Nonlinear stability of the Muskat problem with capillary pressure at the free boundary
type: Item
instance of: scholarly article
title: Nonlinear stability of the Muskat problem with capillary pressure at the free boundary
zbMATH Open document ID: 1028.35123
DOI: 10.1016/S0362-546X(02)00286-9
author: A. Friedman
author: Youshan Tao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-04-28T00:00:00Z
review text: The paper deals with the 2D Muskat problem which models the motion of two immiscible fluids (water and oil) in a porous medium. Starting from Darcy's law and from the equation of continuity, the authors prove that the potential of velocity of the two immiscible fluids satisfy Laplace's equation with appropriate conditions on the boundaries and on the moving free interface (the capillary pressure is proportional to the mean curvature of the interface and the velocity of each fluid on the interface coincides with the velocity of the interface). Considering nearly radially symmetric initial data, depending on a small parameter, the authors prove that the problem has a unique global solution which is analytic with respect to the small parameter. The solution converges to the stationary solution (which is not generally asymptotically stable) provided the initial data satisfy a sequence of necessary nonlinear constraints.
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 76B07
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 1901880
zbMATH Keywords: Capillary pressure
zbMATH Keywords: free boundary
zbMATH Keywords: incompressible pressure
zbMATH Keywords: unique global solution
zbMATH Keywords: Muskat problem
zbMATH Keywords: motion of two immiscible fluids
zbMATH Keywords: porous medium
zbMATH Keywords: radially symmetric initial data
zbMATH Keywords: stationary solution
reviewed by: Adrian Carabineanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901880