Archive for Rational Mechanics and Analysis

, Volume 229, Issue 1, pp 339–360 | Cite as

Global Regularity of 2D Density Patches for Inhomogeneous Navier–Stokes

  • Francisco GancedoEmail author
  • Eduardo García-Juárez


This paper is about Lions’ open problem on density patches (Lions in Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture series in mathematics and its applications, Clarendon Press, Oxford University Press, New York, 1996): whether or not inhomogeneous incompressible Navier–Stokes equations preserve the initial regularity of the free boundary given by density patches. Using classical Sobolev spaces for the velocity, we first establish the propagation of \({C^{1+\gamma}}\) regularity with \({0 < \gamma < 1}\) in the case of positive density. Furthermore, we go beyond this to show the persistence of a geometrical quantity such as the curvature. In addition, we obtain a proof for \({C^{2+\gamma}}\) regularity.


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  1. 1.
    Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, vol. 343. Springer, Berlin, 2011Google Scholar
  2. 2.
    Beale J.T.: The initial value problem for the Navier–Stokes equations with a free surface. Comm. Pure Appl. Math. 34(3), 359–392 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertozzi A., Constantin P.: Global regularity for vortex patches. Comm. Math. Phys. 152(1), 19–28 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castro A., Córdoba D., Gancedo F., Fefferman C., López-Fernández M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175(2), 909–948 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Castro A., Córdoba D., Fefferman C., Gancedo F.: Breakdown of smoothness for the Muskat problem. Arch. Ration. Mech. Anal. 208(3), 805–909 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Castro A., Córdoba D., Fefferman C., Gancedo F., Gómez-Serrano J.: Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. (2) 178(3), 1061–1134 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Castro, A., Córdoba, D., Fefferman, C., Gancedo, F., Gómez-Serrano, J.: Splash Singularities for the Free Boundary Navier–Stokes Equations, preprint arXiv:1504.02775, 2015
  8. 8.
    Castro A., Córdoba D., Fefferman C., Gancedo F.: Splash singularities for the one-phase Muskat problem in stable regimes. Arch. Ration. Mech. Anal. 222(1), 213–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chemin J.-Y.: Persistance de structures géométriques dans les fluides incompressibles bidimensionnels. Ann. Sci. École Norm. Sup. (4) 26(4), 517–542 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coutand, D., Shkoller, S.: On the Splash Singularity for the Free-Surface of a Navier–Stokes Fluid, preprint, arXiv:1505.01929, 2015
  13. 13.
    Coutand D., Shkoller S.: On the impossibility of finite-time splash singularities for vortex sheets. Arch. Ration. Mech. Anal. 221(2), 987–1033 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Danchin R., Mucha P.B.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Comm. Pure. Appl. Math. 65, 1458–1480 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Danchin R., Mucha P.B.: Incompressible flows with piecewise constant density. Arch. Ration. Mech. Anal. 207(3), 991–1023 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Danchin R., Zhang X.: On the persistence of Hölder regular patches of density for the inhomogeneous Navier–Stokes equations. J. Ec. Polytech. Math. 4, 781–811 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Denisova, I.V.: Evolution of a Closed Interface Between Two Liquids of Different Types. European Congress of Mathematics, Vol. II (Barcelona, 2000), pp. 263–272, Progr. Math., 202, Birkhuser, Basel, 2001Google Scholar
  18. 18.
    Fefferman C., Ionescu A.D., Lie V.: On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. J. 165(3), 417–462 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gancedo F., García-Juárez E.: Global regularity for 2D Boussinesq temperature patches with no diffusion. Ann. PDE 3, 14 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo Y., Tice I.: Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Ration. Mech. Anal. 207(2), 459–531 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guo Y., Tice I.: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429–1533 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guo Y., Tice I.: Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hataya Y.: Decaying solution of a Navier–Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Eqs. 120, 215–254 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hoff D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Comm. Pure Appl. Math. 55, 1365–1407 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Huang J., Paicu M., Zhang P.: Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-lipschitz velocity. Arch. Ration. Mech. Anal. 209(2), 631–682 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ladyzhenskaja, O.-A., Solonnikov, V.-A.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. (Russian) Boundary value problems of mathematical physics, and related questions of the theory of functions, 8. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 52, 52–109, 218–219, 1975Google Scholar
  29. 29.
    Liao, X., Zhang, P.: On the global regularity of the two-dimensional density patch for inhomogeneous incompressible viscous flow. Arch. Ration. Mech. Anal. 220(3), 220–3, 937–981, 2016Google Scholar
  30. 30.
    Liao, X., Zhang, P.: Global Regularities of Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow with General Density, 2016 preprint, arXiv:1604.07922v1
  31. 31.
    Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and Its Applications. Clarendon Press, Oxford University Press, New York, 1996. Incompressible Models, Oxford Science Publications 1996Google Scholar
  32. 32.
    Lunardi, A.: Interpolation Theory, 2nd edn. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). Edizioni della Normale, Pisa, 2009Google Scholar
  33. 33.
    Masmoudi N., Rousset F.: Uniform regularity and vanishing viscosity limit for the free surface NavierStokes equations. Arch. Ration. Mech. Anal. 223(1), 301–417 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Paicu M., Zhang P., Zhang Z.: Globalunique solvability of inhomogeneous Navier–Stokes equations with bounded density. Comm. Partial Differ. Equ. 38(7), 1208–1234 (2013)CrossRefzbMATHGoogle Scholar
  35. 35.
    Robinson, J.C., Rodrigo, J.L., Sadowski, W.: The Three-Dimensional Navier–Stokes Equations. Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, 2016Google Scholar
  36. 36.
    Simon J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21(5), 1093–1117 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Solonnikov, V.: A. Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 41(6), 1388–1424, 1448, 1977Google Scholar
  38. 38.
    Sylvester D.L.G.: Large time existence of small viscous surface waves without surface tension. Comm. Partial Differ. Equ. 15(6), 823–903 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, 1970Google Scholar
  40. 40.
    Tani A., Tanaka N.: Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130(4), 303–314 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wang Y., Tice I., Kim C.: The viscous surface-internal wave problem: global well-posedness and decay. Arch. Ration. Mech. Anal. 212(1), 1–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wehausen, J.V., Laitone, E.V.: Surface waves. Handbuch der Physik, Vol. 9, Part 3 pp. 446–778. Springer, Berlin, 1960Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, and IMUSUniversidad de SevillaSevilleSpain

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