### Abstract

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L^{2}(R) maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance ||f||_{1} ≤ 1/5. Previous results of this sort used a small constant ε ≪ 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy ||f _{0}||_{L∞} < ∞ and ||δ_{x}f _{0}||_{L∞} < 1. We take advantage of the fact that the bound ||δ_{x}f_{0}||_{L∞} < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

Original language | English (US) |
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Pages (from-to) | 201-227 |

Number of pages | 27 |

Journal | Journal of the European Mathematical Society |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Fluid interface
- Global existence
- Incompressible flows
- Porous media

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## Cite this

*Journal of the European Mathematical Society*,

*15*(1), 201-227. https://doi.org/10.4171/JEMS/360