The Muskat problem for a viscoelastic filtration

A free boundary problem describing joint filtration of two immiscible incompressible liquids is derived from homogenization theory. We start with a mathematical model on the microscopic level, which consists of the stationary Stokes system for an incompressible inhomogeneous viscous liquid, occupying a pore space, the stationary Lamé equations for an incompressible elastic solid skeleton, coupled with suitable boundary conditions on the common boundary “solid skeleton – pore space”, and a transport equation for the unknown liquid density. Next we prove the solvability of this model and rigorously perform the homogenization procedure as the dimensionless size of pores tends to zero, while the porous body is geometrically periodic. As a result, we prove the solvability of the Muskat problem for viscoelastic filtration.