Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line

Journal of Differential Equations

Volumn 185, Issue 1, 2002, Pages 25-53

  Goubet, Olivier ^a   Rosa, Ricardo M S ^b  

^a UNIVERSITÉ DE PICARDIE JULES VERNE   (France)

^b FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Asymptotic smoothing; Bourgain spaces; Dispersive regularization; Global attractor; Korteweg de vries equation; Noncompact system; Weak damping

Indexed keywords

EID: 0037058009     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.2001.4163     Document Type: Article

References (30)

1. F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990), 85-108.
2. J. M. Ball, A proof of the existence of global attractors for damped semilinear wave equations, unpublished (cited in [7]).
3. J. Bona and R. S. Smith, The initial value problem for Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 555-604.
4. J. Bona and B.-Y. Zhang, The initial-value problem for the forced Korteweg-de Vries equation, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 571-598.
5. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), 209-262.
6. J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differential Equations 74 (1988), 369-390.
7. J. M. Ghidaglia, A note on the strong convergence towards attractors of damped forced KdV equations, J. Differential Equations 110 (1994), 356-359.
8. J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Sém. Bourbaki 796 (1995), 163-187.
9. O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equation, Appl. Anal. 60 (1996), 99-119.
10. O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Continuous Dynamical Systems 6 (2000), 625-644.
11. D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, New York, 1981.
12. J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, Vol. 25, Amer. Math. Soc., Providence, RI. 1988.
13. A. Haraux, Two remarks on hyperbolic dissipative problems, in "Nonlinear Partial Differential Equations and their Applications" (H. Brezis and J. L. Lions, Eds.). College de France Seminar, Vol. VII, pp. 161-179, Pitman, London, 1985.
14. C. E. Kenig, G. Ponce, and L. Vega, Well posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347.
15. C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1-21.
16. C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573-603.
17. D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Philos. Mag. 39 (1895), 422-443.
18. O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Lezioni Lincei, Cambridge Univ. Press, Cambridge, UK, 1991.
19. P. Laurençot, Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line, Czechoslovak Math. J. 48(123) (1998), 85-94.
20. R. M. Miura, C. S. Gardner, and M. D. Kruskal, Kortweg-de Vries equation and generalizations II. existence of conservation laws and constants of motion, J. Math. Phys. 9 (1968), 1204-1209.
21. I. Moise and R. Rosa, On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation, Adv. Differential Equations 2 (1997), 257-296.
22. I. Moise, R. Rosa, and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity 11 (1998), 1369-1393.
23. E. Ott and R. N. Sudan, Damping of solitary waves, Phys. Fluids 13 (1970), 1432-1434.
24. R. Rosa, The global attractor for the 2D Navier-Stokes flow on some unbounded domains, Nonlinear Analysis TMA 32 (1998), 71-85.
25. 1(ℝ), Mat. Contemp. 19 (2000), 129-152.
26. J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976), 78-87.
27. R. Temam, Sur un probleme non-linéaire, J. Math. Pures Appl. 48 (1969), 159-172.
28. R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, Vol. 68 (2nd ed., 1997), Springer-Verlag, New York, 1988.
29. E. Titi and M. Ziane, in preparation.
30. X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its applications, Physica D 88 (1995), 167-175.