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

@article{Escher2010OnTP,
  title={On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results},
  author={Joachim Escher and Bogdan-Vasile Matioc},
  journal={arXiv: Analysis of PDEs},
  year={2010}
}
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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