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Multiscale stochastic homogenization of convection-diffusion equations. (English) Zbl 1199.35017
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 ${\partial u^\omega _{\varepsilon }}/{\partial t} +{1}/{\varepsilon _3}\, \mathcal C\bigl (T_3({x}/{\varepsilon _3}) \omega _3\bigr ) \cdot \nabla u^\omega _{\varepsilon }- \operatorname {div} \bigl ( \alpha \bigl (T_1({x}/{\varepsilon _1})\omega _1, T_2({x}/{\varepsilon _2})\omega _2 ,t\bigr ) \nabla u^\omega _{\varepsilon }\bigr )=f.$ 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 $$\{u^\omega _\varepsilon \}$$ of solutions converges in the sense of G-convergence of parabolic operators to the solution  $$u$$ of the homogenized problem $${\partial u}/{\partial t} - \operatorname {div} ( \mathcal B(t)\nabla u ) = f$$.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
multiscale; stochastic; homogenization; convection-diffusion
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##### References:
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