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

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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: 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_{ε}^{ω}}{\partial t} -

$\frac{\partial u^\omega_\varepsilon}{\partial t}-$ div

(a (T_{1} (\frac{x}{ε_{1}}) ω_{1}, T_{2} (\frac{x}{ε_{2}}) ω_{2}, t, D u_{ε}^{ω})) = f .

$(a(T_1(\frac{x}{\varepsilon_1})\omega_1, T_2(\frac{x}{\varepsilon_2})\omega_2 ,t, D u^\omega_\varepsilon))=f.$
It is shown, under certain structure assumptions on the random map

a (ω_{1}, ω_{2}, t, ξ)

$a(\omega_1,\omega_2,t,\xi)$ , that the sequence

\{u^\omega_\e}

$\{u^\omega_\e}$ of solutions converges weakly in

L^{p} (0, T; W_{0}^{1, p} (Ω))

$L^p(0,T;W^{1,p}_0(\Omega))$ to the solution

u

$u$ of the homogenized problem

\frac{\partial u}{\partial t} -

$\frac{\partial u}{\partial t} -$ div

(b (t, D u)) = f

$( b( t,D u )) = f$ .

Keywords:

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

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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 $( b( t,D u )) = f$ .

This article has been cited by:

1.	Xin Wang, Liqun Cao, Yaushu Wong, Multiscale approach for stochastic elliptic equations in heterogeneous media, 2014, 85, 01689274, 54, 10.1016/j.apnum.2014.06.008
2.	NILS SVANSTEDT, CONVERGENCE OF QUASI-LINEAR HYPERBOLIC EQUATIONS, 2007, 04, 0219-8916, 655, 10.1142/S0219891607001306
3.	Nils Svanstedt, 2008, Chapter 44, 978-3-540-71991-5, 318, 10.1007/978-3-540-71992-2_44
4.	Nils Svanstedt, Multiscale stochastic homogenization of convection-diffusion equations, 2008, 53, 0862-7940, 143, 10.1007/s10492-008-0017-x
5.	Gabriel Nguetseng, Hubert Nnang, Nils Svanstedt, Deterministic homogenization of quasilinear damped hyperbolic equations, 2011, 31, 02529602, 1823, 10.1016/S0252-9602(11)60364-0
6.	Mogtaba Mohammed, Mamadou Sango, Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains, 2016, 97, 18758576, 301, 10.3233/ASY-151355
7.	Nils Svanstedt, Stochastic homogenization of a class of monotone eigenvalue problems, 2010, 55, 0862-7940, 385, 10.1007/s10492-010-0014-8

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