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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability


Authors: Rafael Granero-Belinchón and Steve Shkoller
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 35R35, 35Q35, 35S10, 76B03
DOI: https://doi.org/10.1090/tran/7335
Published electronically: May 20, 2019
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Abstract: We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across the fixed-in-time curve $ (x_1,-1+f(x_1))$, while the interface separating the fluid from the vacuum region is given by the time-dependent curve $ (x_1,h(x_1,t))$. Our estimates are based on a new methodology that relies upon a careful study of the PDE system, coupling Darcy's law and incompressibility of the fluid, rather than the analysis of the singular integral contour equation for the interface function $ h$. We are able to develop an existence theory for any initial interface given by $ h_0 \in H^2$ and any permeability curve-of-discontinuity that is given by $ f \in H^{2.5}$. In particular, our method allows for both curves to have (pointwise) unbounded curvature. In the case that the permeability discontinuity is the set $ f=0$, we prove global existence and decay to equilibrium for small initial data. This decay is obtained using a new energy-energy dissipation inequality that couples tangential derivatives of the velocity in the bulk of the fluid with the curvature of the interface. To the best of our knowledge, this is the first global existence result for the Muskat problem with discontinuous permeability.


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Additional Information

Rafael Granero-Belinchón
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Avenida los Castros s/n, Santander, Spain
Email: rafael.granero@unican.es

Steve Shkoller
Affiliation: Department of Mathematics, University of California, Davis, Davis, California 95616
Email: shkoller@math.ucdavis.edu

DOI: https://doi.org/10.1090/tran/7335
Received by editor(s): November 18, 2016
Received by editor(s) in revised form: July 15, 2017
Published electronically: May 20, 2019
Additional Notes: The first author was funded by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR) and the Grant MTM2014-59488-P from the Ministerio de Economía y Competitividad (MINECO, Spain).
The second author was supported by the National Science Foundation under grant DMS-1301380.
Article copyright: © Copyright 2019 American Mathematical Society