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

Document Type

Article

Publication Date

10-1-2004

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. © 2004 Wiley Periodicals, Inc.

Identifier

4544228729 (Scopus)

Publication Title

Communications on Pure and Applied Mathematics

External Full Text Location

https://doi.org/10.1002/cpa.20040

ISSN

00103640

First Page

1374

Last Page

1411

Issue

10

Volume

57

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