## Viscous displacement in porous media: the Muskat problem in 2D

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## Abstract:

We consider the Muskat problem describing the viscous displacement in a two-phase fluid system located in an unbounded two-dimensional porous medium or Hele-Shaw cell. After formulating the mathematical model as an evolution problem for the sharp interface between the fluids, we show that the Muskat problem with surface tension is a quasilinear parabolic problem, whereas, in the absence of surface tension effects, the Rayleigh–Taylor condition identifies a domain of parabolicity for the fully nonlinear problem. Based upon these aspects, we then establish the local well-posedness for arbitrary large initial data in $H^s$, $s>2$, if surface tension is taken into account, respectively for arbitrary large initial data in $H^2$ that additionally satisfy the Rayleigh–Taylor condition if surface tension effects are neglected. We also show that the problem exhibits the parabolic smoothing effect and we provide criteria for the global existence of solutions.## References

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## Additional Information

**Bogdan–Vasile Matioc**- Affiliation: Institut for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover, Germany
- Email: matioc@ifam.uni-hannover.de
- Received by editor(s): January 13, 2017
- Received by editor(s) in revised form: May 17, 2017
- Published electronically: June 26, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 7511-7556 - MSC (2010): Primary 35R37, 35K59, 35K93, 35Q35, 42B20
- DOI: https://doi.org/10.1090/tran/7287
- MathSciNet review: 3841857