Interface evolution: the Hele-Shaw and Muskat problems

Abstract

We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.

A survey for the Muskat problem and a new estimate

“…The case with different viscosities and densities was shown to be well-posed in [19]. In that proof it is crucial to get control of the norm of the implicit operator given in (11) involved in the definition of the amplitude of the vorticity ω.…”

Untitled

“…Therefore, in order to simplify our presentation, we shall avoid here many details which were carefully proven there. This is especially the case in Section 6 (control of the Birkhoff-Rott integral) and Section 8 (energy estimates), and also for the approximation schemes which are identical to those developed in [Córdoba et al 2011]. Therefore, in the following, we shall focus our attention on the more innovative parts of the proof, namely the evolution of the Rayleigh-Taylor condition, the non-self-intersecting property of the free boundary, and the needed estimates for double-layer potentials.…”

“…This paper extends to the three-dimensional case the results obtained in [Córdoba et al 2011] for the case of two dimensions, by proving local existence in the scale of Sobolev spaces of the initial value problem if the Rayleigh-Taylor (R-T) condition is initially satisfied (see [Saffman and Taylor 1958], where this issue is studied from a physical point of view). In our case, that condition amounts to the positivity of the function…”

“…This paper is a continuation of [Córdoba et al 2011], where the two-dimensional case was considered. Many of the needed estimates can be obtained following exactly the same methods that were used in [Córdoba et al 2011] for the lower-dimensional case. Therefore, in order to simplify our presentation, we shall avoid here many details which were carefully proven there.…”

“…The Muskat problem and related problems [Saffman and Taylor 1958] have been studied recently [Constantin and Pugh 1993;Siegel et al 2004;Córdoba and Gancedo 2007;2009;Córdoba et al 2011]. The first natural question is about the evolution of the system (existence of solutions), at least for a short time t > 0, and the persistence of smoothness of the interface S(t) if we begin with a smooth enough surface at time t = 0.…”