Advertisement

Journal of Dynamics and Differential Equations

, Volume 31, Issue 2, pp 1029–1039 | Cite as

Determining Nodes for the Damped Forced Periodic Korteweg-de Vries Equation

  • Olivier GoubetEmail author
Article
  • 39 Downloads

Abstract

We show that solutions of the periodic KdV equations
$$\begin{aligned} u_t+\gamma u +u_{xxx}+uu_x=f, \end{aligned}$$
are asymptotically determined by their values at three points. That is if there exists \(x_1,x_2,x_3\) such that \(0< x_3-x_2<<x_3-x_1<<1\) and
$$\begin{aligned} \lim _{t\rightarrow +\infty } |u_1(t,x_j)-u_2(t,x_j)|=0, \; \mathrm{for} \; j=1,2,3, \end{aligned}$$
for two solutions \(u_1,u_2\) of the KdV equation above, then
$$\begin{aligned} \lim _{t\rightarrow +\infty }||u_1(t)-u_2(t)||_{H^1}=0. \end{aligned}$$

Keywords

Damped forced KdV equations Global attractor Determining nodes 

Mathematics Subject Classification

35Q53 37L50 35B41 

Notes

References

  1. 1.
    Ferrari, A., Titi, E.S.: Gevrey regularity of solutions of a class of analytic nonlinear parabolic equations. Commun. Partial Differ. Equ. 23, 1–16 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Foias, C., Kukavica, I.: Determining nodes for the Kuramoto–Sivashinsky equation. J. Dyn. Differ. Equ. 7(2), 365–373 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Foias, C., Temam, R.: Determination of the solutions of the Navier–Stokes equations by a set of nodal values. Math. Comput. 43, 117–133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Foias, C., Temam, R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ghidaglia, J.-M.: Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time. J. Differ. Equ. 74, 369–390 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ghidaglia, J.-M.: A note on the strong convergence towards attractors of damped forced KdV equations. J. Differ. Equ. 110, 356–359 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ginibre, J.: Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain). Séminaire Bourbaki, Vol. 1994/1995. Astérisque No. 237, Exp. No. 796, 4, 163–187 (1996)Google Scholar
  8. 8.
    Goubet, O.: Asymptotic smoothing effect for weakly damped forced KdV equations. Discrete Contin. Dyn. Syst. 6(3), 625–644 (2000)zbMATHGoogle Scholar
  9. 9.
    Goubet, O.: Analyticity of the global attractor for damped forced periodic Korteweg-de Vries equation (to appear)Google Scholar
  10. 10.
    Hale, J.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. AMS, Providence (1988)zbMATHGoogle Scholar
  11. 11.
    Hale, J., Raugel, G.: Regularity, determining modes and Galerkin methods. J. Math. Pures Appl. 82, 1075–1136 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kukavica, I.: On the number of determining nodes for the Ginzburg–Landau equation. Nonlinearity 5, 997–1006 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  14. 14.
    Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations: Evolutionary Partial Differential Equations, vol. 4, pp. 103–200. Elsevier, Amsterdam (2008)Google Scholar
  15. 15.
    Moise, I., Rosa, R.: On the regularity of the global attractor for a weakly damped forced KdV equation. Adv. Differ. Equ. 2(2), 257–296 (1997)zbMATHGoogle Scholar
  16. 16.
    Oliver, M., Titi, E.S.: Analyticity of the attractor and the number of determining nodes for a weakly damped driven NLS equation. Indiana Univ. Math. J. 47(1), 49–73 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Raugel, G.: Global Attractors in Partial Differential Equations. Handbook of Dynamical Systems, vol. 2, pp. 885–982. North-Holland, Amsterdam (2002)zbMATHGoogle Scholar
  18. 18.
    Takens, F.: Detecting strange attractors in turbulence. In: Raud, D.A., Young, L.-S. (eds.) Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, New York (1981)Google Scholar
  19. 19.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.LAMFA, UMR CNRS 7352Université de Picardie Jules VerneAmiens CedexFrance

Personalised recommendations