### Abstract

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class W^{2,p}, 1<p≤∞. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

Original language | English (US) |
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Pages (from-to) | 1041-1074 |

Number of pages | 34 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 34 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2017 |

### All Science Journal Classification (ASJC) codes

- Analysis
- Mathematical Physics
- Applied Mathematics

### Keywords

- Darcy's law
- Maximum principle
- Muskat problem
- Porous medium

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## Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*34*(4), 1041-1074. https://doi.org/10.1016/j.anihpc.2016.09.001