Global existence, singular solutions, and ill‐posedness for the Muskat problem

Abstract

The Muskat, or Muskat-Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher-viscosity fluid expands into the lower-viscosity fluid, we show global-in-time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher-viscosity fluid contracts, we construct singular solutions that start off with smooth initial data but develop a point of infinite curvature at finite time.

A note on interface dynamics for convection in porous media

“…When the denser fluid is above, ρ 2 < ρ 1 , we prove that (4) is ill-posed. We obtain this result in a similar way as in [16], using global solutions of (4) in the stable case, ρ 2 > ρ 1 , for small initial data.…”

“…Fluid interface problems in porous media have been widely considered (see [10,16]), together with the two-phase Hele-Shaw flow (see [11,3]). …”

“…The case without surface tension was studied by Siegel, Caflisch and Howison in [16]. They proved global-in-time existence for near planar initial data in a stable case when the Atwood number…”

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].…”

“…Basically, the unstable case arises in the viscosity jump situation when a less viscous fluid pushes a more viscous one. This case was studied in [48], where the contour dynamic equation is proved to be ill-possed. 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.…”