Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

Abstract

The Muskat problem models the evolution of the interface between two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach linear stability of the Muskat problem. We show that the RayleighTaylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.

A survey for the Muskat problem and a new estimate

“…In the case with small initial data, it is possible to use the parabolic character of the equation in the stable state (see (13) and (17) for the lineal interpretation) to prove global in time regularity in different situations. For purely surface tension driven fluids (g = 0) see results in [29,18].…”

“…A fascinating behavior of Muskat solution, which can be proved analytically, is finite time singularity formation starting from regular stable initial data. In [13] it is proved that in the case µ 1 = µ 2 and τ = 0 there are solutions of the Muskat equation with initial interfaces being certain smooth stable graphs, which enter the unstable regime, where the interface is no longer a graph, in finite time. In particular there exists a time t p in which…”

“…At some branch in the interface it is possible to localize the heavy fluid on top of the lighter one. An important reason why this phenomenon arises is that, even for large initial data, Muskat solutions become instantly analytic [13]. So that, despite the interface is about to reach an unstable regime, the analyticity remains by the time the wave-turning occurs.…”

“…Recently, computer evidence has shown how singularities may developed in Muskat [12]. With recent new techniques, it is now possible to prove different types of finite time singularity formation for those scenarios [13,10,11]. These are the first analytic proofs of blow-up for incompressible fluid in well-posed situations.…”

Formation of Singularities in Fluid Interfaces

“…Our work on Muskat is joint with Angel Castro, Diego Cordoba, Francisco Gancedo and Maria Lopez-Fernandez [7,8]. Regarding alpha-patches, we discuss the work of Diego Cordoba, Marco Fontelos, Ana Mancho and José Rodrigo [16].…”