Nonlinear stability for saddle solutions of ideal flows and symmetry breaking

Communications in Mathematical Physics

Volumn 193, Issue 3, 1998, Pages 713-736

  Wolansky, G ^a   Ghil, M ^b  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032073658     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050345     Document Type: Article

References (29)

1. Abarbanel, H.D.I., Holm, D.D., Marsden, J.E., Ratiu, T.: Nonlinear stability analysis of ideal stratified, incompressible fluid flow. Phys. Rev. Lett. 52, 2352-2355 (1984)
2. Andrews, D.G.: On the existence of non-zonal flows satisfying sufficient conditions for stability. Geophys. Astrophys. Fluid Dyn. 28, 243-256 (1984)
3. Arnol'd, V.I.: Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Sov. Math. Dokl. 6, 773-776 (1965)
4. Arnol'd, V.I.: On an a priori estimate in the theory of hydrodynamical stability. Am. Math. Soc. Transl.: Series 2 79, 267-269 (1969)
5. Arnol'd, V.I.: Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1980
6. Ball, J.M., Marsden, J.E: Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rat. Mech. Anal. 86, 251-277 (1984)
7. Ball, J.M., Knops, J.E., Marsden, J.E.: Two examples in nonlinear elasticity. Lec. Notes in Math., 665, 41-49 (1978)
8. Blumen, W.: On the stability of quasigeostrophic flow. J. Atmos. Sci. 25, 929-931 (1968)
9. Batt, J., Rein, G.: A rigorous stability result for the Vlasov-Poisson system in three dimensions. Anal. Mat. Pura Appl. 164, 133-154 (1993)
10. Carnevale, G.F., Shepherd, T.G.: On the interpretation of Andrews' theorem. Geophys. Astrophys. Fluid Dyn. 51, 1-17 (1990)
11. Casimir, H.G.B.: Üeber die Konstruktion einer zu den irreduziblen Darstellungen halbeinfacher kontinuierlicher Gruppen gehörigen Differentialgleichung. Proc. R. Soc. Amsterdam 34, 844-846 (1931)
12. Chern, S.-J., Marsden, J.E.: A note on symmetry and stability for fluid flows. Geophys. Astrophys. Fluid Dyn. 51, 19-26 (1990)
13. Holm, D.D.: Nonlinear stability of fluid and plasma equilibria. Phys. Reports 123 #1 & 2, 1-116 (1985)
14. Kandrup, H.E., Morrison, P.J.: Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relatevistic star clusters. Ann. Phys. 233, 114-166 (1993)
15. Lewis, D.: Nonlinear stability of a rotating planar liquid drop. Arch. Rat. Mech. Anal. 106, 287-333 (1989)
16. Marsden, J.E.: Lectures on Mechanics. Lond. Math. Soc. Lect. Note Ser. 174, Cambridge: Cambridge Univ. Press, 1992
17. McIntyre, M.E., Shepherd, T.G.: An exact local conservation theory for finite-amplitude disturbance to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527-565 (1987)
18. Mu, M.: Personal communication, Isaac Newton Institute, Cambridge (September 1996)
19. Nycander, J.: Refutation of stability proofs for dipole vortices. Phys. Fluids A 4, 467-476 (1992)
20. Ripa, P.: On the stability of elliptical vortex solutions of the shallow-water equations. J. Fluid Mech. 183, 343-363 (1987)
21. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Vol. IV, New York: Academic Press, 1978
22. Shepherd, T.G.: Rigorous bounds on the nonlinear saturation of instabilities of parallel shear flows. J. Fluid Mech. 196, 291-322 (1988)
23. Sakuma, H., Ghil, M.: Stability of stationary barotropic modons by Lyapunov's direct method. J. Fluid Mech. 211, 393-416 (1990)
24. Sakuma, H., Ghil, M.: Stability of propagating modons for small-amplitude perturbations. Phys. Fluids A 3, 408-414 (1991)
25. Simo, J.C., Lewis, D.R., Marsden, J.E.: Stability of relative equilibria I: The reduced energy momentum method. Arch. Rat. Mech. Anal. 115, 15-59 (1991)
26. Simo, J.C., Posbergh, T.A., Marsden, J.E.: Stability of coupled rigid body and geometrically exact rods: Block diagonalization and the energy-momentum method. Phys. Repts. 193, 280-360 (1990)
27. Simo, J.C., Posbergh, T.A., Marsden, J.E.: Stability of relative equilibria II: Three dimensional elasticity. Arch. Rat. Mech. Anal. 115, 61-100 (1991)
28. Wolansky, G., Ghil, M.: Stability of quasi-geostrophic flows in periodic channels. Phys. Lett. A 202, 111-116 (1995)
29. Wolansky, G., Ghil, M.: An extension of Arnol'd's second stability theorem for the Euler equation. Physica D 94, 161-167 (1996)