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

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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. Allain G.: Small-time existence for the Navier–Stokes equations with a free surface. Appl. Math. Optim. 16(1), 37–50 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bae H.: Solvability of the free boundary value problem of the Navier–Stokes equations. Discrete Contin. Dyn. Syst. 29(3), 769–801 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beale, J. : Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal. 84(4), 307–352 (1983/84)

    Google Scholar 

  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)

  6. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics, Clarendon Press, Oxford, (1961)

  7. Coutand, D., Shkoller, S.: Unique solvability of the free-boundary Navier–Stokes equations with surface tension. Preprint (2003) [arXiv:math/0212116]

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  11. Evans, L.: Partial Differential Equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, (2010)

  12. Guo Y., Tice I.: Linear Rayleigh–Taylor instability for viscous, compressible fluids. SIAM J. Math. Anal. 42(4), 1688–1720 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Guo Y., Tice I.: Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Guo, Y., Tice, I.: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE (to appear)

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Hataya Y.: Global solution of two-layer Navier–Stokes flow. Nonlinear Anal. 63, e1409–e1420 (2005)

    Article  MATH  Google Scholar 

  17. Hataya Y.: Decaying solution of a Navier–Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)

    MATH  MathSciNet  Google Scholar 

  18. Jin B.J., Padula M.: In a horizontal layer with free upper surface. Commun. Pure Appl. Anal. 1(3), 379–415 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, New York, (1969)

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  22. Prüess J., Simonett G.: On the two-phase Navier–Stokes equations with surface tension. Interfaces Free Bound. 12(3), 311–345 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

  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. 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. 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)

    MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tanaka N.: Global existence of two phase non-homogeneous viscous incompressible fluid flow. Comm. Partial Differential Equations 18(1–2), 41–81 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam, (1984)

  36. Wehausen, J., Laitone, E.: Surface waves. Handbuch der Physik Vol. 9, Part 3, pp. 446–778. Springer, Berlin, (1960)

  37. Xu L., Zhang Z.: On the free boundary problem to the two viscous immiscible fluids. J. Differ. Equ. 248(5), 1044–1111 (2010)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Fujian, 361005, China

    Yanjin Wang

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Ian Tice

  3. DPMMS, University of Cambridge, Cambridge, CB3 0WA, UK

    Chanwoo Kim

Authors
  1. Yanjin Wang
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  2. Ian Tice
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  3. Chanwoo Kim
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Correspondence to Ian Tice.

Additional information

Communicated by C. Dafermos

Y. Wang was partially supported by National Natural Science Foundation of China-NSAF (No. 10976026). I. Tice was supported by an NSF Postdoctoral Research Fellowship. C. Kim was supported in part by NSF Grant FRG DMS 07-57227.

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Wang, Y., Tice, I. & Kim, C. The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay. Arch Rational Mech Anal 212, 1–92 (2014). https://doi.org/10.1007/s00205-013-0700-2

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  • DOI: https://doi.org/10.1007/s00205-013-0700-2

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