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Homogenization in perforated domains with rapidly pulsing perforations

Published online by Cambridge University Press:  15 September 2003

Doina Cioranescu  and
Andrey L. Piatnitski
Doina Cioranescu
Affiliation:
Laboratoire Jacques-Louis Lions (Analyse Numérique), Université Paris VI – CNRS, 175 rue du Chevaleret, 75013 Paris, France; cioran@ann.jussieu.fr.
Andrey L. Piatnitski
Affiliation:
Narvik University College HiN, Department of Mathematics, P.O. Box 385, 8505 Narvik, Norway. Lebedev Physical Institute, Russian Academy of Science, Leninski Prospect 53, Moscow 117333, Russia; andrey@sci.lebedev.ru.
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Abstract

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the solutions. We obtain them in a weighted space where the weight is the principal eigenfunction of an “adjoint” periodic time-dependent eigenvalue problem. This problem is not a classical one, and its investigation is an important part of this work. Then, by using the multiple scale method, we construct the leading terms of a formal expansion (with respect to ε) of the solution and give the limit “homogenized” problem. An interesting peculiarity of the model is that, depending on the geometry of the holes, a large convection term may appear in the limit equation.

Keywords

Homogenizationperforated domainspulsing perforationsmultiple scale method.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 9 , January 2003 , pp. 461 - 483
Copyright
© EDP Sciences, SMAI, 2003

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