Inertial manifolds and slow manifolds
Roger Temam
Laboratoire d'Analyse Numerique, CNllS et Universite Paris-Sud
Batiment 425,91405 Orsay
FRANCE
INTRODUCTION
Our aim in this article is to present some receent results on the mathematical theory
of Inertial Manifolds and Slow Manifolds.
Inertial manifolds have been introduced as a tool to study the long time behavior of
infinite dimensional dynamical systems and to reduce them to finite dimensional systems
(see C. Foias, G.R. Sell and R. Temam (1985, 1988) and below). Slow manifolds are
a related concept introduced in meteorology for computational purposes. The study
of slow manifolds from the mathematical viewpoint has been initiated in A. Debussche
and R. Temam (1991a,b). In this article we present a self-contained study of inertial
manifolds and slow manifolds with applications to partial differential equations from
mathematical physics, including equations of Navier-Stokes type.
The article is organized as follows. Section 1 contains the description of the framework and the general hypotheses. Section 2 contains the statement and the proof of the
main results in the Lipschitz case (Theorem 2.1). Section 3 contains some complements
and applications. In particular we consider the locally Lipschitz case (Theorem 3.1) and
show how one can estimate the dimension of the inertial manifold in terms of physically
relevant quantities. Finally in Section 4 we give an application of the general results to
an example related to Slow Manifolds, a related concept appearing in meteorology.
CONTENT
Introduction
1. The Functional Setting
1.1. Notations and Hypotheses
1.2. Construction of the Inertial Manifold
2. The Main Result (Lipschitz case)
2.1. Existence of Inertial Manifolds
2.2. Properties of T
NATO ASI Series, Vol. 148
The Mathematics of Models for Climatology
and Environment
Edited by Jesus Ildefonso Dial.
© Springcr- Verlag Berlin Heidelberg 1997
Roger Temam
Laboratoire d'Analyse Numerique, CNllS et Universite Paris-Sud
Batiment 425,91405 Orsay
FRANCE
INTRODUCTION
Our aim in this article is to present some receent results on the mathematical theory
of Inertial Manifolds and Slow Manifolds.
Inertial manifolds have been introduced as a tool to study the long time behavior of
infinite dimensional dynamical systems and to reduce them to finite dimensional systems
(see C. Foias, G.R. Sell and R. Temam (1985, 1988) and below). Slow manifolds are
a related concept introduced in meteorology for computational purposes. The study
of slow manifolds from the mathematical viewpoint has been initiated in A. Debussche
and R. Temam (1991a,b). In this article we present a self-contained study of inertial
manifolds and slow manifolds with applications to partial differential equations from
mathematical physics, including equations of Navier-Stokes type.
The article is organized as follows. Section 1 contains the description of the framework and the general hypotheses. Section 2 contains the statement and the proof of the
main results in the Lipschitz case (Theorem 2.1). Section 3 contains some complements
and applications. In particular we consider the locally Lipschitz case (Theorem 3.1) and
show how one can estimate the dimension of the inertial manifold in terms of physically
relevant quantities. Finally in Section 4 we give an application of the general results to
an example related to Slow Manifolds, a related concept appearing in meteorology.
CONTENT
Introduction
1. The Functional Setting
1.1. Notations and Hypotheses
1.2. Construction of the Inertial Manifold
2. The Main Result (Lipschitz case)
2.1. Existence of Inertial Manifolds
2.2. Properties of T
NATO ASI Series, Vol. 148
The Mathematics of Models for Climatology
and Environment
Edited by Jesus Ildefonso Dial.
© Springcr- Verlag Berlin Heidelberg 1997