Homogenization and localization in locally periodic transport
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 1-30.

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 $\sqrt{\epsilon }$ 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 $\sqrt{\epsilon }$, i.e., of the form $exp\left(-\frac{1}{2\epsilon }M\left(x-{x}_{0}\right)·\left(x-{x}_{0}\right)\right)$, 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.

DOI : https://doi.org/10.1051/cocv:2002016
Classification : 35B27,  82D75
Mots clés : homogenization, localization, transport
@article{COCV_2002__8__1_0,
author = {Allaire, Gr\'egoire and Bal, Guillaume and Siess, Vincent},
title = {Homogenization and localization in locally periodic transport},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--30},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002016},
zbl = {1065.35042},
mrnumber = {1932943},
language = {en},
url = {www.numdam.org/item/COCV_2002__8__1_0/}
}
Allaire, Grégoire; Bal, Guillaume; Siess, Vincent. Homogenization and localization in locally periodic transport. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002) , pp. 1-30. doi : 10.1051/cocv:2002016. http://www.numdam.org/item/COCV_2002__8__1_0/

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