A global attracting set for nonlocal Kuramoto-Sivashinsky equations arising in interfacial electrohydrodynamics

European Journal of Applied Mathematics

Volumn 17, Issue 6, 2006, Pages 677-703

  Tseluiko, Dmitri ^a   Papageorgiou, Demetrios T ^a  

^a NEW JERSEY INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33744902440     PISSN: 09567925     EISSN: 14694425     Source Type: Journal    
DOI: 10.1017/S0956792506006760     Document Type: Article

References (34)

1. ABDELOUHAB, A., BONA, J. L., FELLAND, M. & SAUT, J.-C. (1989) Nonlocal models for nonlinear dispersive waves. Phys. D, 40, 360-392.
2. BENNEY D. J. (1966) Long waves on liquid films. J. Math. Phys. 45, 150-155.
3. COHEN, B. I., KROMMES, J. A., TANG, W. M. & AND ROSENBLUTH, M. N. (1976) Non-linear saturation of the dissipative trapped ion mode by mode coupling. Nucl. Fusion, 16, 971-992.
4. COLLET, P., ECKMANN, J.-P., EPSTEIN, H. & STUBBE, J. (1993) Analyticity for the Kuramoto-Sivashinsky equation. Phys. D, 67, 321-326.
5. COLLET, P., ECKMANN, J.-P., EPSTEIN, H. & STUBBE, J. (1993) A global attracting set for the Kuramoto-Sivashinsky equation. Commun. Math. Phys. 152, 203-214.
6. COWARD, A. V., PAPAGEORGIOU, D. T. & SMYRLIS, Y. S. (1995) Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients. Zeit. Angew. Math. Phys. (ZAMP), 46, 1-39.
7. DUAN, J. & ERVIN, V. J. (1998) Dynamics of a nonlocal Kuramoto-Sivashinsky equation. J. Diff. Eq. 143, 243-266.
8. FRISCH, U., SHE, Z. S. & THUAL, O. (1986) Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech. 186, 221-240.
9. GONZALEZ, A. & CASTELLANOS, A. (1996) Nonlinear electrohydrodynamic waves on films falling down an inclined plane. Phys. Rev. E, 53(4), 3573-3578.
10. GOODMAN, J. (194) Stability of the Kuramoto-Sivashinsky and related systems. Commun. Pure Appl. Math. 47, 293-306.
11. GREENE, J. M. & KIM, J. S. (1988) The steady states of the Kuramoto-Sivashinsky equation. Phys. D, 33, 99-120.
12. HENRY, D. (1981) Geometric Theory of Semilinear Parabolic Equations, volume 840 of Lecture notes in mathematics. Springer-Verlag.
13. HOOPER, A. P. & GRIMSHAW, R. (1985) Nonlinear instability at the interface between two fluids. Phys. Fluids, 28, 37-45.
14. HYMAN, J. M. & NIKOLAENKO, B. (1986) The Kuramoto-Sivashinsky equations, a bridge between PDEs and dynamical systems. Phys. D, 18, 113-126.
15. HYMAN, J. M., NIKOLAENKO, B. & ZALESKI, S. (1986) Order and complexity in the Kuramoto-Sivashinsky model of turbulent interfaces. Phys. D, 23, 265-292.
16. IL'YASHENKO, J. S. (1992) Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation. J. Dyn. Diff. Equations, 4(4), 585-615.
17. JOLLY, M. S., ROSA, R. & TEMAM, R. (2000) Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation. Adv. Differ. Equ. 5, 31-66.
18. KEVREKIDIS, I. G., NICOLAENKO, B. & SCOVEL, C. (1990) Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation. SIAM J. Appl. Math. 50(3), 760-790.
19. KURAMOTO, Y. (1978) Diffusion-induced chaos in reaction systems. Prog. Theoret. Phys. Suppl. 64, 346-367.
20. KURAMOTO, Y. & TSUZUKI, T. (1975) On the formation of dissipative structures in reaction diffusion systems. Prog. Theoret. Phys. 54, 687-699.
21. KURAMOTO, Y. & TSUZUKI, T. (1976) Persistent propogation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theoret. Phys. 55, 356-369.
22. NICOLAENKO, B., SCHEURER, B. & TEMAM, R. (1985) Some global dynamical properties of the Kuramoto-Sivashinsky equation: Nonlinear stability and attractors. Phys. D, 16(2), 155-183.
23. PAPAGEORGIOU, D. T., MALDARELLI, C. & RUMSCHITZKI, D. S. (1990) Nonlinear interfacial stability of core-annular film flow. Phys. Fluids A, 2(3), 340-352.
24. PAPAGEORGIOU, D. T. & SMYRLIS, Y. S. (1991) The route to chaos for the Kuramoto-Sivashinsky equation. Theoret. Comput. Fluid Dynamics, 3, 15-42.
25. SELL, G. R. & YOU, S. (2002) Dynamics of Evolutionary Equations, volume 143 of Applied Mathematical Sciences. Springer.
26. SHLANG, T. & SIVASHINSKY, G. I. (1982) Irregular flow of a liquid film down a vertical column. J. Phys. 43, 459-466.
27. SIVASHINSKY, G. I. (1977) Nonlinear analysis of hydrodynamic instability in laminar flames, part 1. Acta Astronautica, 4, 1176-1206.
28. SIVASHINSKY, G. I. (1983) Instabilities, pattern formation, and turbulence in flames. Ann. Rev. Fluid Mech. 15, 179-199.
29. SIVASHINSKY, G. I. & MICHELSON, D. M. (1980) On irregular wavy flow on liquid film down a vertical plane. Prog. Theor. Phys. 63(6), 2112-2114.
30. SMYRLIS, Y. S. & PAPAGEORGIOU, D. T. (1991) Predicting chaos for the infinite dimensional dynamical systems: The kuramoto-sivashinsky equation, a case study. Proc. Natl. Acad. Sci. USA, 88(24), 11129-11132.
31. SMYRLIS, Y. S. & PAPAGEORGIOU, D. T. (1996) Computational study of chaotic and ordered solutions of the Kuramoto-Sivashinsky equation. ICASE Report No. 96-12, pp. 1-32.
32. TEMAM, R. (1988) Infinite-dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag.
33. TILLEY, B. S., DAVIS, S. H. & BANKOFF, S. G. (1994) Nonlinear long-wave stability of superposed fluids in an inclined chanel. J. Fluid Mech. 227, 55-83.
34. TSELUIKO, D. & PAPAGEORGIOU, D. T. (2005) Wave evolution on electrified falling films. J. Fluid Mech. (to appear).