Attracting curves on families of stationary solutions in two-dimensional Navier-Stokes and reduced magnetohydro dynamics

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 454, Issue 1973, 1998, Pages 1407-1444

  Berks, Gianne ^a   Ratiu, Tudor ^b,c  

^a UNIVERSITY OF SURREY   (United Kingdom)

^b ISAAC NEWTON INSTITUTE FOR MATHEMATICAL SCIENCES   (United Kingdom)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Attractors; Euler's equations; Magnetohydrodynamics; Navier stokes; Shadowing; Stability

Indexed keywords

EID: 0242313024     PISSN: 13645021     EISSN: None     Source Type: Journal    
DOI: 10.1098/rspa.1998.0214     Document Type: Article

References (27)

1. Arnold, V. I. 1972 Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field. J. Appl. Math. Mech. 36, 236-242.
2. Aubin, T. 1982 Nonlinear analysis on manifolds. Mange-Ampere equations. Grundlehren der mathematischen Wissenschaften, vol. 252. New York: Springer.
3. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. & Ratiu, T. S. 1994 Dissipation induced instabilities. Ann. Inst. H. Poincaré, A. Nonlin. 11, 37-90.
4. Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. & Ratiu, T. S. 1996 The Euler-Poincaré equations and Lie-Poisson dissipation. Commun. Math. Phys. 175, 1-42.
5. Burton, G. R. 1987 Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276, 225-253.
6. Burton, G. R. & McLeod, J. B. 1991 Maximisation and minimisation on classes of rearrangements. Proc. R. Soc. Edinb. A 119, 287-300.
7. Chern, S. J. & Marsden, J. E. 1990 A note on symmetry and stability for fluid flows. Astr. Fluid. Dyn. 51, 1-4.
8. Constantin, P., Foias, C. & Temam, R. 1988 On the dimension of the attractors in two-dimensional turbulence. Physica D 30, 284-296.
9. Derks, G., Lewis, D. & Ratiu, T. 1995 Approximations with curves of relative equilibria in Hamiltonian systems with dissipation. Nonlinearity 8, 1087-1114.
10. Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353-391.
11. Foias, C. & Saut. J. C. 1984 Asymptotic behavior, as t → ∞ of solutions of Navier-Stokes equation and nonlinear spectral manifolds. Indiana Univ. Math. Jl 33, 459-477.
12. Foias, C., Sell, G. & Titi, E. S. 1989 Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dynamics Differ. Equat. 1, 199-244.
13. Friedlander, S., Gilbert, A. D. & Vishik, M. 1993 Hydrodynamic instability for certain ABC flows. Geophys. Astrophys. Fluid Dynam. 73, 97-107.
14. Galloway, D. & Frisch, U. 1987 A note on the stability of a family of space-periodic Beltrami flows. J. Fluid Mech. 180, 557-564.
15. Ghidaglia, J. M. 1986 Long time behavior of solutions of abstract inequalities: applications to thermo-hydraulic and magnetohydrodynamic equations. J. Differ. Equat. 61, 268-294.
16. Hasegawa, A. 1985 Self-organization process in continuous media. Adv. Phys. 34, 1-42.
17. Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1-116.
18. Kato, T. 1967 On classical solutions of the two-dimensional non-stationary Euler equation. Arch. Ration Mech. Analysis 25, 188-200.
19. Kozono, H. 1989 Weak and classical solutions of the two-dimensional magnetohydrodynamic equations. Tohoku Math. J/41, 471-488.
20. Marchioro, C. 1986 An example of absence of turbulence for any Reynolds number. Commun. Math. Phys. 105, 99-106.
21. Marsden, J. E. & Morrison, P. J. 1984 Noncanonical Hamiltonian field theory and reduced MHD. Contemp. Math. 28, 133-150.
22. Morrison, P. J. & Eliezer, S. 1986 Spontaneous symmetry breaking and neutral stability in the noncanonical Hamiltonian formalism. Phys. Rev. A 33, 4205-4213.
23. Morrison, P. J. & Hazeltine, R. D. 1984 Hamiltonian formulation of reduced magnetohydrodynamics. Physics Fluids 27, 886-897.
24. Simö, J. C., Lewis, D. K. & Marsden, J. E. 1991 Stability of relative equilibria. I. the reduced energy momentum method. Arch. Ration Mech. Analysis 115, 15-60.
25. Sermange, M. & Temam, R. 1983 Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635-664.
26. Temam, R. 1988 Infinite dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68. New York: Springer.
27. van Groesen, E. 1988 Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations. Physica A148, 312-330.