Correctors for Parabolic Equations in a Heterogeneous Fibered Medium

Mourad Sfaxi; Ali Sili

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 1025-1053
  • ISSN: 0392-4033

Abstract

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We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium Ω made of two materials. The first one is located in a set F ϵ of cylindrical parallel fibers periodically distributed with a period of size ϵ , and the second one is located in the "matrix" M ϵ = Ω F ϵ . The ratio between the conductivity coefficients of the two materials is of order 1 / ϵ 2 . After writing the homogenized problem, we give a corrector result and prove that the solution ue of the starting problem is of the form u ϵ = u ~ ϵ + u ^ ϵ , where u ~ ϵ is a corrector for u ϵ and u ^ ϵ is a time boundary layer. In contrast to the known results for parabolic equations, this boundary layer is not concentrated about the time origin t = 0 , but it remains at least for all t ( 0 , m ) with some m > 0 . The proof of the latter is based on the fact that ue does not converge, in general, in L 2 ( Ω × ( 0 , T ) ) for the strong topology.

How to cite

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Sfaxi, Mourad, and Sili, Ali. "Correctors for Parabolic Equations in a Heterogeneous Fibered Medium." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 1025-1053. <http://eudml.org/doc/290374>.

@article{Sfaxi2007,
abstract = {We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\Omega$ made of two materials. The first one is located in a set $F_\epsilon$ of cylindrical parallel fibers periodically distributed with a period of size $\epsilon$, and the second one is located in the "matrix" $M_\epsilon = \Omega \setminus F_\epsilon$. The ratio between the conductivity coefficients of the two materials is of order $1/\epsilon^2$. After writing the homogenized problem, we give a corrector result and prove that the solution ue of the starting problem is of the form $u_\epsilon = \tilde\{u\}_\epsilon + \hat\{u\}_\epsilon$, where $\tilde\{u\}_\epsilon$ is a corrector for $u_\{\epsilon\}$ and $\hat\{u\}_\epsilon$ is a time boundary layer. In contrast to the known results for parabolic equations, this boundary layer is not concentrated about the time origin $t = 0$, but it remains at least for all $t \in (0, m)$ with some $m > 0$. The proof of the latter is based on the fact that ue does not converge, in general, in $L^\{2\}(\Omega \times (0, T))$ for the strong topology.},
author = {Sfaxi, Mourad, Sili, Ali},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {1025-1053},
publisher = {Unione Matematica Italiana},
title = {Correctors for Parabolic Equations in a Heterogeneous Fibered Medium},
url = {http://eudml.org/doc/290374},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Sfaxi, Mourad
AU - Sili, Ali
TI - Correctors for Parabolic Equations in a Heterogeneous Fibered Medium
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 1025
EP - 1053
AB - We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\Omega$ made of two materials. The first one is located in a set $F_\epsilon$ of cylindrical parallel fibers periodically distributed with a period of size $\epsilon$, and the second one is located in the "matrix" $M_\epsilon = \Omega \setminus F_\epsilon$. The ratio between the conductivity coefficients of the two materials is of order $1/\epsilon^2$. After writing the homogenized problem, we give a corrector result and prove that the solution ue of the starting problem is of the form $u_\epsilon = \tilde{u}_\epsilon + \hat{u}_\epsilon$, where $\tilde{u}_\epsilon$ is a corrector for $u_{\epsilon}$ and $\hat{u}_\epsilon$ is a time boundary layer. In contrast to the known results for parabolic equations, this boundary layer is not concentrated about the time origin $t = 0$, but it remains at least for all $t \in (0, m)$ with some $m > 0$. The proof of the latter is based on the fact that ue does not converge, in general, in $L^{2}(\Omega \times (0, T))$ for the strong topology.
LA - eng
UR - http://eudml.org/doc/290374
ER -

References

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  7. SFAXI, M. - SILI, A., Correcteurs pour une équation parabolique dans une structure mince hétérogène, to appear. 
  8. SILI, A., Corrector results for Nonlinear Parabolic Monotone Problems in a Thin Domain, Com. Appl. Nonlin. Anal., (9) (2002), 1-22. Zbl1144.35024MR1916297
  9. SILI, A., Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Mod. Meth. Appl. Sci., Vol. 14, (3) (2004), 329-353. Zbl1085.35029MR2047574DOI10.1142/S0218202504003258
  10. SIMON, J., Compact Sets in L p ( 0 , T , B ) , Ann. Mat. Pura Appl., 146 (1987), 65-96. MR916688DOI10.1007/BF01762360
  11. TARTAR, L., Cours Peccot, Collège de France, 1977, unpublished, partially written in 

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