Structural evolution of the Taylor vortices

 Access to full text

 Full (PDF)

1. [1] R. Abraham and J. Marsden, Foundations of Mechanics, Addison-Wesley: Reading, MA (1978). Zbl0393.70001MR515141
2. [2] D. V. Anosov and V. Arnold, Dynamical Systems I, Springer-Verlag, New York, Heidelberg, Berlin (1985). Zbl0658.00008MR970793
3. [3] V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, Heidelberg, Berlin (1978). Zbl0386.70001MR690288
4. [4] Alain Bensoussan, Jacques-Louis Lions and Papanicolaou George, Asymptotic analysis for periodic structures, Ser. Studies in Mathematics and its Applications. 5; North-Holland Publishing Co., Amsterdam (1978) 700. Zbl0404.35001MR503330
5. [5] D. Chillingworth, Differential topology with a view to applications. Pitman, London, San Francisco, Melbourne. Research Notes in Mathematics, 9 (1976). Zbl0336.58001MR646088
6. [6] A. Chorin, Vorticity and Turbulence, Springer-Verlag (1994). Zbl0795.76002MR1281384
7. [7] P. Constantin and C. Foias, The Navier-Stokes Equations, Univ. of Chicago Press, Chicago (1988). Zbl0687.35071MR972259
8. [8] L. Caffarelli and R. Kohn and L. Nirenberg, On the regularity of the solutions of Navier-Stokes Equations. Comm. Pure Appl. Math. 35 (1982) 771-831. Zbl0509.35067MR673830
9. [9] Strebel, Kurt, Quadratic differentials, Springer-Verlag, Berlin (1984) 184. Zbl0547.30001MR743423
10. [10] A. Fathi, F. Laudenbach and V. Poénaru, Travaux de Thurston sur les surfaces. Asterisque 66-67 (1979). Zbl0446.57010MR568308
11. [11] A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 (1994) 333-408. Zbl0796.76084MR1265233
12. [12] H. Hopf, Abbildungsklassen in-dirnensionaler mannigfaltigkeiten. Math. Annalen 96 (1926) 225-250. Zbl52.0571.01JFM52.0571.01
13. [13] D. Gottlieb, Vector fields and classical theorems of topology. Rendiconti del Seminario Matematico e Fisico, Milano 60 (1990) 193-203. Zbl0810.57020MR1229491
14. [14] J. Milnor, Topology from the differentiable viewpoint. University Press of Virginia, based on notes by D.W. Weaver, Charlott-seville (1965). Zbl0136.20402MR226651
15. [15] J. Guckenheimer and P. J. Holmes, Nonlinear oscillations, dynamical Systems, and bifurcations of vector fields, Springer-Verlag, New York, Heidelberg, Berlin (1983). Zbl0515.34001MR709768
16. [16] J. K. Haie, Ordinary differential equations, Robert E. Krieger Publishing Company, Malabar, Florida (1969). Zbl0433.34003
17. [17] M. W. Hirsch, Differential topology, Springer-Verlag, New York, Heidelberg, Berlin (1976). Zbl0356.57001MR448362
18. [18] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris (1969). Zbl0189.40603MR259693
19. [19] A. Katok and B. Hasselblatt, Introduction to the Modem Theory of Dynamical Systems, Cambridge University Press (1995). Zbl0878.58020MR1326374
20. [20] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que posent l'hydrodynamique. J. Math. Pures et Appl. XII (1933) 1-82. Zbl0006.16702
21. [21] J. L. Lions, R. Temam and S. Wang, New formulations of the primitive equations of the atmosphere and applications. Nonlinearity 5 (1992) 237-288. Zbl0746.76019MR1158375
22. [22] J. L. Lions, R. Temam and S. Wang, On the Equations of Large-Scale Ocean. Nonlinearity 5 (1992) 1007-1053. Zbl0766.35039MR1187737
23. [23] J. L. Lions, R. Temam and S. Wang, Models of the coupled atmosphere and ocean (CAO I). Computational Mechanics Advance, 1 (1993) 3-54. Zbl0805.76011MR1252502
24. [24] J. L. Lions, R. Temam and S. Wang, Geostrophic Asymptotics of the Primitive Equations of the Atmosphere. Topological Methods in Nonlinear Analysis 4; note "Special issue dedicated to J. Leray" (1994) 253-287. Zbl0846.35106MR1350974
25. [25] J. L. Lions, R. Temam and S. Wang, Mathematical study of the coupled models of atmosphere and ocean (CAO III). J. Math. Pures Appl. 73 (1995) 105-163. Zbl0866.76025MR1325825
26. [26] J. L. Lions, R. Temam and S. Wang, A Simple Global Model for the General Circulation of the Atmosphere, "Dedicated to Peter D. Lax and Louis Nirenberg on the occasion of their 70th birthdays". Comm. Pure. Appl. Math. 50 (1997) 707-752. Zbl0992.86001MR1454171
27. [27] P. L. Lions, Mathematical Topics in Fluid Mechanics, Oxford science Publications (1996). Zbl0866.76002
28. [28] A. Majda, Vorticity and the mathematical theory of incompressible fluid flow. Frontiers of the mathematical sciences: 1985 (New York). Comm. Pure Appl. Math. 39 (1986) S187-S220. Zbl0595.76021MR861488
29. [29] T. Ma and S. Wang, Dynamics of Incompressible Vector Fields. Appl. Math. Lett. 12 (1999) 39-42. Zbl0989.37012MR1750594
30. [30] T. Ma and S. Wang, Dynamics of 2-D Incompressible Flows. Proceedings of the International Conferences on Differential Equations and Computation (1999). Zbl0957.37046MR1774477
31. [31] T. Ma and S. Wang, The Geometry of the Stream Lines of Steady States of the Navier-Stokes Equations. Contemporary Mathematics, AMS 238 (1999) 193-202. Zbl0947.35110MR1724664
32. [32] T. Ma and S. Wang, Block structure and stability of 2-D Incompressible Flows (in preparation, 1999). 
33. [33] T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields. Nonlinearity (revised, 1999). Zbl1012.57042
34. [34] A. Majda, The interaction of nonlinear analysis and modern applied mathematics. Proc. Internat. Congress Math., Kyoto, 1990, Springer-Verlag, New York, Heidelberg, Berlin (1991) Vol. 1. Zbl1153.76300MR1127159
35. [35] N. Markley, The Poincaré-Bendixson theorem for Klein bottle. Trans. AMS 135 (1969). Zbl0175.50101MR234442
36. [36] L. Markus and R. Meyer, Generic Hamiltoman Systems are neither integrable nor ergodic. Memoirs of the American Mathematical Society 144 (1974). Zbl0291.58009
37. [37] J. Moser, Stable and Random Motions in Dynamical Systems. Ann. Math. Stud. No. 77 Princeton (1973). Zbl0271.70009MR442980
38. [38] J. Palis and W. de Melo, Geometric theory of dynamical Systems, Springer-Verlag, New York, Heidelberg, Berlin (1982). Zbl0491.58001MR669541
39. [39] J. Palis and S. Smale, Structural stability theorem. Global Analysis. Proc. Symp. in Pure Math. XIV (1970). Zbl0214.50702MR267603
40. [40] M. Peixoto, Structural stability on two dimensional manifolds. Topology 1 (1962) 101-120. Zbl0107.07103MR142859
41. [41] C. Pugh, The closing lemma. Amer. J. Math. 89 (1967) 956-1009. Zbl0167.21803MR226669
42. [42] Shub, Michael, Stabilité globale des systèmes dynamiques. Société Mathématique de France. Note With an English preface and summary Astérisque 56 (1978) iv+211. Zbl0396.58014MR513592
43. [43] C. Robinson, Generic properties of conservative systems, I, II. Amer. J. Math. 92 (1970) 562-603 and 897-906. Zbl0212.56601MR273640
44. [44] C. Robinson, Structure stability of vector fields. Ann. of Math. 99 (1974) 154-175. Zbl0275.58012MR334283
45. [45] C. Robinson, Structure stability of C1 diffeomorphisms. J. Differential Equations 22 (1976) 28-73. Zbl0343.58009MR474411
46. [46] G. Schwartz, Hodge decomposition-A method for solving boundary value problems. Lecture Notes in Mathematics 1607 Springer-Verlag (1995). Zbl0828.58002MR1367287
47. [47] S. Smale, Differential dynamical systems. Bull. AMS 73 (1967) 747-817. Zbl0202.55202MR228014
48. [48] F. Takens, Hamiltonian systems: generic properties of closed orbits and local perturbations. Math. Ann. 188 (1970) 304-312. Zbl0191.21602MR284670
49. [49] G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. A 223 (1923) 289-343. Zbl49.0607.01JFM49.0607.01
50. [50] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd edition, North Holland, Amsterdam (1984). Zbl0568.35002MR769654
51. [51] R. Thom, Structural Stability and Morphogenesis, Benjamin-Addison Wesley (1975). Zbl0303.92002MR488156
52. [52] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. AMS 19 (1988) 417-431. Zbl0674.57008MR956596
53. [53] V. Trofimov, Introduction to Geometry on Manifolds with Symmetry, MIA Kluwer Academic Publishers (1994). Zbl0804.53002MR1367288
54. [54] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, Heidelberg, Berlin (1990). Zbl0701.58001MR1056699
55. [55] J. C. Yoccoz, Recent developments in dynamics, in Proc. Internat. Congress. Math., Zurich (1994), Birkhauser Verlag, Basel, Boston, Berlin (1994) 246-265 Vol. 1. Zbl0844.58001MR1403926