### 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) |
---|---|

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 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

**Global existence, singular solutions, and ill-posedness for the Muskat problem.** / Siegel, Michael; Caflisch, Russel; Howison, Sam.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 57, no. 10, pp. 1374-1411. https://doi.org/10.1002/cpa.20040

}

TY - JOUR

T1 - Global existence, singular solutions, and ill-posedness for the Muskat problem

AU - Siegel, Michael

AU - Caflisch, Russel

AU - Howison, Sam

PY - 2004/10

Y1 - 2004/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=4544228729&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544228729&partnerID=8YFLogxK

U2 - 10.1002/cpa.20040

DO - 10.1002/cpa.20040

M3 - Article

VL - 57

SP - 1374

EP - 1411

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 10

ER -