Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Frontmatter

General Introduction. The User’s Guide

Abstract

In this General Introduction we intend to focus on the general motivations and general ideas underlying this work, separating them from the mathematical technicalities, and thus developing further the presentation in the Preface.

Roger Temam

Chapter I. General Results and Concepts on Invariant Sets and Attractors

Abstract

This chapter is devoted to the presentation of general results and concepts on invariant sets and attractors. Although the presentation is sufficiently general to include ordinary differential equations, the main objective is, of course, to consider infinite-dimensional dynamical systems. The general results, which will be applied in the following chapters to infinite-dimensional systems of physical interest, are already illustrated in this chapter with simpler examples which do not necessitate any knowledge of partial differential equations, i.e., they are limited to some ordinary differential equations and to some iterative processes.

Roger Temam

Chapter II. Elements of Functional Analysis

Abstract

In this chapter we present some general tools and basic results of functional analysis which will be used frequently in the sequel; most results are recalled without proofs. This chapter is not conceived as an introduction to the next chapters; it is not necessary, and it is not suggested that this chapter be read completely before reading the subsequent ones. It should be viewed as a technical reference to be read “locally” as needed.

Roger Temam

Chapter III. Attractors of the Dissipative Evolution Equation of the First Order in Time: Reaction—Diffusion Equations. Fluid Mechanics and Pattern Formation Equations

Abstract

Our aim in this chapter is to describe some nonlinear evolution equations of the first order in time, which arise in mechanics and physics. In each case, we present briefly the physical model and the governing equations; then we present the mathematical setting of the equations which leads to the introduction of the corresponding semigroup {S(t)} _t≥0. Once the semigroup is defined, we address the following questions:

Roger Temam

Chapter IV. Attractors of Dissipative Wave Equations

Abstract

Our aim in this chapter is to study some nonlinear wave equations and a nonlinear Schrödinger equation, the Ginzburg—Landau equation. The wave equations that we consider are the sine—Gordon equation, a nonlinear wave equation of relativisitic quantum mechanics, and some nonlinear vibration equations in solid mechanics that involve fourth-order differential operators in space variables. Strictly speaking, the nonlinear Schrödinger equation is an evolution equation of the first order in time, and it is studied by the methods of Chapter III; however, this equation is related to wave phenomena and from the physical point of view has some properties in common with wave equations.

Roger Temam

Chapter V. Lyapunov Exponents and Dimension of Attractors

Abstract

This chapter contains the essential definitions and results which concern the study of the geometry of the attractors and functional sets, viz. the concept of Lyapunov exponents and Lyapunov numbers and general abstract results concerning the dimensions of attractors and functional sets. The Lyapunov numbers have a geometrical interpretation. They indicate how volumes are distorted in dimension m by the semigroup: the semigroup on the attractor is contracting in some directions and expanding in other directions, leading to a dynamics which can be complicated. Because the Lyapunov numbers (or the Lyapunov exponents which are the logarithms of the Lyapunov numbers) indicate the exponential rates of variation of lengths, surfaces, volumes in dimension 1, 2, 3,..., they provide valuable information about the dynamics.

Roger Temam

Chapter VI. Explicit Bounds on the Number of Degrees of Freedom and the Dimension of Attractors of Some Physical Systems

Abstract

This chapter is aimed at applying the general results of Chapter V to the attractors of all the physical equations that we have considered in Chapters III and IV. It appears as one of the culminating points of the theory of attractors for dissipative partial differential equations presented in this book.

Roger Temam

Chapter VII. Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions

Abstract

This corresponds to the case where the semigroup S(t) is not defined everywhere, or is not defined for all times t. In that case, a concept of functionalinvariant sets can be introduced by modifying slightly that given in Chapter I. Of course, we can no longer expect to prove the existence of absorbing sets and attractors, and the results of Chapter I do not apply anymore. However, the results of Chapter V concerning the dimensions of invariant sets have been proved at a level of generality which makes them applicable to such situations. Some examples are given without any attempt at generality.

Roger Temam

Chapter VIII. The Cone and Squeezing Properties. Inertial Manifolds

Abstract

Our aim in this final chapter is to introduce the concept of inertial manifolds for dissipative dynamical systems. This is a new concept which has recently emerged in nonlinear dynamics and, here, we shall restrict ourselves to describing a few typical results. Our presentation follows with some slight simplifications and generalizations C. Foias, G. Sell, and R. Temam [1], [2], C. Foias, B. Nicolaenko, G. Sell, and R. Temam [1], [2]. The reader is referred to Remark 4.3 for some bibliographical references on this rapidly expanding subject.

Roger Temam

Chapter IX. Inertial Manifolds and Slow Manifolds. The Non-self-adjoint Case

Abstract

This chapter and the following chapter are based on a presentation of Inertial Manifolds (I.M.) which is different from that contained in Chapter VIII. As we said in the Preface to the Second Edition, Chapters IX and X can be read independently of Chapter VIII. The present chapter contains a presentation of inertial manifolds which is self-contained. The main result of this chapter is an existence result for inertial manifolds (see Theorem 2.1 and also Theorem 3.1) which generalizes Theorem 3.1 of Chapter VIII. Although the results are very similar in cases where both theorems apply, the approach in this chapter offers several advantages.

Roger Temam

Chapter X. Approximation of Attractors and Inertial Manifolds. Convergent Families of Approximate Inertial Manifolds

Abstract

Our aim in this chapter is to study the approximation of attractors and inertial manifolds by smooth finite-dimensional manifolds, called approximate inertial manifolds. There is a vast and growing literature on approximation of attractors and inertial manifolds, approximate dynamics, and related algorithms. These important topics could justify, by themselves, the writing of one or several books; we will not try to give an exhaustive presentation or even an exhaustive bibliography on this question (see, however, a few references in Remark 3.2 at the end of this chapter).

Roger Temam

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