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 rootepsilon 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 rootepsilon, i.e., of the form exp( -1/2epsilon M(x - x(0)) . (x - x(0))), 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.