Homogenization and localization in locally periodic transport

Free access article

Issue		ESAIM: COCV Volume 8, 2002 A tribute to JL Lions


Page(s)		1 - 30
DOI		10.1051/cocv:2002016

ESAIM: COCV, June 2002, Vol. 8, pp. 1-30
DOI: 10.1051/cocv:2002016

Homogenization and localization in locally periodic transport

Grégoire Allaire¹, Guillaume Bal² and Vincent Siess³

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 December 14, 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 $\varepsilon$ -periodic functions modulated by a macroscopic variable, where $\varepsilon$ is a small parameter. The mean free path of the particles is also of order $\varepsilon$ . 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 $\varepsilon$ goes to 0, at a new scale of the order of $\sqrt{\varepsilon}$ which is superimposed with the usual $\varepsilon$ 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 $\sqrt{\varepsilon}$ , i.e., of the form $\exp \left (- \frac {1} {2 \varepsilon} M (x-x_0)\cdot (x-x_0) \right )$ , 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.

Mathematics Subject Classification. 35B27, 82D75.

Key words: Homogenization, localization, transport.

© EDP Sciences, SMAI 2002

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