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

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 (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

top
  1. N.S. Bakhvalov, Averaging of partial differential equations with rapidly oscillating coefficients. Soviet Math. Dokl.16 (1975).  
  2. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland (1978).  
  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.  
  4. D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590-607.  
  5. D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer-Verlag (1999).  
  6. P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains. Ricerche di Mat.L (2001) 115-144.  
  7. A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs (1964).  
  8. U. Hornung, Homogenization and porous media. Springer-Verlag, IAM6 (1997).  
  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.  
  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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.