### Abstract

The Muskat, or Muskat-Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction. In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem. For the stable, forward Muskat problem, in which the higher-viscosity fluid expands into the lower-viscosity fluid, we show global-in-time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher-viscosity fluid contracts, we construct singular solutions that start off with smooth initial data but develop a point of infinite curvature at finite time.

Original language | English (US) |
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Pages (from-to) | 1374-1411 |

Number of pages | 38 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 57 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2004 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*57*(10), 1374-1411. https://doi.org/10.1002/cpa.20040