×

Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. (English) Zbl 0668.35084

The Korteweg-de Vries equation was originally derived as a model for unidirectional waves of small amplitude, occurring in a large variety of physical situations in which nonlinearity and dispersion are important and have comparable effect. In many real problems, one cannot neglect energy dissipation mechanisms and external excitation. In this paper, the equation \[ u_ t+uu_ x+u_{xxx}+\gamma u=f \] is considered to possess space-periodic solutions in the case where the external excitation f is either time-independent or time-periodic. The paper is organized as follows. The second section deals with the construction of the universal attractor for a nonlinear group, based on time-uniform estimates that follow from the use of the first three classical polynomial multiplies. The third section contains a result on the finite dimension of the universal attractor. Finally, an Appendix is devoted to an abstract result on the evolution of Gram determinants.
Reviewer: L.Y.Shih

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076
[3] Bona, J. L.; Smith, R., The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278, 555-604 (1975) · Zbl 0306.35027
[4] Bourbaki, N., Espaces vectoriels topologiques (1981), Masson: Masson Paris · Zbl 0482.46001
[5] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Mem. Amer. Math. Soc., 53, No. 314 (1985) · Zbl 0567.35070
[6] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, Vol. I (1966), Interscience: Interscience New York) · Zbl 0729.00007
[7] Ghidaglia, J. M., C.R. Acad. Sci. Paris Sér. I Math., 305, 291-294 (1987), see also · Zbl 0638.35020
[9] Ghidaglia, J. M.; Temam, R., Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., 66, 273-319 (1987) · Zbl 0572.35071
[11] Hurewitz, W.; Wallman, H., Dimension Theory (1941), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[12] Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary wave, Phil. Mag., 39, 422-443 (1895)
[13] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[14] Lions, J. L.; Magenes, E., Nonhomogeneous Boundary Value Problems and Applications (1972), Springer-Verlag: Springer-Verlag Berlin/New York, [translated from Dunod, Paris, 1968] · Zbl 0223.35039
[15] Mandelbrot, B., Fractals: Form, Chance and Dimensions (1977), Freeman: Freeman San Francisco · Zbl 0376.28020
[16] Miles, J. W., The Korteweg-de Vries equation: A historical essay, J. Fluid Mech., 106, 131-147 (1981) · Zbl 0468.76003
[17] Ott, E.; Sudan, R. N., Damping of solitary waves, Phys. Fluids, 13, 1432-1434 (1970)
[18] Saut, J. C., Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl., 58, 21-61 (1979) · Zbl 0449.35083
[19] Temam, R., Sur un problème non linéaire, J. Math. Pures Appl., 48, 159-172 (1969) · Zbl 0187.03902
[20] Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics (1988), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0662.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.