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

By M. Siegel, Russel E. Caflisch, Sam D. Howison. 2004

142 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/56d47a2da164b55c215b6593c1d7d48753eddad7

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Citations in the corpus

Interface evolution: the Hele-Shaw and Muskat problems (2008, background): “In order to frame our result let us point out that in [17] is treated the case where both densities are equal, showing global existence for small data in the stable case and ill-possednes in the unstable case.”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “lts on singularity formation and to [34, 30] for recent reviews on the Muskat problem. In the case of small data and inﬁnite depth, global strong solutions have been constructed in subcritical spaces [11, 14, 15, 16, 17, 18, 21, 24, 52] and in critical spaces [33]. We note in particular that [24] al”
On the global existence for the Muskat problem (2010, background): “The key point for our result, in comparison to previous work [7, 19, 9, 14], is that the constant c0 can be easily explicitly computed (see (8)).”
Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “Inspired by the analysis of free boundary flows, different approaches succeeded to establish local well-posedness results for smooth enough initial data (see [6,7,31,33,51,59]).”
On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data (2016, background): “For the case Aμ 6= 0, we refer to [1, 25].”
On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results (2010, background): “Furthermore, if the lower-viscosity fluid expands into the higher-viscosity fluid, then the linearised problem is ill-posed, a result which is related with the investigation in [21].”
Relaxation of the incompressible porous media equation (2011, background): “If initially the two fluids form a horizontal interface, with the heavier fluid on top, it is well known that the initial value problem is ill-posed in classical function spaces [22, 19, 2].”
Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities (2007, methodology): “[18] M. Siegel, R. Caflisch and S. Howison.”