Homogenization of the Stokes equations with a random potential

Homogenization of the Stokes equations in a random porous medium is considered. Instead of the homogeneous Dirichlet condition on the boundaries of numerous small pores, used in the existing work on the subject, we insert a term with a positive rapidly oscillating potential into the equations. Physically, this corresponds to porous media whose rigid matrix is slightly permeable to fluid. This relaxation of the boundary value problem permits one to study the asymptotics of the solutions and to justify the Darcy law for the limit functions under much fewer restrictions. Specifically, homogenization becomes possible without any connectedness conditions for the porous domain, whose verification would lead to problems of percolation theory that are insufficiently studied.