Correctors for Parabolic Equations in a Heterogeneous Fibered Medium

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 -