Nonlinear instability in an ideal fluid

Susan Friedlander; Walter Strauss; Misha Vishik

Annales de l'I.H.P. Analyse non linéaire (1997)

  • Volume: 14, Issue: 2, page 187-209
  • ISSN: 0294-1449

How to cite


Friedlander, Susan, Strauss, Walter, and Vishik, Misha. "Nonlinear instability in an ideal fluid." Annales de l'I.H.P. Analyse non linéaire 14.2 (1997): 187-209. <>.

author = {Friedlander, Susan, Strauss, Walter, Vishik, Misha},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {linearized instability; Euler equations; space periodic flows without stagnation points; space-periodic shear flows},
language = {eng},
number = {2},
pages = {187-209},
publisher = {Gauthier-Villars},
title = {Nonlinear instability in an ideal fluid},
url = {},
volume = {14},
year = {1997},

AU - Friedlander, Susan
AU - Strauss, Walter
AU - Vishik, Misha
TI - Nonlinear instability in an ideal fluid
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1997
PB - Gauthier-Villars
VL - 14
IS - 2
SP - 187
EP - 209
LA - eng
KW - linearized instability; Euler equations; space periodic flows without stagnation points; space-periodic shear flows
UR -
ER -


  1. [BB] J. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Funct. Anal., Vol. 15, 1974, 4, pp. 341-363. Zbl0279.58005MR344713
  2. [B] F. Browder, On the spectral theory of elliptic differential operators, I, Math. Annalen, Vol. 142, 1961, pp. 22-130. Zbl0104.07502MR209909
  3. [FV] S. Friedlander, and M. Vishik, Instability criteria for steady flows of a perfect fluid, Chaos, Vol. 2, 1992, No. 3, pp. 455-460. Zbl1055.76518MR1184488
  4. [G] N. Gunther, On the motion of fluid in a moving container, Isvestia Akad. Nauk SSSR. Ser. Fiz.-Mat., Vol. 20, 1927, pp. 1323-1348, 1503-1532; Ibid. Vol. 21, 1927, pp. 521-556, 735-756; Vol. 22, 1928, pp. 9-30. 
  5. [GS] Y. Guo, and W. Strauss, Nonlinear instability of double-humped equilibria, Ann. IHP (An. NL) (to appear). Zbl0836.35130MR1340268
  6. [K] T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rat. Mech. Anal., Vol. 27, 1968, pp. 188-200. Zbl0166.45302MR211057
  7. [L] L. Lichtenstein, Über einige Existenzprobleme der Hydrodynamik homogener unzusammendrückbarer, reibunglosser Flussikeiten und die Helmgoltzschen Wirbelsalte, Math. Zeit., Vol. 23, 1925, pp. 89-154; Ibid. Vol. 26, 1927, pp. 196-323, 387-415, 723; Vol. 32, 1930, pp. 608. Zbl51.0658.01MR1544857JFM51.0658.01
  8. [Li] V. Liu, An example of instability for the Navier-Stokes equations on the 2-dimensional torus, Comm. in PDE's, Vol. 17, 1992, 11-12, pp. 1995-2012. Zbl0781.35054MR1194747
  9. [N] R. Nussbaum, The radius of the essential spectrum, Duke Math. J., Vol. 37, 1970, pp. 473-478. Zbl0216.41602MR264434
  10. [MS] L. Meshalkin, and Y. Sinai, Investigation of stability for a system of equations describing plane motion of a viscous incompressible fluid, Applied Math. and Mech., Vol. 25, 1961, pp. 1140-1143. Zbl0108.39501MR137419
  11. [T] R. Temam, Local existence of C∞ solutions of the Euler equations of incompressible perfect fluids, Lectures Notes in Math., 1976, Vol. 565, pp. 184-194. Zbl0355.76017MR467033
  12. [To] W. Tollmien, Ein allgemeines Kriterium der Instabilitat laminar geschwindigkeits verteilungen, Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl. Fachgruppe I, Vol. 1, 1935, pp. 79-114. JFM61.0925.01
  13. [Y] V. Yudovich, The Linearization Method in Hydrodynamical Stability Theory, Translations of Math. Monographs, Vol. 74, 1989, AMS170 pp. Zbl0727.76039MR1003607
  14. [Yu] V. Yudovich, Example of secondary stationary flow or periodic flow appearing while a laminar flow of a viscous incompressible fluid loses its stability, Applied Math. and Mech., Vol. 29, 1965, No. 3, pp. 453-467. Zbl0148.22307
  15. [V] M. Vishik, Spectrum of small oscillations of an ideal fluid and Lyapunov exponents preprint. Zbl0870.76017MR1423046
  16. [W] W. Wolibner, Un théorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z., Vol. 37, 1933, pp. 698-726. Zbl0008.06901MR1545430JFM59.1447.02

Citations in EuDML Documents

  1. F. Rousset, N. Tzvetkov, Transverse nonlinear instability for two-dimensional dispersive models
  2. Emmanuel Grenier, On the derivation of homogeneous hydrostatic equations
  3. B. Desjardins, E. Grenier, Linear instability implies nonlinear instability for various types of viscous boundary layers
  4. H. Kalisch, Nonexistence of Coherent Structures in Two-dimensional Inviscid Channel Flow
  5. Christophe Cheverry, Cascade of phases in turbulent flows
  6. Emmanuel Grenier, Non dérivation des équations de Prandtl
  7. Guy Métivier, Exemples d’instabilités pour des équations d’ondes non linéaires
  8. Christophe Cheverry, Olivier Guès, Guy Métivier, Oscillations fortes sur un champ linéairement dégénéré
  9. Christophe Cheverry, Sur la propagation de quasi-singularités
  10. Christophe Cheverry, Sur un problème de stabilité posé en optique géométrique non linéaire surcritique

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