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

Hakima Bessaih; Yalchin Efendiev; Florin Maris

doi:10.3934/nhm.2015.10.343

Article Contents

2015, Volume 10, Issue 2: 343-367. Doi: 10.3934/nhm.2015.10.343

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

1.
University of Wyoming, Department of Mathematics, Dept. 3036, 1000 East University Avenue, Laramie, WY 82071
2.
Department of Mathematics & ISC, Texas A&M University, 3404 TAMU, College Station, TX 77843-3404
3.
Numerical Porous Media SRI Center, CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900

Received: February 2014

Revised: October 2014

Published: June 2015

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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 two-scale 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.

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Mathematics Subject Classification: Primary: 60H15, 76M50, 76D07; Secondary: 76S05, 60H30.

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