Spectral barriers and inertial manifolds for dissipative partial differential equations. (English) Zbl 0701.35024

The authors consider the dissipative partial differential equation (1) \(u_ t+A(u)+R(u)=0\), where A is the dissipative part which is usually a self-adjoint unbounded positive operator, in a Hilbert space H and the nonlinear inhomogeneous term R(u) is defined, the domain of A, and has some local Lipschitz property there.
An inertial manifold is an invariant manifold that is finite dimensional, Lipschitz, and attracts exponentially all trajectories. The authors introduce the notion of a spectral barrier for a nonlinear dissipative partial differential equation. Using this notion, the authors present a proof of existence of inertial manifolds that requires easily verifiable conditions, namely the existence of large enough spectral barriers.
Most equations of the type (1) possess absorbing sets, i.e., sets Y in which all the trajectories enter in finite time, never to leave again. Some of the proofs of existence of inertial manifolds assume that the nonlinear term R(u) is equal to zero outside Y. This procedure is called “Preparation”.
In section 3 the authors consider Kuramoto-Sivashinskij equation \[ (2)\quad u_ t+u_{xxxx}+u_{xx}+uu_ x=0, \] while in section 6 they consider reaction-diffusion systems \[ (3)\quad \partial u/\partial t- \Delta u+f(u)=0. \]
Reviewer: F.M.Ragab


35B40 Asymptotic behavior of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
Full Text: DOI


[1] Constantin, P., and Foias, C. (1985). Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractor for 2D Navier-Stokes equations.Commun. Pure Appl. Math. 38, 1-27. · Zbl 0582.35092
[2] Constantin, P., Foias, C., and Témam, R. (1985). Attractors representing turbulent flows.Mem. Am. Math. Soc. 314, 53.
[3] Constantin, P., Foias, C., Nicolaenko, B., and Témam, R. (1988).Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations (Applied Mathematical Sciences Series No. 70), Springer-Verlag, New York. · Zbl 0701.35024
[4] Constantin, P. (1989). A Construction of Inertial Manifolds. In B. Nicolaenko (ed.) Proceedings of AMS Conference, Boulder, Colorado, 1987.
[5] Conway, E., Hoff, D. and Smoller, J. (1978). Large time behavior of solutions of nonlinear reaction-diffusion equations.SIAM J. Appl. Math. 35, 1-16. · Zbl 0383.35035
[6] Doering, C. R., Gibbon, J. D., Holm, D. D., and Nicolaenko, B. (1988). Low dimensional behavior in the complex Ginzburg-Landau equation: Los Alamos report LA-UR-87-1546.J. Nonlinearity (in press).
[7] Foias, C., Nicolaenko, B., and Témam, R. (1986). Asymptotic study of an equation of G. I. Sivashinsky for two-dimensional turbulences of the Kolmogorov flow.Proc. Acad. Sci. Paris 303, 717-720. · Zbl 0628.35014
[8] Foias, C., Sell, G. R., and Témam, R. (1988)a. Inertial manifolds for nonlinear evolutionary equations.J. Differential Equations (in press). · Zbl 0643.58004
[9] Foias, C., Nicolaenko, B., Sell, G. R., and Témam, R. (1988)b. Inertial manifolds for the Kuramoto-Sivashinsky equation.J. Math. Pures Appl. (in press). · Zbl 0694.35028
[10] Ghidaglia, J. M. (1988). Dimensions of attractors for the Ginzburg-Landau equation.Physica D (in press). · Zbl 0705.35108
[11] Henry, D. (1983). Geometric theory of parabolic equations.Lect. Notes Math. 840.
[12] Mallet-Paret, J., and Sell, G. R. (1987). Inertial manifolds for reaction-diffusion equations in higher space dimensions. IMA preprint series, no. 331, University of Minnesota, Minneapolis.J. Amer. Math. Soc. 1, 805-866 (1988). · Zbl 0674.35049
[13] Mañé, R. (1977). Reduction of semilinear parabolic equations to finite dimensionalC 1 flows.Lect. Notes Math. 597, 361-378.
[14] Mora, X. (1983). Finite-dimensional attracting manifolds in reaction-diffusion equations.Contemp. Math. 17, 353-360 · Zbl 0525.35046
[15] Nicolaenko, B., Scheurer, B., and Témam, R. (1985). Some global dynamical properties of the Kuramoto-Sivashinsky equation: nonlinear stability and attractors.Physica D 16, 155-183. · Zbl 0592.35013
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.