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Transactions of the American Mathematical Society

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


Author: Bogdan–Vasile Matioc
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
Published electronically: June 26, 2018
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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.


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Bogdan–Vasile Matioc
Affiliation: Institut for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover, Germany
Email: matioc@ifam.uni-hannover.de

DOI: https://doi.org/10.1090/tran/7287
Keywords: Muskat problem, Rayleigh--Taylor condition, surface tension, singular integral operator
Received by editor(s): January 13, 2017
Received by editor(s) in revised form: May 17, 2017
Published electronically: June 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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