Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 4, pp 1667–1680 | Cite as

Eigenvalues of the Linearized 2D Euler Equations via Birman–Schwinger and Lin’s Operators

  • Yuri LatushkinEmail author
  • Shibi Vasudevan


We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman–Schwinger type operators \(K_{\lambda }(\mu )\) and their associated 2-modified perturbation determinants \(\mathcal D(\lambda ,\mu )\). Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator \(L_\mathrm{vor}\) in terms of zeros of the 2-modified Fredholm determinant \(\mathcal D(\lambda ,0)={\text {det}}_{2}(I-K_{\lambda }(0))\) associated with the Hilbert Schmidt operator \(K_{\lambda }(\mu )\) for \(\mu =0\). As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for \(L_\mathrm{vor}\) to the number of negative eigenvalues of a limiting elliptic dispersion operator \(A_{0}\).


2D Euler equations Instability Birman–Schwinger operators 2-modified perturbation determinants 


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Conflict of interest

The authors thereby declare that they have no conflicts of interests.


  1. 1.
    Belenkaya, L., Friedlander, S., Yudovich, V.: The unstable spectrum of oscillating shear flows. SIAM J. Appl. Math. 59(5), 1701–1715 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Butta, P., Negrini, P.: On the stability problem of stationary solutions for the Euler equation on a 2-dimensional torus. Reg. Chaotic Dyn. 15, 637–645 (2010)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence (1999)CrossRefGoogle Scholar
  4. 4.
    Cox, G.: The \(L^2\) essential spectrum of the 2D Euler operator. J. Math. Fluid Mech. 16, 419–429 (2014)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  6. 6.
    Faddeev, L.D.: On the theory of the stability of stationary plane parallel flows of an ideal fluid. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 21, 164–172 (1971)MathSciNetGoogle Scholar
  7. 7.
    Friedlander, S.: Lectures on stability and instability of an ideal fluid. In: Hyperbolic Equations and Frequency Interactions (Park City, UT, 1995). IAS/Park City Mathematics Series, vol. 5, pp. 227-304. American Mathematical Society, Providence (1999)Google Scholar
  8. 8.
    Friedlander, S., Schnirelman, A.: Instability of Steady Flows of an Ieal Incompressible Fluid. Mathematical Fluid Mechanics, pp. 143–172. Birkhauser, Basel (2001)Google Scholar
  9. 9.
    Friedlander, S., Howard, L.: Instability in parallel flows revisited. Stud. Appl. Math. 101(1), 1–21 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Friedlander, S., Shvydkoy, R.: On recent developments in the spectral problem for the linearized Euler equation. Contemp. Math. 371, 271–295 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Friedlander, S., Strauss, W., Vishik, M.: Nonlinear instability in an ideal fluid. Ann. Inst. Poincare 14(2), 187–209 (1997)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Friedlander, S., Vishik, M., Yudovich, V.: Unstable eigenvalues associated with inviscid fluid flows. J. Math. Fluid Mech. 2(4), 365–380 (2000)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Friedlander, S., Yudovich, V.: Instabilities in fluid motion. Not. AMS 46(11), 1358–1367 (1999)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gallaire, F., Grard-Varet, D., Rousset, F.: Three-dimensional instability of planar flows. Arch. Ration. Mech. Anal. 186(3), 423–475 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gohberg, I., Krein, M.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969)Google Scholar
  16. 16.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980)zbMATHGoogle Scholar
  17. 17.
    Lin, Z.: Instability of periodic BGK waves. Math. Res. Lett. 8(4), 521–534 (2001)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Lin, Z.: Instability of some ideal plane flows. SIAM J. Math. Anal. 35(2), 318–356 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lin, Z.: Nonlinear instability of ideal plane flows. Int. Math. Res. Not. 41, 2147–2178 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lin, Z.: Some stability and instability criteria for ideal plane flows. Commun. Math. Phys. 246, 87–112 (2004)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  22. 22.
    Shvydkoy, R., Latushkin, Y.: Essential spectrum of the linearized 2D Euler equation and Lyapunov-Oseledets exponents. J. Math. Fluid Mech. 7, 164–178 (2005)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Simon, B.: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, 2nd edn. American Mathematical Society, Providence (2005)Google Scholar
  24. 24.
    Simonnet, E.: On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains. Physica D 237, 2539–2552 (2008)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Swaters, G.: Introduction to Hamiltonian Fluid Dynamics and Stability Theory: Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematic, vol. 102. Chapman & Hall/CRC, Boca Raton (2000)Google Scholar
  26. 26.
    Yafaev, D.: Mathematical Scattering Theory. General Theory. Translations of Mathematical Monographs, vol. 105. American Mathematical Society, Providence (1992)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBengaluruIndia

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