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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1675–1743 | Cite as

Sharp Nonlinear Stability Criterion of Viscous Non-resistive MHD Internal Waves in 3D

  • Yanjin WangEmail author
Article
  • 94 Downloads

Abstract

We consider the dynamics of two layers of incompressible electrically conducting fluid interacting with a magnetic field, which are confined within a 3D horizontally infinite slab and separated by a free internal interface. We assume that the upper fluid is heavier than the lower fluid so that the fluids are susceptible to Rayleigh–Taylor instability, yet we show that the viscous and non-resistive problem around the equilibrium is nonlinearly stable provided that the strength of the vertical component of the steady magnetic field, \({|{\bar{B}_3}|}\), is greater than the critical value, \({\mathcal{M}_c}\), which we identify explicitly. We also prove that the problem is nonlinearly unstable if \({|{\bar{B}_3}| < \mathcal{M}_c}\). Our results indicate that the non-horizontal magnetic field has a strong stabilizing effect on the Rayleigh–Taylor instability but the horizontal one does not have this in 3D.

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Notes

Acknowledgements

The author is deeply grateful to the referees for the valuable comments and suggestions.

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Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Beale, J.: The initial value problem for the Navier-Stokes equations with a free surface. Commun. Pure Appl. Math. 34(3), 359–392 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cabannes, H.: Theoretical Magnetofludynamics. Academic Press, New York (1970)Google Scholar
  3. 3.
    Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Clarendon Press, Oxford (1961)zbMATHGoogle Scholar
  4. 4.
    Cowling, T.G.: Magnetohydrodynnamics. Institute of Physics Publishing, 1976Google Scholar
  5. 5.
    Ebin, D.: Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible fluids. Commun. Partial Differ. Equ. 13(10), 1265–1295 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gerbeau, J.-F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, Y., Strauss, W.: Instability of periodic BGK equilibria. Commun. Pure Appl. Math. 48(8), 861–894 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guo, Y., Tice, I.: Compressible, inviscid Rayleigh-Taylor instability. Indiana Univ. Math. J. 60, 677–712 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo, Y., Tice, I.: Linear Rayleigh-Taylor instability for viscous, compressible fluids. SIAM J. Math. Anal. 42(4), 1688–1720 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    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
  12. 12.
    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
  13. 13.
    Hwang, H., Guo, Y.: On the dynamical Rayleigh-Taylor instability. Arch. Ration. Mech. Anal. 167(3), 235–253 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jiang, F., Jiang, S.: On instability and stability of three-dimensional gravity driven viscous flows in a bounded domain. Adv. Math. 264, 831–863 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jiang, F., Jiang, S.: On linear instability and stability of the Rayleigh-Taylor problem in magnetohydrodynamics. J. Math. Fluid Mech. 17(4), 639–668 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Jiang, F., Jiang, S., Wang, Y.J.: On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations. Commun. Partial Differ. Equ. 39(3), 399–438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jang, J., Tice, I.: Instability theory of the Navier-Stokes-Poisson equations. Anal. PDE 6(5), 1121–1181 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jang, J., Tice, I., Wang, Y.J.: The compressible viscous surface-internal wave problem: nonlinear Rayleigh-Taylor instability. Arch. Ration. Mech. Anal. 221(1), 215–272 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jang, J., Tice, I., Wang, Y.J.: The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit. Commun. Math. Phys. 343(3), 1039–1113 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jang, J., Tice, I., Wang,Y.J.: The compressible viscous surface-internalwave problem: local well-posedness. SIAM J. Math. Anal. 48(4), 2602–2673, 2016Google Scholar
  21. 21.
    Kull, H.: Theory of the Rayleigh-Taylor instability. Phys. Rep. 206(5), 197–325 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    Ladyzhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, New York (1969)zbMATHGoogle Scholar
  23. 23.
    Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media, 2nd edn. Pergamon, New York (1984)Google Scholar
  24. 24.
    Lin, F.H.: Some analytical issues for elastic complex fluids. Commun. Pure Appl. Math. 65(7), 893–919 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Padula, M., Solonnikov, V.A.: On the free boundary problem of magnetohydrodynamics. J. Math. Sci. 178, 313–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Prüss, J., Simonett, G.: On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations. Indiana Univ. Math. J. 59, 1853–1872 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rayleigh, L.: Analytic solutions of the Rayleigh equation for linear density profiles. Proc. Lond. Math. Soc. 14, 170–177 (1883)MathSciNetzbMATHGoogle 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)MathSciNetGoogle Scholar
  29. 29.
    Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Commun. Partial Differ. Equ. 31(1–3), 417–429 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Temam, R.: Navier-Stokes Equations: Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  32. 32.
    Wang, Y.J.: Critical magnetic number in the magnetohydrodynamic Rayleigh-Taylor instability. J. Math. Phys. 53(7), 073701 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Wang, Y.J., Tice, I.: The viscous surface-internal wave problem: nonlinear Rayleigh-Taylor instability. Commun. Partial Differ. Equ. 37(11), 1967–2028 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Y.J., 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
  35. 35.
    Wehausen, J., Laitone, E.: Surface waves. Handbuch der Physik, Vol. 9, Part 3. Springer, Berlin, 446–778, 1960Google Scholar

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

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina

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