the whole space with quadratic potential, rescaled by a factor $\sqrt{\epsilon }$, i.e., of the form $exp\left(-\frac{1}{2\epsilon }M(x-x_0)·(x-x_0)\right)$, where $M$ is a positive definite matrix. Furthermore, the eigenvalue corresponding to this second term gives a first-order correction to the eigenvalue of the heterogeneous spectral transport problem.