Advertisement

On the Linear Stability of Vortex Columns in the Energy Space

  • Thierry Gallay
  • Didier SmetsEmail author
Article
  • 53 Downloads

Abstract

We investigate the linear stability of inviscid columnar vortices with respect to finite energy perturbations. For a large class of vortex profiles, we show that the linearized evolution group has a sub-exponential growth in time, which means that the associated growth bound is equal to zero. This implies in particular that the spectrum of the linearized operator is entirely contained in the imaginary axis. This contribution complements the results of our previous work Gallay and Smets (Spectral stability of inviscid columnar vortices, 2018. arXiv:1805.05064), where spectral stability was established for the linearized operator in the enstrophy space.

Notes

Acknowledgements

This work was partially supported by grants ANR-18-CE40-0027 (Th.G.) and ANR-14-CE25-0009-01 (D.S.) from the “Agence Nationale de la Recherche”. The authors warmly thank an anonymous referee for suggesting a more natural way to prove compactness of the operator \(B_{m,k}\), which is now implemented in Sect. 3.2.

Compliance with ethical standards

Conflict of interest

The author(s) declares that they have no competing interests.

References

  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)zbMATHGoogle Scholar
  2. 2.
    Alekseenko, S.V., Kuibin, P.A., Okulov, V.L.: Theory of Concentrated Vortices. An Introduction. Springer, Berlin (2007)zbMATHGoogle Scholar
  3. 3.
    Arnold, V.I.: Conditions for the nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Dokl. Mat. Nauk. 162, 773–777 (1965)Google Scholar
  4. 4.
    Drazin, P., Reid, W.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  5. 5.
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators, 2nd edn. Oxford University Press, Oxford (2018)CrossRefGoogle Scholar
  6. 6.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, Berlin (1999)Google Scholar
  7. 7.
    Fabre, D., Sipp, D., Jacquin, L.: Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274 (2006)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems. Springer Monographs in Mathematics. Springer, Berlin (2011)Google Scholar
  9. 9.
    Gallay, Th., Smets, D.: Spectral stability of inviscid columnar vortices (2018). arXiv:1805.05064
  10. 10.
    Gallay, Th., Smets, D.: On the linear stability of vortex columns in the energy space (first version) (2018). arXiv:1811.07584v1
  11. 11.
    Howard, L.N., Gupta, A.S.: On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476 (1962)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Le Dizès, S., Lacaze, L.: An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96 (2005)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Marchioro, C., Pulvirenti, M.: Some considerations on the nonlinear stability of stationary planar Euler flows. Commun. Math. Phys. 100, 343–354 (1985)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rayleigh, L.: On the dynamics of revolving fluids. Proc. R. Soc. A 93, 148–154 (1917)ADSCrossRefGoogle Scholar
  15. 15.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  16. 16.
    Roy, A., Subramanian, G.: Linearized oscillations of a vortex column: the singular eigenfunctions. J. Fluid Mech. 741, 404–460 (2014)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Thomson (Lord Kelvin), Sir W.: Vibrations of a columnar vortex. Proc. R. Soc. Edinb. 10, 443–456 (1880) [The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science X (1880), 153–168]Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble AlpesCNRSGièresFrance
  2. 2.Laboratoire Jacques-Louis LionsSorbonne UniversitéParisFrance

Personalised recommendations