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.
Similar content being viewed by others
References
Billotti, J. E., and LaSalle, J. P. (1971). Dissipative periodic processes.Bull. Am. Math. Soc. 77, 1082–1088.
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.
Constantin, P., and Foias, C. (1988).Navier-Stokes Equations, University of Chicago Press, Chicago.
Constantin, P., Foias, C., and Témam, R. (1985). Attractors representing turbulent flows.Mem. Am. Math. Soc. 314.
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.
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.
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.
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.
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.
Foias, C., Sell, G. R., and Témam, R. (1988c). Inertial manifolds for nonlinear evolutionary equations.J. Differential Equations 73, 309–353.
Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs, vol. 25), American Mathematical Society, Providence.
Henry, D. B. (1981).Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics, no. 840), Springer-Verlag, New York.
Luskin, M., and Sell, G. R. (1989). Parabolic regularization and inertial manifolds. In preparation.
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.
Minea, Gh. (1976). Remarques sur l'unicité de la stationnaire d'une équation de type NavierStokes,Rev. Roum. Math. Pures Appl. 21, 1071–1075.
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.
Témam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Foias, C., Sell, G.R. & Titi, E.S. Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J Dyn Diff Equat 1, 199–244 (1989). https://doi.org/10.1007/BF01047831
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01047831