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Homogenization in perforated domainswith 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 whosegeometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodicperforations, 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 nothold in this model, a first difficulty is to get a priori estimates of the solutions. We obtain them in a weighted space where theweight is the principal eigenfunction of an “adjoint” periodictime-dependent eigenvalue problem. This problem is not aclassical one, and its investigation is an importantpart of this work. Then, by using the multiple scale method, we construct theleading 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.

Information

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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