Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T19:24:50.662Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

N.S. Bakhvalov, Averaging of partial differential equations with rapidly oscillating coefficients. Soviet Math. Dokl. 16 (1975).
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland (1978).
Campillo, F., Kleptsyna, M.L. and Piatnitski, A.L., Homogenization of random parabolic operators with large potential. Stochastic Process. Appl. 93 (2001) 57-85. CrossRef
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. CrossRef
D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag (1999).
P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Ricerche di Mat. L (2001) 115-144.
A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964).
U. Hornung, Homogenization and porous media. Springer-Verlag, IAM 6 (1997).
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994).
M.L. Kleptsyna and A.L. Piatnitski, Homogenization of random parabolic operators. Gakuto International Series. Math. Sci. Appl. 9 (1997), Homogenization and Appl. to Material Sciences, 241-255.
M.L. Kleptsyna and A.L. Piatnitski, Averaging of non selfadjoint parabolic equations with random evolution (dynamics), Preprint INRIA. J. Funct. Anal. (submitted).
M.A. Krasnosel'skii, E.A. Lifshits and E.A. Sobolev, Positive linear systems. The method of positive linear operators.Heldermann Verlag , Sigma Ser. Appl. Math. 5 (1989)
Piatnitsky, A.L., Parabolic equations with rapidly oscillating coefficients (In Russian). English transl. Moscow Univ. Math. Bull. 3 (1980) 33-39.