Nonlinear instability in an ideal fluid

 Access to full text

 Full (PDF)

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

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