Homogenization in perforated domains with rapidly pulsing perforations

Doina Cioranescu; Andrey L. Piatnitski

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 9, page 461-483
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Cioranescu, Doina, and Piatnitski, Andrey L.. "Homogenization in perforated domains with rapidly pulsing perforations." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 461-483. <http://eudml.org/doc/90706>.

@article{Cioranescu2010,
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. },
author = {Cioranescu, Doina, Piatnitski, Andrey L.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Homogenization; perforated domains; pulsing perforations; multiple scale method.; multiple scale method; homogeneous Neumann condition; weighted space},
language = {eng},
month = {3},
pages = {461-483},
publisher = {EDP Sciences},
title = {Homogenization in perforated domains with rapidly pulsing perforations},
url = {http://eudml.org/doc/90706},
volume = {9},
year = {2010},
}

TY - JOUR
AU - Cioranescu, Doina
AU - Piatnitski, Andrey L.
TI - Homogenization in perforated domains with rapidly pulsing perforations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 9
SP - 461
EP - 483
AB - 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.
LA - eng
KW - Homogenization; perforated domains; pulsing perforations; multiple scale method.; multiple scale method; homogeneous Neumann condition; weighted space
UR - http://eudml.org/doc/90706
ER -

References

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  9. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994).  
  10. 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.  Zbl0892.35019
  11. M.L. Kleptsyna and A.L. Piatnitski, Averaging of non selfadjoint parabolic equations with random evolution (dynamics), Preprint INRIA. J. Funct. Anal. (submitted).  
  12. 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)  
  13. A.L. Piatnitsky, Parabolic equations with rapidly oscillating coefficients (In Russian). English transl. Moscow Univ. Math. Bull.3 (1980) 33-39.  

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