Article
MSC:
35B27,
35B40,
35K57,
60H15,
76M35,
76M50 | MR 2399903 | Zbl 1199.35017 | DOI: 10.1007/s10492-008-0017-x
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Keywords:
multiscale; stochastic; homogenization; convection-diffusion
multiscale; stochastic; homogenization; convection-diffusion
Summary:
Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ${\partial u^\omega _{\varepsilon }}/{\partial t} +{1}/{\epsilon _3}\,\mathcal C\bigl (T_3({x}/{\varepsilon _3}) \omega _3\bigr )\cdot \nabla u^\omega _{\varepsilon }- \div \bigl ( \alpha \bigl (T_1({x}/{\varepsilon _1})\omega _1, T_2({x}/{\varepsilon _2})\omega _2 ,t\bigr ) \nabla u^\omega _{\varepsilon }\bigr )=f$. It is shown, under certain structure assumptions on the random vector field ${\mathcal C}(\omega _3)$ and the random map $\alpha (\omega _1,\omega _2,t)$, that the sequence $\lbrace u^\omega _\epsilon \rbrace $ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem ${\partial u}/{\partial t} - \div ( \mathcal B(t)\nabla u ) = f$.
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