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

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

2010-05-14

arxiv.org/abs/1005.2512v2

Mathematics

Analysis of PDEs

24 pages, 3 figures

Scientific paper

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.

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