Contour dynamics of incompressible 3-D fluids in a porous medium with different densities Diego C´ordoba and Francisco Gancedo September 4, 2006 Abstract 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. 1 Introduction The evolution of a fluid in a porous medium is an important and interesting topic of fluid mechanics (see [3]). This phenomena is based on an experimental physical principle given by H. Darcy in 1856. Darcy’s law for a 3-D fluid is given by the momentum equation μ κ v = −∇p − (0, 0, g ρ), where v is the incompressible velocity, p is the pressure, μ is the dynamic viscosity, κ is the permeability of the medium, ρ is the liquid density and g is the acceleration due to gravity. A different problem is the motion of a 2-D fluid in a Hele–Shaw cell (see [12]). In this case the fluid is set between two fixed parallel plates. These plates are close enough in such a way that the mean velocity is described by 12μ b 2 v = −∇p − (0, g ρ), where b denotes the distance between the plates. Considering that the fluid in the porous medium only moves in two directions suppressing one of the variables in the horizontal plane, these two different physical phenomena of fluid dynamics become nevertheless mathematically analogous if we identify the permeability of the medium κ and the constant b 2 /12. The Muskat problem (see [14]) and the two-phase Hele-Shaw flow (see [17]) model the evolution of an interface between two fluids (in a porous medium and in a Hele–Shaw cell respectively) with different viscosities and densities. A lot of information can be found in the 1