Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition

Hakima Bessaih, Yalchin R. Efendiev, Florin Maris

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3 Scopus citations

Abstract

The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the twoscale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508-1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. © American Institute of Mathematical Sciences.
Original languageEnglish (US)
Pages (from-to)343-367
Number of pages25
JournalNetworks and Heterogeneous Media
Volume10
Issue number2
DOIs
StatePublished - Apr 2015

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This project started when Hakima Bessaih was visiting the Numerical Porous Media (NumPor) SRI Center at KAUST. She would like to thank the NumPor for the financial support, the excellent working atmosphere and the warm hospitality. H.B. was supported in part by the Simons Foundation grant #283308, the NSF grants DMS-1416689 and DMS-1418838. YE's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project.

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