Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

P. Constantin, C. Foias, B. Nicolaenko, Roger Temam

122 pages, parution le 01/01/1989

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Résumé

This work, the main results of which were announced in (CFNT), focuses on a new geometric explicit construction of inertial manifolds from integral manifolds generated by some initial dimensional surface. The method covers a large class of dissipative PDEs. The existence of a smooth integral manifold the closure of which in an inertial manifold M (i.E. containing X and uniformly exponentially attracting) requires a more detailed analysis of the geometric properties of the infinite dimensional flow. The method is explicity constructive, integrating forward in time and avoiding any fixed point theorems. The key geometric property upon which we base the construction of our integral inertial manifold M is a Spectral Blocking Property of the flow, which controls the evolution of the position of surface elements relative to the fixed reference frame associated to the linear principal part of the PDE.

Contents

Introduction.

Presentation of the Approach and of the Main Results.
The Transport of Finite Dimensional Contact Elements.
Spectral Blocking Property.
Strong Squeezing Property.
Cone Invariance Properties.
Consequences Regarding the Global Attractor.
Local Exponential Decay Toward Blocked Integral Surfaces.
Exponential Decay of Volume Elements and the Dimension of the Global Attractor.
Choice of the Initial Manifold.
Construction of the Inertial Mainfold.
Lower Bound for the Exponential Rate of Convergence to the Attractor.
Asymptotic Completeness : Preparation.
Asymptotic Completeness : Proof of Theorem 12.1.
Stability with Respect to Perturbations.
Application : The Kuramoto-Sivashinsky Equation.
Application : A Nonlocal Burgers Equation.
Application : The Cahn-Hilliard Equation.
Application : A parabolic Equation in Two Space Variables.
Application : The Chaffee-Infante Reaction Diffusion Equation.

References.
Index.

L'auteur - Roger Temam

Temam, R., University de Paris Sud, Orsay, France

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	P. Constantin, C. Foias, B. Nicolaenko, Roger Temam
Parution	01/01/1989
Nb. de pages	122
Format	16 x 24
Couverture	Relié
Poids	361g
Intérieur	Noir et Blanc
EAN13	9780387967295
ISBN13	978-3-540-96729-3