Journal of Dynamics and Differential Equations

, Volume 1, Issue 2, pp 199–244

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

  • Ciprian Foias
  • George R. Sell
  • Edriss S. Titi
Article

Abstract

In this paper, we study the long-time behavior of a class of nonlinear dissipative partial differential equations. By means of the Lyapunov-Perron method, we show that the equation has an inertial manifold, provided that certain gap condition in the spectrum of the linear part of the equation is satisfied. We verify that the constructed inertial manifold has the property of exponential tracking (i.e., stability with asymptotic phase, or asymptotic completeness), which makes it a faithful representative to the relevant long-time dynamics of the equation. The second feature of this paper is the introduction of a modified Galerkin approximation for analyzing the original PDE. In an illustrative example (which we believe to be typical), we show that this modified Galerkin approximation yields a smaller error than the standard Galerkin approximation.

key words

Dissipation exponential tracking inertial manifolds nonlinear equations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Billotti, J. E., and LaSalle, J. P. (1971). Dissipative periodic processes.Bull. Am. Math. Soc. 77, 1082–1088.Google Scholar
  2. Constantin, P., and Foias, C. (1985). Global Lyapunov exponents, Kaplan-York formulas and the dimension of the attractor for 2D Navier-Stokes equation.Commun. Pure Appl. Math. 38, 1–27.Google Scholar
  3. Constantin, P., and Foias, C. (1988).Navier-Stokes Equations, University of Chicago Press, Chicago.Google Scholar
  4. Constantin, P., Foias, C., and Témam, R. (1985). Attractors representing turbulent flows.Mem. Am. Math. Soc. 314.Google Scholar
  5. Constantin, P., Foias, C., Nicolaenko, B., and Témam, R. (1986). Nouveaux résultats sur les variétés inertiélles pour les équations différentielles dissipative.C. R. Acad. Sci. [I] 302, 375–378.Google Scholar
  6. Constantin, P., Foias, C., Nicolaenko, B., and Témam, R. (1988).Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations (Applied Mathematical Sciences, no. 70), Springer-Verlag, New York.Google Scholar
  7. Foias, C., Nicolaenko, B., Sell, G. R., and Témam, R. (1985). Variétés inertiélles pour l'équation de Kuramoto-Sivashinsky.C. R. Acad. Sci. [I] 301, 285–288.Google Scholar
  8. Foias, C., Jolly, M. S., Kevrekidis, I. G., Sell, G. R., and Titi, E. S. (1988a). On the computation of inertial manifolds.Phys. Lett. A 131, 433–436.Google Scholar
  9. Foias, C., Nicolaenko, B., Sell, G. R., and Témam, R. (1988b). Inertial manifolds for the Kuramoto Sivashinsky equation and an estimate of their lowest dimensions.J. Math. Pures Appl. 67, 197–226.Google Scholar
  10. Foias, C., Sell, G. R., and Témam, R. (1988c). Inertial manifolds for nonlinear evolutionary equations.J. Differential Equations 73, 309–353.Google Scholar
  11. Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs, vol. 25), American Mathematical Society, Providence.Google Scholar
  12. Henry, D. B. (1981).Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics, no. 840), Springer-Verlag, New York.Google Scholar
  13. Luskin, M., and Sell, G. R. (1989). Parabolic regularization and inertial manifolds. In preparation.Google Scholar
  14. Mallet-Paret, J., and Sell, G. R. (1988). Inertial manifolds for reaction diffusion equations in higher space dimensions.J. Am. Math. Soc. 1, 805–866.Google Scholar
  15. Minea, Gh. (1976). Remarques sur l'unicité de la stationnaire d'une équation de type NavierStokes,Rev. Roum. Math. Pures Appl. 21, 1071–1075.Google Scholar
  16. Nicolaenko, B., Scheurer, B., and Témam, R. (1985). Some global dynamical properties of the Kuramoto Sivashinsky equations: nonlinear stability and attractors.Physica D 16, 155–183.Google Scholar
  17. Témam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Ciprian Foias
    • 1
  • George R. Sell
    • 2
  • Edriss S. Titi
    • 3
  1. 1.Department of MathematicsIndiana UniversityBloomington
  2. 2.Institute for Mathematics and Its ApplicationsUniversity of MinnesotaMinneapolis
  3. 3.Department of MathematicsUniversity of ChicagoChicago

Personalised recommendations