×

Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. (English) Zbl 0692.35053

Summary: 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.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
37-XX Dynamical systems and ergodic theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Billotti, J. E., and LaSalle, J. P. (1971). Dissipative periodic processes.Bull. Am. Math. Soc. 77, 1082-1088. · Zbl 0274.34061
[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. · Zbl 0582.35092
[3] Constantin, P., and Foias, C. (1988).Navier-Stokes Equations, University of Chicago Press, Chicago. · Zbl 0687.35071
[4] Constantin, P., Foias, C., and Témam, R. (1985). Attractors representing turbulent flows.Mem. Am. Math. Soc. 314.
[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.
[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. · Zbl 0701.35024
[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.
[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.
[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. · Zbl 0694.35028
[10] Foias, C., Sell, G. R., and Témam, R. (1988c). Inertial manifolds for nonlinear evolutionary equations.J. Differential Equations 73, 309-353. · Zbl 0643.58004
[11] Hale, J. K. (1988).Asymptotic Behavior of Dissipative Systems (Mathematical Surveys and Monographs, vol. 25), American Mathematical Society, Providence. · Zbl 0642.58013
[12] Henry, D. B. (1981).Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics, no. 840), Springer-Verlag, New York. · Zbl 0456.35001
[13] Luskin, M., and Sell, G. R. (1989). Parabolic regularization and inertial manifolds. In preparation. · Zbl 0688.58035
[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. · Zbl 0674.35049
[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. · Zbl 0365.76027
[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. · Zbl 0592.35013
[17] Témam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.