Journal of Dynamics and Differential Equations

, Volume 1, Issue 2, pp 199–244

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

Authors

  • Ciprian Foias
    • Department of MathematicsIndiana University
  • George R. Sell
    • Institute for Mathematics and Its ApplicationsUniversity of Minnesota
  • Edriss S. Titi
    • Department of MathematicsUniversity of Chicago
Article

DOI: 10.1007/BF01047831

Cite this article as:
Foias, C., Sell, G.R. & Titi, E.S. J Dyn Diff Equat (1989) 1: 199. doi:10.1007/BF01047831

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

Dissipationexponential trackinginertial manifoldsnonlinear equations
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Copyright information

© Plenum Publishing Corporation 1989