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A Maximum Principle for the Muskat Problem for Fluids with Different Densities

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Abstract

We study the fluid interface problem given by two incompressible fluids with different densities evolving by Darcy’s law. This scenario is known as the Muskat problem for fluids with the same viscosities, being in two dimensions mathematically analogous to the two-phase Hele-Shaw cell. We prove in the stable case (the denser fluid is below) a maximum principle for the L norm of the free boundary.

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References

  1. Ambrose D.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Euro. J. Appl. Math. 15, 597–607 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bear J.: Dynamics of Fluids in Porous Media. American Elsevier, New York (1972)

    Google Scholar 

  3. Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Constantin P., Pugh M.: Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393–415 (1993)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Córdoba A., Córdoba D.: A maximum principle applied to Quasi-geostrophic equations. Commun. Math. Phys. 249(3), 511–528 (2004)

    Article  MATH  ADS  Google Scholar 

  6. Córdoba D., Fontelos M.A., Mancho A.M., Rodrigo J.L.: Evidence of singularities for a family of contour dynamics equations. Proc. Natl. Acad. Sci. USA 102, 5949–5952 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Córdoba D., Gancedo F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273(2), 445–471 (2007)

    Article  MATH  ADS  Google Scholar 

  8. Dombre T., Pumir A., Siggia E.: On the interface dynamics for convection in porous media. Physica D 57, 311–329 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Escher J., Simonett G.: Classical solutions for Hele- Shaw models with surface tension. Adv. Differ. Eqs. 2, 619–642 (1997)

    MATH  MathSciNet  Google Scholar 

  10. Gancedo F.: Existence for the α-patch model and the QG sharp front in Sobolev spaces. Adv. Math. 217/6, 2569–2598 (2008)

    Article  MathSciNet  Google Scholar 

  11. Hele-Shaw, H.S.: The flow of water. Nature 58 (1489), 34-36 (1898); Ibid. 58 (1509), 520 (1898)

  12. Hou T.Y., Lowengrub J.S., Shelley M.J.: Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114, 312–338 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Muskat M.: The flow of homogeneous fluids through porous media. McGraw-Hill, New York (1937)

    MATH  Google Scholar 

  14. Saffman P.G., Taylor G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. London, Ser. A 245, 312–329 (1958)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Siegel M., Caflisch R., Howison S.: Global existence, singular solutions, and Ill-Posedness for the Muskat problem. Comm. Pure and Appl. Math. 57, 1374–1411 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Stein E.: Harmonic Analysis. Princeton University Press, Princeton, NJ (1993)

    MATH  Google Scholar 

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Correspondence to Francisco Gancedo.

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Communicated by P. Constantin

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Córdoba, D., Gancedo, F. A Maximum Principle for the Muskat Problem for Fluids with Different Densities. Commun. Math. Phys. 286, 681–696 (2009). https://doi.org/10.1007/s00220-008-0587-1

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  • DOI: https://doi.org/10.1007/s00220-008-0587-1

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