Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities

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.

A note on interface dynamics for convection in porous media

“…We know that the interface can be parameterized as a function (see [6]). In order to get this, we need to show that ∂ t z 1 (α, t) = 0 and that if the initial interface is given by z(α, 0) = (α, f 0 (α)) then we get z(α, t) = (α, f (α, t)).…”

“…Changes in the tangential component of the velocity only introduce changes in the parametrization of the curve. In [6] two of the authors use this property to parameterize the interface as a function (α, f (α, t)), getting the following evolution equation…”

A survey for the Muskat problem and a new estimate

“…For purely surface tension driven fluids (g = 0) see results in [29,18]. Without surface tension (τ = 0), global existence for the viscosity jump case was proven in [48] and extended to the density jump case in [21], showing in both papers instant analyticity of the solutions. For gravity and surface tension interaction with boundary values see [34].…”

“…In the density jump case (µ 2 = µ 1 ), the unstable regime holds when the more dense fluid lies above the interface and the less dense fluid lies below it. The contour dynamics equation is shown to be ill-posed in this scenario [21]. On the other hand, the lost of derivative in the contour equation is of order one, so that it is possible to find solutions of the system with analytic initial data even in the unstable case [32,12].…”

“…Without surface tension (τ = 0), the positivity of the nonlinear RayleighTaylor function have been shown to be crucial to disclose a local-in-time existence result [1,51]. The system is proved to be well-posed in the case of equal viscosity µ 1 = µ 2 = µ for the stable case [21] by using energy estimates on the contour dynamics equation through the chain of Sobolev norms. In this situation, the Rayleigh-Taylor condition is satisfied only if the free boundary is represented by the graph of a function (α, f (α, t)) and ρ 2 > ρ 1 .…”