Homogenization in perforated domains with rapidly pulsing perforations

Doina Cioranescu; Andrey L. Piatnitski

ESAIM: Control, Optimisation and Calculus of Variations (2003)

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

Abstract

top
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

top

Cioranescu, Doina, and Piatnitski, Andrey L.. "Homogenization in perforated domains with rapidly pulsing perforations." ESAIM: Control, Optimisation and Calculus of Variations 9 (2003): 461-483. <http://eudml.org/doc/244795>.

@article{Cioranescu2003,
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 $\varepsilon $ 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 $\varepsilon $) 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; homogeneous Neumann condition; weighted space},
language = {eng},
pages = {461-483},
publisher = {EDP-Sciences},
title = {Homogenization in perforated domains with rapidly pulsing perforations},
url = {http://eudml.org/doc/244795},
volume = {9},
year = {2003},
}

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
PY - 2003
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 $\varepsilon $ 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 $\varepsilon $) 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; homogeneous Neumann condition; weighted space
UR - http://eudml.org/doc/244795
ER -

References

top
  1. [1] N.S. Bakhvalov, Averaging of partial differential equations with rapidly oscillating coefficients. Soviet Math. Dokl. 16 (1975). Zbl0331.35009
  2. [2] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland (1978). Zbl0404.35001MR503330
  3. [3] F. Campillo, M.L. Kleptsyna and A.L. Piatnitski, Homogenization of random parabolic operators with large potential. Stochastic Process. Appl. 93 (2001) 57-85. Zbl1099.35009MR1819484
  4. [4] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. Zbl0427.35073MR548785
  5. [5] D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag (1999). Zbl0929.35002MR1676922
  6. [6] P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Ricerche di Mat. L (2001) 115-144. Zbl1102.35305MR1941824
  7. [7] A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964). Zbl0144.34903MR181836
  8. [8] U. Hornung, Homogenization and porous media. Springer-Verlag, IAM 6 (1997). Zbl0872.35002MR1434315
  9. [9] V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of differential operators and integral functionals. Springer-Verlag (1994). Zbl0838.35001MR1329546
  10. [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. Zbl0892.35019MR1473991
  11. [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. [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) Zbl0674.47036
  13. [13] A.L. Piatnitsky, Parabolic equations with rapidly oscillating coefficients (In Russian). English transl. Moscow Univ. Math. Bull. 3 (1980) 33-39. Zbl0466.35005

NotesEmbed ?

top

You must be logged in to post comments.