Interface evolution: the Hele-Shaw and Muskat problems

By A. Córdoba, D. C'ordoba, Francisco Gancedo. 2008

168 citations recorded by the source corpus.

We study the dynamics of the interface between two incompressible 2-D ows 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 uids. This physical scenario is known as the two-dimensional Muskat problem or the two-phase Hele-Shaw ow. We prove local-existence in Sobolev spaces when, initially, the dierence 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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ced78b4e98e4209f29a093ac06dfce90ce25195c

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In Applied Mathematics

Citations in the corpus

Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, background): “The latter work opened the door to the solution of many important problems concerning the Cauchy problem or blow-up solutions (see [19, 9, 10, 11], more references are given below as well as in the survey papers [30, 31]).”
On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results (2010, background): “Further local well-posedness results have been established [1] and more recently in [5, 6] by means of energy estimates in the absence of surface tension forces.”
Well-posedness of the Muskat problem with H2 initial data (2014, unknown): “[7] A. Castro, D. Córdoba, and F. Gancedo.”
On the global existence for the Muskat problem (2010, methodology): “Recently, the well-posedness has been shown without surface tension in [8] (for previous work on this topic see [1], [21], [2] and [9]) using arguments that rely upon the boundedness properties of the Hilbert transforms associated to C1,γ curves.”
A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “ontour dynamic formulation for the Muskat problem without viscosity jump and with inﬁnite depth, and proved local well-posedness in Hd+2(Rd), d= 1;2; here the interface is the graph of a function. In [23, 24], Cordoba, C´ ordoba and Gancedo extended this result to the case of viscosity jump and nong”
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]).”
Finite time singularity for the modified SQG patch equation (2016, background): “, [13] where the Muskat problem is discussed.”
Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves (2011, background): “using (23), and therefore (see section 9 in [11] for more details) d dt ‖z‖RT (t) ≤ C(‖z‖RT (t) + 1) .”