Skip to main content
SpringerLink
Log in

Splash Singularities for the One-Phase Muskat Problem in Stable Regimes

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

In this paper we show the existence in finite time of splash singularities for the one-phase Muskat problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrose D.M.: Well-posedness of two-phase Hele-Shaw flow without surface tension. European J. Appl. Math. 15(5), 597–607 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrose D.M.: The zero surface tension limit of two-dimensional interfacial Darcy flow. J. Math. Fluid Mech. 16(1), 105–143 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bear, J.: Dynamics of fluids in porous media. Dover Publications, 1988

  4. Beck T., Sosoe P., Wong P.: Duchon-Robert solutions for the Rayleigh-Taylor and Muskat problems. Journal of Differential Equations 256(1), 206–222 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Castro A., Córdoba D., Fefferman C., Gancedo F.: Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal., 208(3), 805–909 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Castro A., Córdoba D., Fefferman C., Gancedo F., Gómez-Serrano J.: Splash singularity for water waves. Proc. Natl. Acad. Sci., 109(3), 733–738 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Castro A., Córdoba D., Fefferman C., Gancedo F., Gómez-Serrano J.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. 178(3), 1061–1134 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castro A., Córdoba D., Fefferman C., Gancedo F., López-Fernández M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175(2), 909–948 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin P., Córdoba D., Gancedo F., Strain R.M.: On the global existence for the Muskat problem. J. Eur. Math. Soc. 15, 201–227 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Córdoba A., Córdoba D.: A pointwise estimate for fractionary derivatives with applications to P.D.E. Proc. Natl. Acad. Sci. 100(26), 15316–15317 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Córdoba A., Córdoba D., Gancedo F.: Interface evolution: the Hele-Shaw and Muskat problems. Ann. Math., 173(1), 477–542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Córdoba D., Gancedo F.: A maximum principle for the Muskat problem for fluids with different densities. Comm. Math. Phys. 286(2), 681–696 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Córdoba D., Granero R., Orive R.: The confined Muskat problem: differences with the deep water regime. Comm. Math. Sci. 12(3), 423–455 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Córdoba, D., Pernas-Castaño, T.: Non-splat singularity for the one-phase Muskat problem. Trans. Am. Math. Soc. (2014). arXiv:1409.2483

  16. Coutand D., Shkoller S.: On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Comm. Math. Phys. 325(1), 143–183 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Escher J., Matioc B.V.: On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results. Z. Anal. Anwend. 30(2), 193–218 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Evans L.C.: Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence (1998)

    Google Scholar 

  20. Fefferman C., Ionescu A., Lie V.: On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. J. 165(3), 417–462 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gancedo F., Strain R.M.: Absence of splash singularities for SQG sharp fronts and the Muskat problem. Proc. Natl. Acad. Sci. 111(2), 635–639 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  22. Gómez-Serrano J., Granero R.: On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof. Nonlinearity 27(6), 1471–1498 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Granero R.: Global existence for the confined Muskat problem. SIAM J. Math. Anal. 46(2), 1651–1680 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guo Y., Hallstrom C., Spirn D.: Dynamics near unstable, interfacial fluids. Comm. Math. Phys. 270(3), 635–689 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Knüpfer H., Masmoudi N.: Darcy’s flow with prescribed contact angle–Well-posedness and lubrication approximation. Arch. Ration. Mech. Anal. 218(2), 589–646 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Muskat M.: Two Fluid systems in porous media. The encroachment of water into an oil sand. Physics 5, 250–264 (1934)

    Article  ADS  MATH  Google Scholar 

  27. Otto F.: Evolution of microstructure in unstable porous media flow: a relaxational approach. Comm. Pure Appl. Maths 52, 873–915 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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  ADS  MathSciNet  MATH  Google Scholar 

  29. Siegel M., Caflisch R., Howison S.: Global Existence, Singular Solutions, and Ill-Posedness for the Muskat Problem. Comm. Pure Appl. Math. 57, 1374–1411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Székelyhidi L. Jr.: Relaxation of the incompressible porous media equation. Ann. Sci. Éc. Norm. Supér. 45(3), 491–509 (2012)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas and Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid and CSIC-UAM-UC3M-UCM, C/ Nicolas Cabrera, 13-15, 28049, Madrid, Spain

    Angel Castro

  2. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera, 13-15, 28049, Madrid, Spain

    Diego Córdoba

  3. Department of Mathematics, Princeton University, 1102 Fine Hall, Washington Rd, Princeton, NJ, 08544, USA

    Charles Fefferman

  4. Departamento de Análisis Matemático and IMUS, Universidad de Sevilla, C/ Tarfia s/n, Campus Reina Mercedes, 41012, Seville, Spain

    Francisco Gancedo

Authors
  1. Angel Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diego Córdoba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Charles Fefferman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Francisco Gancedo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diego Córdoba.

Additional information

Communicated by Peter Constantin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castro, A., Córdoba, D., Fefferman, C. et al. Splash Singularities for the One-Phase Muskat Problem in Stable Regimes. Arch Rational Mech Anal 222, 213–243 (2016). https://doi.org/10.1007/s00205-016-0999-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-0999-6

Keywords

Search

Navigation