Contour dynamics of incompressible 3-d fluids in a porous medium with different densities

We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically analogous to the two- phase Hele - Shaw cell. We focus on a fluid interface given by a jump of densities, being the equation of the evolution obtained using Darcy's law. We prove local well- posedness when the smaller density is above ( stable case) and in the unstable case we show ill- posedness.