Issue |
ESAIM: COCV
Volume 8, 2002
A tribute to JL Lions
|
|
---|---|---|
Page(s) | 1 - 30 | |
DOI | https://doi.org/10.1051/cocv:2002016 | |
Published online | 15 August 2002 |
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
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.
Mathematics Subject Classification: 35B27 / 82D75
Key words: Homogenization / localization / transport.
© EDP Sciences, SMAI, 2002
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