Homogenization and localization in locally periodic transport

¹ Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France, and CEA Saclay, DEN/DM2S, 91191 Gif-sur-Yvette, France; gregoire.allaire@polytechnique.fr.
² Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA; gb2030@columbia.edu.
³ CEA Saclay, DEN/DM2S, 91191 Gif-sur-Yvette, France; siess@soleil.serma.cea.fr.

Received: 14 December 2001

Abstract

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 ε-periodic functions modulated by a macroscopic variable, where ε is a small parameter. The mean free path of the particles is also of order ε. 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 definite. This assumption yields a concentration phenomenon around x₀, as ε goes to 0, at a new scale of the order of which is superimposed with the usual ε 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 , i.e., of the form , 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.