Issue |
ESAIM: COCV
Volume 9, February 2003
|
|
---|---|---|
Page(s) | 461 - 483 | |
DOI | https://doi.org/10.1051/cocv:2003023 | |
Published online | 15 September 2003 |
Homogenization in perforated domains with rapidly pulsing perforations
1
Laboratoire Jacques-Louis Lions (Analyse Numérique),
Université Paris VI – CNRS,
175 rue du Chevaleret,
75013 Paris,
France;
cioran@ann.jussieu.fr.
2
Narvik University College HiN,
Department of Mathematics,
P.O. Box 385,
8505 Narvik,
Norway.
3
Lebedev Physical Institute,
Russian Academy of Science,
Leninski Prospect 53,
Moscow 117333,
Russia;
andrey@sci.lebedev.ru.
Received:
28
May
2002
Revised:
3
March
2003
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.
Mathematics Subject Classification: 35B27 / 74Q10 / 76M50
Key words: Homogenization / perforated domains / pulsing perforations / multiple scale method.
© EDP Sciences, SMAI, 2003
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