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Well-posedness of two-phase Hele–Shaw flow without surface tension

Published online by Cambridge University Press:  04 March 2005

DAVID M. AMBROSE
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA email: ambrose@courant.nyu.edu

Abstract

We prove short-time well-posedness of a Hele–Shaw system with two fluids and no surface tension (this is also known as the Muskat problem). We restrict our attention here to the stable case of the problem. That is, in order for the motion to be well-posed, the initial data must satisfy a sign condition which is a generalization of a condition of Saffman and Taylor. This sign condition essentially means that the more viscous fluid must displace the less viscous fluid. The proof uses the formulation introduced in the numerical work of Hou, Lowengrub, and Shelley, and relies on energy methods.

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Papers
Copyright
2004 Cambridge University Press

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