On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results

By Joachim Escher, Bogdan–Vasile Matioc. 2010

91 citations recorded by the source corpus.

We consider in this paper the Muskat problem in a periodic geometry and incorporate capillary as well as gravity effects in the modelling. The problem re-writes as an abstract evolution equation and we use this property to prove well-posedness of the problem and to establish exponential stability of some flat equilibrium. Using bifurcation theory we also find finger shaped steady-states which are all unstable.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/7f5a87a274dcc62b0b94092b05c01a8e9a939f67

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Well-posedness of the Muskat problem with H2 initial data (2014, background): “[31] J. Escher and B.-V. Matioc.”
Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves (2011, background): “There are several results on global existence for small initial data (small compared to 1 in several norms more regular than Lipschitz [9, 35, 32, 13, 19]) taking advantage of the parabolic character of the equation for small initial data.”