The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

Article

Abstract

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.

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References

  1. 1.
    Allain G.: Small-time existence for the Navier–Stokes equations with a free surface. Appl. Math. Optim. 16(1), 37–50 (1987)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bae H.: Solvability of the free boundary value problem of the Navier–Stokes equations. Discrete Contin. Dyn. Syst. 29(3), 769–801 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beale J.: The initial value problem for the Navier–Stokes equations with a free surface. Comm. Pure Appl. Math. 34(3), 359–392 (1981)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Beale, J. : Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. 84(4), 307–352 (1983/84)Google Scholar
  5. 5.
    Beale, J.,Nishida, T. : Large-time behavior of viscous surface waves. Recent topics in nonlinear PDE, II (Sendai, 1984), pp. 1–14, North-Holland Math. Stud., vol. 128. North-Holland, Amsterdam, (1985)Google Scholar
  6. 6.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics, Clarendon Press, Oxford, (1961)Google Scholar
  7. 7.
    Coutand, D., Shkoller, S.: Unique solvability of the free-boundary Navier–Stokes equations with surface tension. Preprint (2003) [arXiv:math/0212116]Google Scholar
  8. 8.
    Denisova I.V.: Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Acta Appl. Math. 37(1–2), 31–40 (1994)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Denisova I.V.: Global solvability of the problem on the motion of two fluids without surface tension. J. Math. Sci. 152(5), 625–637 (2008)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Denisova I.V., Solonnikov V.A.: Classical solvability of the problem on the motion of two viscous incompressible fluids. St.Petersburg Math. J. 7(5), 755– (1996)MathSciNetGoogle Scholar
  11. 11.
    Evans, L.: Partial Differential Equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, (2010)Google Scholar
  12. 12.
    Guo Y., Tice I.: Linear Rayleigh–Taylor instability for viscous, compressible fluids. SIAM J. Math. Anal. 42(4), 1688–1720 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Guo Y., Tice I.: Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Guo, Y., Tice, I.: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE (to appear)Google Scholar
  15. 15.
    Guo Y., Tice I.: Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Rational Mech. Anal. 207(2), 459–531 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hataya Y.: Global solution of two-layer Navier–Stokes flow. Nonlinear Anal. 63, e1409–e1420 (2005)CrossRefMATHGoogle Scholar
  17. 17.
    Hataya Y.: Decaying solution of a Navier–Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)MATHMathSciNetGoogle Scholar
  18. 18.
    Jin B.J., Padula M.: In a horizontal layer with free upper surface. Commun. Pure Appl. Anal. 1(3), 379–415 (2002)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, New York, (1969)Google Scholar
  20. 20.
    Ladyzhenskaya O.A., Rivkind J., Ural’ceva N.: Solvability of diffraction problems in the classical sense. Trudy Mat. Inst. Steklov. 92, 116–146 (1966)MATHMathSciNetGoogle Scholar
  21. 21.
    Nishida T., Teramoto Y., Yoshihara H.: Global in time behavior of viscous surface waves: horizontally periodic motion. J. Math. Kyoto Univ. 44(2), 271–323 (2004)MATHMathSciNetGoogle Scholar
  22. 22.
    Prüess J., Simonett G.: On the two-phase Navier–Stokes equations with surface tension. Interfaces Free Bound. 12(3), 311–345 (2010)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Rayleigh L.: Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177 (1883)MATHMathSciNetGoogle Scholar
  24. 24.
    Shibata, Y., Shimizu, S.: Free boundary problems for a viscous incompressible fluid. Kyoto Conference on the Navier–Stokes Equations and their Applications, pp. 356–358, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, (2007)Google Scholar
  25. 25.
    Solonnikov, V.A.: Solvability of a problem on the motion of a viscous incompressible fluid that is bounded by a free surface. Math. USSR-Izv. 11(6): 1323–1358 (1977/1978)Google Scholar
  26. 26.
    Solonnikov, V.A.: On an initial boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Proc. Steklov Inst. Math. 3, 191–239 (1991)Google Scholar
  27. 27.
    Solonnikov V.A.: Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersburg Math. J. 3(1), 189–220 (1992)MathSciNetGoogle Scholar
  28. 28.
    Solonnikov V.A., Skadilov V.E.: On a boundary value problem for a stationary system of Navier–Stokes equations. Proc. Steklov Inst. Math. 125, 186–199 (1973)MATHGoogle Scholar
  29. 29.
    Sylvester D.L.G.: Large time existence of small viscous surface waves without surface tension. Comm. Partial Differential Equations 15(6), 823–903 (1990)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Tanaka N.: Global existence of two phase non-homogeneous viscous incompressible fluid flow. Comm. Partial Differential Equations 18(1–2), 41–81 (1993)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Tanaka N.: Two-phase free boundary problem for viscous incompressible thermo-capillary convection. Jpn. J. Math. 21(1), 1-42 (1995)Google Scholar
  32. 32.
    Tani, A.: Small-time existence for the three-dimensional Navier–Stokes equations for an incompressible fluid with a free surface. Arch. Rational Mech. Anal. 133(4), 299–331 (1996)Google Scholar
  33. 33.
    Tani A., Tanaka N.: Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Rational Mech. Anal. 130(4), 303–314 (1995)ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Taylor G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. Roy. Soc. London Ser. A. 201, 192–196 (1950)ADSCrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam, (1984)Google Scholar
  36. 36.
    Wehausen, J., Laitone, E.: Surface waves. Handbuch der Physik Vol. 9, Part 3, pp. 446–778. Springer, Berlin, (1960)Google Scholar
  37. 37.
    Xu L., Zhang Z.: On the free boundary problem to the two viscous immiscible fluids. J. Differ. Equ. 248(5), 1044–1111 (2010)ADSCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityFujianChina
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  3. 3.DPMMSUniversity of CambridgeCambridgeUK

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