Well-posedness of the Muskat problem with H2 initial data

By C. H. Arthur Cheng, Rafael Granero-Belinch'on, Steve Shkoller. 2014

109 citations recorded by the source corpus.

Abstract We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small H 2 perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for H 2 initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e881066af4e4f6e00050b8761cc97d6f41068376

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, background): “olutions have been constructed in subcritical spaces [13, 17, 18, 19, 20, 21, 22, 25, 3 28, 59] and in critical spaces [39]. We note in particular that [28] allows for interfaces with large slope and [18, 39] allow for viscosity jump. Global weak solutions were obtained in [20, 29]. As noted earlier”
Global regularity for 2D Muskat equations with finite slope (2015, background): “Recently, results involving solely the second derivative of f appeared in the work [CGBS14], where the authors explore the PDE structure of the original Darcy’s law in the bulk rather than the corresponding integral equation for contour dynamics (1.”