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

Michael Siegel, Russel Caflisch, Sam Howison

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1374-1411
Number of pages38
JournalCommunications on Pure and Applied Mathematics
Volume57
Issue number10
DOIs
StatePublished - Oct 2004

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Ill-posedness
Singular Solutions
Global Existence
Fluids
Fluid
Viscosity
Hele-Shaw
Curvature
Immiscible Fluids
Forward Problem
Pressure Gradient
Pressure gradient
Small Perturbations
Expand
Porous Media
Porous materials
Injection
Exact Solution
Unstable
Singularity

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Communications on Pure and Applied Mathematics, Vol. 57, No. 10, 10.2004, p. 1374-1411.

Research output: Contribution to journalArticle

Siegel, Michael ; Caflisch, Russel ; Howison, Sam. / Global existence, singular solutions, and ill-posedness for the Muskat problem. In: Communications on Pure and Applied Mathematics. 2004 ; Vol. 57, No. 10. pp. 1374-1411.
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