Homogenization and localization in locally periodic transport

Abstract. In this paper, we study the homogenization and localization of a spectral transport equa-tion 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 coecients of the domain are "-periodic functions modu-lated by a macroscopic variable, where " is a small parameter. The mean free path of the particles is also of order ". We assume that the leading eigenvalue of the periodicity cell problem admits a unique minimum in the domain at a point x0 where its Hessian matrix is positive denite. This assumption yields a concentration phenomenon around x0, as " goes to 0, at a new scale of the order of p " which is superimposed with the usual " oscillations of the homogenized limit. More precisely, we prove that the particle density is asymptotically the product of two terms. The rst one is the leading eigenvector of a cell transport equation with periodic boundary conditions. The second term is the rst eigenvector of a homogenized diusion equation in the whole space with quadratic potential, rescaled by a facto