The WKB method for conjugate points in the volumorphism group

Indiana University Mathematics Journal

Volumn 57, Issue 7, 2008, Pages 3303-3327

  Preston, Stephen C ^a  

^a UNIVERSITY OF COLORADO   (United States)

Author keywords

Conjugate point; Diffeomorphism; Epiconjugate; Fredholm; Ideal fluid; Monoconjugate; Volume preserving; Volumorphism; WKB

Indexed keywords

EID: 62649158334     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2008.57.3413     Document Type: Article

References (18)

1. V.I. ARNOLD, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), 319-361. MR 0202082 (34 #1956) (French)
2. V.I. ARNOLD and B.A. KHESIN, Topolagical Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998, ISBN 0-387-94947-X. MR 1612569 (99b:58002)
3. B.J. BAYLY, Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows, Phys. Fluids 31 (1988), 56-64, http://dx.doi.org/10.1063/1.867002.
4. L. BILIOTTI, R. EXEL, P. PICCIONE, and D.V. TAUSK, On the singularities of the exponential map in infinite dimensional Riemannian manifolds, Math. Ann. 336 (2006), 247-267, http://dx.doi.org/10.1007/s00208-006-0001-2. MR 2244372 (2007e:58007)
5. D.G. EBIN, G. MISIOLEK, and S.C. PRESTON, Singularities of the exponential map on the volume-preserving diffeomorphism group, Geom. Funct. Anal. 16 (2006), 850-868. MR 2255383 (2008j:58009)
6. D.G. EBIN and J. MARSDEN, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102-163, http://dx.doi.org/10.2307/1970699. MR 0271984 (42 #6865)
7. N. GROSSMAN, Hilbert manifolds without epiconjugatepoints, Proc. Amer. Math. Soc. 16 (1965), 1365-1371, http://dx.doi.org/10. 2307/2035933. MR 0188943 (32 #6370)
8. B. KHESIN, Topological fluid dynamics, Notices Amer. Math. Soc. 52 (2005), 9-19. MR 2105567 (2005h:37206)
9. A. LIFSCHITZ and E. HAMEIRI, Local stability conditions in fluid dynamics, Phys. Fluids A 3 (1991), 2644-2651, http://dx.doi.org/10.1063/1.858153. MR 1137216 (92j:76041)
10. G. MISIOLEK, Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J. 42 (1993), 215-235, http://dx.doi.org/10.1512/iumj.1993.42.42011. MR 1218713 (94j:58027)
11. 2), Proc. Amer. Math. Soc. 124 (1996), 977-982, http://dx.doi.org/10.1090/S0002- 9939-96-03149-8. MR 1301038 (96f:58009)
12. S.C. PRESTON, For ideal fluids, Eulerian and Lagrangian instabilities are equivalent, Geom. Funct. Anal. 14 (2004), 1044-1062, http://dx.doi.org/10.1007/s00039-004-0482-7. MR 2105952 (2005f:7165)
13. -, On the volumorphism group, the first conjugate point is always the hardest, Comm. Math. Phys. 267 (2006), 493-513, http://dx.doi.org/10. 1007/s00220-006-0070-9. MR 2249777 (2007f:7127)
14. W.T. REID, Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences, vol. 31, Springer-Verlag, New York, 1980, ISBN 0-387-90542-1, With a preface by John Burns. MR 606199 (82f:4002)
15. P. ROUCHON, The Jacobi equation, Riemannian curvature and the motion of a perfect incompressible fluid, European J. Mech. B Fluids 11 (1992), 317-336. MR 1168147 (93a:76047)
16. M.E. TAYLOR, Partial Differential Equations. I. Basic Theory. Basic Theory, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996, ISBN 0-387-94653-5. MR 1395148 (98b:35002b)
17. M.M. VISHIK and S. FRIEDLANDER, Dynamo theory methods for hydrodynamic stability, J. Math. Pures Appl. (9) 72 (1993), 145-180. MR 1216094 (95a:76036)
18. W. WOLIBNER, Un theorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Madi. Z. 37 (1933), 698-726, http://dx.doi.org/10. 1007/BF01474610. MR 1545430 (French)