Multiscale stochastic homogenization of monotone operators

Nils Svanstedt

doi:10.3934/nhm.2007.2.181

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

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Multiscale stochastic homogenization of monotone operators

Nils Svanstedt^1,

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

Received: February 2006

Revised: September 2006

Published: March 2007

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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$.

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Mathematics Subject Classification: 35B27, 35B40.

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