Homogenization in perforated domains with rapidly pulsing perforations
Laboratoire Jacques-Louis Lions (Analyse Numérique),
Université Paris VI – CNRS,
175 rue du Chevaleret,
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; firstname.lastname@example.org.
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