The confined Muskat problem: Differences with the deep water regime

Rafael Orive-Illera (Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain; Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Spain)

Abstract

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.

Keywords

Darcy’s law, Hele-Shaw cell, Muskat problem, maximum principle, well-posedness, blow-up, ill-posedness

2010 Mathematics Subject Classification

35Q35, 35R35, 76D27, 76S05, 76Txx

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Published 26 November 2013