Multiscale stochastic homogenization of monotone operators

Nils Svanstedt; Nils Svanstedt

doi:10.3934/nhm.2007.2.181

Networks and Heterogeneous Media

2007, Volume 2, Issue 1: 181-192. doi: 10.3934/nhm.2007.2.181

Previous Article Next Article

Multiscale stochastic homogenization of monotone operators

Nils Svanstedt ^{1
,}

1.
Department of Computational Mathematics, Chalmers University, SE-412 96 Göteborg

Received: 01 February 2006 Revised: 01 September 2006
35B27, 35B40.

Multiscale stochastic homogenization is studied for divergence structure parabolic problems. More specifically we consider the asymptotic behaviour of a sequence of realizations of the form
$\frac{\partial u^\omega_\varepsilon}{\partial t}- $div$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$
It is shown, under certain structure assumptions on the random map $a(\omega_1,\omega_2,t,\xi)$, that the sequence $\{u^\omega_\e}$ of solutions converges weakly in $ L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem $ \frac{\partial u}{\partial t} - $div$( b( t,D u )) = f$.
- multiscale,
- stochastic,
- homogenization,
- G-convergence,
- elliptic,
- parabolic
Citation: Nils Svanstedt. Multiscale stochastic homogenization of monotone operators[J]. Networks and Heterogeneous Media, 2007, 2(1): 181-192. doi: 10.3934/nhm.2007.2.181

Related Papers:
Abstract

Multiscale stochastic homogenization is studied for divergence structure parabolic problems. More specifically we consider the asymptotic behaviour of a sequence of realizations of the form
$\frac{\partial u^\omega_\varepsilon}{\partial t}- $div$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$
It is shown, under certain structure assumptions on the random map $a(\omega_1,\omega_2,t,\xi)$, that the sequence $\{u^\omega_\e}$ of solutions converges weakly in $ L^p(0,T;W^{1,p}_0(\Omega))$ to the solution $u$ of the homogenized problem $ \frac{\partial u}{\partial t} - $div$( b( t,D u )) = f$.
Reader Comments

Your name:*

Email:*
© 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)