# Homogenization and localization in locally periodic transport

Grégoire Allaire; Guillaume Bal; Vincent Siess

ESAIM: Control, Optimisation and Calculus of Variations (2002)

- Volume: 8, page 1-30
- ISSN: 1292-8119

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topAllaire, Grégoire, Bal, Guillaume, and Siess, Vincent. "Homogenization and localization in locally periodic transport." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 1-30. <http://eudml.org/doc/244643>.

@article{Allaire2002,

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 $x_0$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around $x_0$, 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.},

author = {Allaire, Grégoire, Bal, Guillaume, Siess, Vincent},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {homogenization; localization; transport; spectral transport equation; concentration phenomenon},

language = {eng},

pages = {1-30},

publisher = {EDP-Sciences},

title = {Homogenization and localization in locally periodic transport},

url = {http://eudml.org/doc/244643},

volume = {8},

year = {2002},

}

TY - JOUR

AU - Allaire, Grégoire

AU - Bal, Guillaume

AU - Siess, Vincent

TI - Homogenization and localization in locally periodic transport

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2002

PB - EDP-Sciences

VL - 8

SP - 1

EP - 30

AB - 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 $x_0$ where its hessian matrix is positive definite. This assumption yields a concentration phenomenon around $x_0$, 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.

LA - eng

KW - homogenization; localization; transport; spectral transport equation; concentration phenomenon

UR - http://eudml.org/doc/244643

ER -

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