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

Siegel, Michael and Caflisch, Russell E. and Howison, Sam (2004) Global existence, singular solutions, and ill-posedness for the Muskat problem. Communications on Pure and Applied Mathematics, LVII . pp. 1-38. (In Press)

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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.

Item Type:Article
Additional Information:This is a preprint of an article accepted for publication in Communications on Pure and Applied Mathematics, Copyright 2004 Wiley Periodicals Inc.
Uncontrolled Keywords:Muskat problem, Hele-Shaw problem, free boundary problem, blow-up, singularity formation
Subjects:O - Z > Partial differential equations
D - G > Fluid mechanics
Research Groups:Oxford Centre for Industrial and Applied Mathematics
ID Code:102
Deposited By:Sam Howison
Deposited On:15 Jun 2004
Last Modified:20 Jul 2009 14:18

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