Inertial manifolds for nonlinear evolutionary equations. (English) Zbl 0643.58004

The authors introduce the concept of an inertial manifold for nonlinear evolutionary equations, including ordinary and partial differential equations. These finite dimensional Lipschitz-manifolds turn out to be appropriate to study the long-time behaviour of solutions of the evolutionary equations. In particular they contain the global attractor, attract exponentially all solutions and they are stable with respect to certain perturbations.
In the infinite dimensional case they allow the reduction of the dynamics to a finite dimensional ordinary differential equation.
Reviewer: N.Jacob


37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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[1] Babin, A.V; Vishik, M.I, Regular attractors of semigroups and evolution equations, J. math. pures appl., 62, 441-491, (1983) · Zbl 0565.47045
[2] Ball, J.M, Saddle point analysis…, (), 93-160
[3] Billotti, J.E; LaSalle, J.P, Dissipative periodic processes, Bull. amer. math. soc., 77, 1082-1088, (1971) · Zbl 0274.34061
[4] Carr, J, Applications of centre manifold theory, () · Zbl 0464.58001
[5] Chafee, N, The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. differential equations, 4, 661-679, (1968) · Zbl 0169.11301
[6] Conway, E; Hoff, D; Smoller, J, Large time behavior of solutions of non-linear reaction-diffusion equations, SIAM J. appl. math., 35, No. 11, 1-16, (1978) · Zbl 0383.35035
[7] Constantin, P; Foias, C; Manley, O.P; Temam, R, Connexion entre la théorie mathématique des équations de Navier-Stokes et le théorie conventionelle de la turbulence, C. R. acad. sci. ser. I, 297, 599-602, (1983) · Zbl 0531.76066
[8] Constantin, P; Foias, C; Nicolaenko, B; Temam, R, Integral manifolds and inertial manifolds for dissipative partial differential equations, () · Zbl 0683.58002
[9] Constantin, P; Foias, C; Temam, R, Attractors representing turbulent flows, Memoirs amer. math. soc., 314, (1985) · Zbl 0567.35070
[10] Coppel, W.A, Stability and asymptotic behavior of differential equations, (1965), Heath Boston · Zbl 0154.09301
[11] Foias, C; Manley, O; Temam, R, Attractors for the benard problem: existence and physical bounds of their fractal dimension, Nonlinear analysis: theory, methods and applications, 11, 939-967, (1987) · Zbl 0646.76098
[12] Foias, C; Nicolaenko, B; Sell, G.R; Temam, R, Variétés inertielles pour l’équation de Kuramoto-Sivashinsky, C. R. acad. sci. Paris ser. 1, 301, 285-288, (1985) · Zbl 0591.35063
[13] {\scC. Foias, B. Nicolaenko, G. R. Sell, and R. Temam}, Inertial manifold for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., to appear. · Zbl 0694.35028
[14] Foias, C; Sell, G.R; Temam, R, Variétés inertielles des équations differentielles dissipatives, C. R. acad. sci. Paris ser. 1, 301, 139-141, (1985) · Zbl 0591.35062
[15] Foias, C; Temam, R, Some analytic and geometric properties of the solutions of the Navier-Stokes equations, J. math. pures appl., 58, 339-368, (1979) · Zbl 0454.35073
[16] Hale, J.K, Ordinary differential equations, (1968), Wiley New York · Zbl 0191.15504
[17] Hale, J.K; Magalhães, L.T; Oliva, W.M, An introduction to infinite dimensional dynamical systems—geometric theory, () · Zbl 0533.58001
[18] Hardy, G.H; Wright, E.M, An introduction to the theory of numbers, (1962), Oxford Univ. Press London/New York · Zbl 0020.29201
[19] Hartman, P, Ordinary differential equations, (1964), Wiley New York · Zbl 0125.32102
[20] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[21] Hirsch, M; Pugh, C; Shub, M, Invariant manifolds, () · Zbl 0355.58009
[22] Kelley, A, The stable, center-stable, center, center-unstable, unstable manifolds, J. differential equations, 3, 546-570, (1967) · Zbl 0173.11001
[23] Kurzweil, J, Invariant manifolds I, Comm. math. univ. carolinae, 11, 336-390, (1970) · Zbl 0197.47702
[24] Lions, J.L, Quelques methodes de resolution des problemes aux limites nonlineaires, (1969), Dunod Paris · Zbl 0189.40603
[25] Mallet-Paret, J, Negatively invariant sets of compact maps and an extension of a theorem of cartwright, J. differential equations, 22, 331-348, (1976) · Zbl 0354.34072
[26] Mallet-Paret, J; Sell, G, Inertial manifolds for reaction diffusion equations in higher space dimension, () · Zbl 0674.35049
[27] Mañé, R, On the dimension of the compact invariant sets of certain nonlinear aps, (), 230-242
[28] Metivier, G, Valeurs propres d’opérateurs definis sur la restriction de systems variation-nels a des sous-espaces, J. math. pures appl., 57, 133-156, (1978) · Zbl 0328.35029
[29] Nicolaenko, B; Scheurer, B; Temam, R, Some global dynamical properties of the Kuramoto Sivashinsky equations: nonlinear stability and attractors, Physica, 16D, 155-183, (1985) · Zbl 0592.35013
[30] Pliss, V.A, A reduction principle in the theory of the stability of motion, Izv. akad. nauk SSSR mat. ser., 28, 1297-1324, (1964), [in Russian] · Zbl 0131.31505
[31] Sacker, R.J; Sell, G.R, The spectrum of an invariant submanifold, J. differential equations, 38, 135-160, (1980) · Zbl 0415.58015
[32] Sacker, R.J, On invariant surfaces and bifucation of periodic solutions of ordinary differential equations, (1964), New York University, IMM-NYU
[33] Richards, J, On the gaps between numbers which are the sum of two squares, Adv. math., 46, 1-2, (1982) · Zbl 0501.10047
[34] Sell, G.R, The structure of a flow in the vicinity of an almost periodic motion, J. differential equations, 27, 359-393, (1978) · Zbl 0382.34017
[35] Temam, R, Navier-Stokes equations and nonlinear functional analysis, () · Zbl 0833.35110
[36] Temam, R, Attractors for Navier-Stokes equations, (), 272-292 · Zbl 0572.35083
[37] Kamaev, D.A, Hopf’s conjecture for a class of chemical kinetics equations, J. soviet math., 25, 836-849, (1984) · Zbl 0531.35040
[38] Mañé, R, Reduction of semilinear parabolic equations to finite dimensional C1 flows, (), 361-378
[39] Mora, X, Finite dimensional attracting manifolds in reaction-diffusion equations, Contemporary math., 17, 353-360, (1983) · Zbl 0525.35046
[40] {\scX. Mora}, Finite dimensional attracting manifolds for damped semilinear wave equations, in /ldContributions to Nonlinear Partial Differential Equations, II,/rd to appear.
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