Methods and Applications of Analysis

Volume 20 (2013)

Number 1

Steady-state fingering patterns for a periodic Muskat problem

Pages: 33 – 46

DOI: http://dx.doi.org/10.4310/MAA.2013.v20.n1.a2

Authors

Mats Ehrnström (Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)

Joachim Escher (Institut für Angewandte Mathematik, Leibniz Universität Hannover, Hannover, Germany)

Bogdan-Vasile Matioc (Institut für Angewandte Mathematik, Leibniz Universität Hannover, Hannover, Germany)

Abstract

We study global bifurcation branches consisting of stationary solutions of the Muskat problem. It is proved that the steady-state fingering patterns blow up as the surface tension increases: we find a threshold value for the cell height with the property that below this value the fingers will touch the boundaries of the cell when the surface tension approaches a finite value from below; otherwise, the maximal slope of the fingers tends to infinity.

Keywords

Muskat problem, fingering patterns, existence, steady-state solutions, periodic solutions

2010 Mathematics Subject Classification

34A12, 34C23, 34C25, 70K42

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