Multiscale stochastic homogenization of convection-diffusion equations. (English) Zbl 1199.35017

Svanstedt, Nils

Appl. Math., Praha 53, No. 2, 143-155 (2008).

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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