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

By A. Castro, D. C'ordoba, Charles Fefferman, Francisco Gancedo, M. López-Fernández. 2011

175 citations recorded by the source corpus.

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 Rayleigh-Taylor 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.

Subfields: Applied Mathematics, Computational Mechanics, Earth-Surface Processes

View Paper: https://www.semanticscholar.org/paper/794916e5b6a3366a392fd74608940418e969d079

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In Applied Mathematics

In Computational Mechanics

In Earth-Surface Processes

Citations in the corpus

Global solutions for the gravity water waves system in 2d (2013, background): “er the reade r to [ABZ11, ABZ12, AM03, BG98, CHS10, CL00, CS07, Lin05, Lan05, ShZ12] for some of the works on the local well-posedness of (E). Recently, blow-up solutions in the form of turning waves [CCFGL13] and“splash” solutions [CCFGG13, CS14] have been constructed. The question of existence of ”
Well-posedness of the Muskat problem with H2 initial data (2014, unknown): “[5] A. Castro, D. Cordoba, C. Fefferman, F. Gancedo, and M. Lopez-Fernandez.”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “in [10], [9] showed that there exist initial data for which the interface is the graph of a function, but after a finite time the water wave “turns over” and the interface is no longer a graph.”
Finite time singularity for the modified SQG patch equation (2016, result): “We refer to [5], [4], [54], [55] for other recent advances in some of the interface problems of fluid dynamics.”
On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data (2016, background): “Introduction The Muskat problem [5, 21, 26], mathematically analogous to the vertical Hele-Shaw model in two-dimensional case, describes the motion of two incompressible immiscible fluids in porous media, has attracted a great deal of attentions in recent years, see for examples [8, 10, 11, 13, 14, ”
Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “In contrast to the positive results, [1] shows that there is an open subset of initial data inH(4) such that the Rayleigh-Taylor condition breaks down in finite time.”
Endpoint Sobolev Theory for the Muskat Equation (2020, background): “To explain this, we first of all recall that Castro, Córdoba, Fefferman, Gancedo and López-Fernández proved that the Cauchy problem is not well posed globally in time for some large data ([16, 17, 18]).”