# Homogenization and localization in locally periodic transport

Grégoire Allaire; Guillaume Bal; Vincent Siess

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- 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 (2010): 1-30. <http://eudml.org/doc/90646>.

@article{Allaire2010,

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 $\sqrt\{\varepsilon\}$ 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 $\sqrt\{\varepsilon\}$, 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.; transport; spectral transport equation; concentration phenomenon},

language = {eng},

month = {3},

pages = {1-30},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/90646},

volume = {8},

year = {2010},

}

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

DA - 2010/3//

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
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 $\sqrt{\varepsilon}$ 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 $\sqrt{\varepsilon}$, 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.; transport; spectral transport equation; concentration phenomenon

UR - http://eudml.org/doc/90646

ER -

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