Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations

Ciprian Foias, George R. Sell, Edriss S. Titi

Research output: Contribution to journalArticlepeer-review

175 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)199-244
Number of pages46
JournalJournal of Dynamics and Differential Equations
Volume1
Issue number2
DOIs
StatePublished - Apr 1 1989

Keywords

  • Dissipation
  • exponential tracking
  • inertial manifolds
  • nonlinear equations

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