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The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability

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Abstract

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and t he upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces.We are concerned with the Rayleigh–Taylor instability when the upper fluid is heavier than the lower fluid along the equilibrium interface. When the surface tension at the free internal interface is below the critical value, we prove that the problem is nonlinear unstable.

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References

  1. Beale, J.: Large-time regularity of viscous surface waves. Arch. Rat. Mech. Anal. 84(4), 307–352 (1983/1984)

  2. Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. The International Series of Monographs on Physics. Clarendon Press, Oxford, 1961

  3. Ebin D.: Ill-posedness of the Rayleigh–Taylor and Helmholtz problems for incompressible fluids. Commun. Part. Differ. Equ. 13(10), 1265–1295 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo Y., Strauss W.: Instability of periodic BGK equilibria. Commun. Pure Appl. Math. 48(8), 861–894 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guo Y., Tice I.: Compressible, inviscid Rayleigh–Taylor instability. Indiana Univ. Math. J. 60, 677–712 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Guo Y., Tice I.: Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Rat. Mech. Anal. 207(2), 459–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hwang H., Guo Y.: On the dynamical Rayleigh–Taylor instability. Arch. Rat. Mech. Anal. 167(3), 235–253 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jang J., Tice I.: Instability theory of the Navier–Stokes–Poisson equations. Anal. PDE 6(5), 1121–1181 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jang, J., Tice, I., Wang, Y.J.: The compressible viscous surface-internal wave problem: local well-posedness (2015). (Preprint)

  11. Jang, J., Tice, I., Wang, Y.J.: The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit (2015). (Preprint)

  12. Kull H.: Theory of the Rayleigh–Taylor instability. Phys. Rep. 206(5), 197–325 (1991)

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

  14. Matsumura A., Nishida T.: Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Commun. Math. Phys. 89(4), 445–464 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Prüss J., Simonett G.: On the Rayleigh–Taylor instability for the two-phase Navier-Stokes equations. Indiana Univ. Math. J. 59, 1853–1872 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rayleigh L.: Analytic solutions of the Rayleigh equation for linear density profiles. Proc. Lond. Math. Soc. 14, 170–177 (1883)

    MathSciNet  MATH  Google Scholar 

  17. Taylor G.I.: The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proc. R. Soc. Lond. Ser. A. 201, 192–196 (1950)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Wang Y.J., Tice I.: The viscous surface-internal wave problem: nonlinear Rayleigh–Taylor instability. Commun. Part. Differ. Equ. 37(11), 1967–2028 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, Y.J., Tice I., Kim C.: The viscous surface-internal wave problem: global well-posedness and decay. Arch. Rat. Mech. Anal. 212(1), 1–92 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wehausen, J., Laitone, E.: Surface waves. Handbuch der Physik, vol. 9, Part 3, pp. 446–778. Springer, Berlin, 1960

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Juhi Jang

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

    Ian Tice

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

    Yanjin Wang

Authors
  1. Juhi Jang
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  2. Ian Tice
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  3. Yanjin Wang
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Corresponding author

Correspondence to Ian Tice.

Additional information

Communicated by P. Constantin

J. Jang was supported in part by NSF Grants DMS-1212142 and DMS-1351898.

Y. J. Wang was supported by the National Natural Science Foundation of China (Nos.11201389, 11531010), the Fujian Province Natural Science Funds for Distinguished Young Scholar (No. 2015J06001), a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 201418), the Specialized Research Fund for the Doctoral Program of Higher Education (20120121120023), Xiamen University President Fund (20720150211), and Program for New Century Excellent Talents in Fujian Province University.

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Jang, J., Tice, I. & Wang, Y. The Compressible Viscous Surface-Internal Wave Problem: Nonlinear Rayleigh–Taylor Instability. Arch Rational Mech Anal 221, 215–272 (2016). https://doi.org/10.1007/s00205-015-0960-0

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  • DOI: https://doi.org/10.1007/s00205-015-0960-0

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