Homogenization of the Stokes equations with general random coefficients (Q1332669)

Summary: When an attempt is made to model fluid flow in a porous medium, one is often lead to the homogenization problem for the Stokes system. One considers for each value of a small positive parameter \(\varepsilon\) the solution \((u_ \varepsilon, p_ \varepsilon)\) of a Stokes system with coefficients and boundary conditions depending randomly on \(\varepsilon\), and one seeks to prove that \((u_ \varepsilon, p_ \varepsilon)\) converges in some sense as \(\varepsilon \to 0\) to a limit, and to derive equations which this homogenized limit satisfies. In this paper, we concentrate on the special but interesting case of \(\varepsilon\)- independent Dirichlet boundary conditions and general random coefficients for \(n\)-dimensional Stokes systems, and we use the method of stochastic two-scale convergence in the mean, introduced and studied in [\textit{A. Bourgeat}, \textit{A. Mikelić} and \textit{S. Wright}, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math. (to appear)], to solve in a fairly direct and elegant way both of these problems.