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Asymptotic analysis of the equations of motion for viscoelastic oldroyd fluid
Closed trajectories on symmetric convex Hamiltonian energy surfaces
1. | Key Laboratory of Pure and Applied Mathematics, School of Mathematical Science, Peking University, Beijing, 100871, China |
References:
[1] |
A. Borel, "Seminar on Transformation Groups," Annals of Mathematics Studies, No. 46, Princeton Univ. Press, Princeton, 1960. |
[2] |
I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 1 (1984), 19-78. |
[3] |
I. Ekeland, "Convexity Methods in Hamiltonian Mechanics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19, Springer-Verlag, Berlin, 1990. |
[4] |
I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[5] |
I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 4 (1987), 307-335. |
[6] |
E. Fadell and P. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[7] |
H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.
doi: 10.2307/120994. |
[8] |
Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149. |
[9] |
Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[10] |
Y. Long, "Index Theory for Symplectic Paths with Applications," Progress in Math., 207, Birkhäuser Verlag, Basel, 2002. |
[11] |
C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[12] |
Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[13] |
P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[14] |
A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197. |
[15] |
C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.
doi: 10.1090/S0002-9947-1989-0978370-5. |
[16] |
W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[17] |
W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$, J. Eur. Math. Soc., 11 (2009), 575-596.
doi: 10.4171/JEMS/161. |
[18] |
W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$, J. Diff. Equa., 246 (2009), 4322-4331.
doi: 10.1016/j.jde.2008.10.003. |
[19] |
A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
show all references
References:
[1] |
A. Borel, "Seminar on Transformation Groups," Annals of Mathematics Studies, No. 46, Princeton Univ. Press, Princeton, 1960. |
[2] |
I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 1 (1984), 19-78. |
[3] |
I. Ekeland, "Convexity Methods in Hamiltonian Mechanics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 19, Springer-Verlag, Berlin, 1990. |
[4] |
I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[5] |
I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltoniens convexes, Ann. IHP. Anal. non Linéaire, 4 (1987), 307-335. |
[6] |
E. Fadell and P. Rabinowitz, Generalized comological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[7] |
H. Hofer, K. Wysocki, and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math., 148 (1998), 197-289.
doi: 10.2307/120994. |
[8] |
Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math., 187 (1999), 113-149. |
[9] |
Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics, Advances in Math., 154 (2000), 76-131.
doi: 10.1006/aima.2000.1914. |
[10] |
Y. Long, "Index Theory for Symplectic Paths with Applications," Progress in Math., 207, Birkhäuser Verlag, Basel, 2002. |
[11] |
C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $\mathbfR^{2n}$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[12] |
Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $\mathbfR^{2n}$, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[13] |
P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[14] |
A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France, 116 (1988), 171-197. |
[15] |
C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math. Soc., 311 (1989), 621-655.
doi: 10.1090/S0002-9947-1989-0978370-5. |
[16] |
W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[17] |
W. Wang, Stability of closed characteristics on compact convex hypersurfaces in $R^6$, J. Eur. Math. Soc., 11 (2009), 575-596.
doi: 10.4171/JEMS/161. |
[18] |
W. Wang, Symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$, J. Diff. Equa., 246 (2009), 4322-4331.
doi: 10.1016/j.jde.2008.10.003. |
[19] |
A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
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