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

Michael Siegel, Russel E. Caflisch, Sam Howison

Research output: Contribution to journalArticle


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 languageEnglish (US)
Pages (from-to)1374-1411
Number of pages38
JournalCommunications on Pure and Applied Mathematics
Issue number10
StatePublished - Oct 1 2004


ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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