Chueshov. Introduction to infinite-dimensional dynamical and dissipative systems(419s).pdf

Author: I. D. Chueshov

I. D. Chueshov

This book provides an exhau -

stive introduction to the scope

of main ideas and methods of the

theory of infinite-dimensional dis -

sipative dynamical systems which

has been rapidly developing in re -

cent years. In the examples

sys tems generated by nonlinear

partial differential equations

arising in the different problems

of modern mechanics of continua

are considered. The main goal

of the book is to help the reader

to master the basic strategies used

in the study of infinite-dimensional

dissipative systems and to qualify

him/her for an independent scien -

tific research in the given branch.

Experts in nonlinear dynamics will

find many fundamental facts in the

convenient and practical form

in this book.

The core of the book is com -

posed of the courses given by the

author at the Department

of Me chanics and Mathematics

at Kharkov University during

a number of years. This book con -

tains a large number of exercises

which make the main text more

complete. It is sufficient to know

the fundamentals of functional

analysis and ordinary differential

equations to read the book.

Title:

Introduction to the Theory

of InfiniteDimensional

Dissipative Systems

ISBN: 966–7021–64–5

Dissipative Systems

I. D. Chueshov

Chueshov

I ntroduction

ntroduction

Theory

to the

of Infinite-Dimensional

Infinite-Dimensional

D issipative

issipative

S ystems

ystems

Universitylecturesincontemporarymathematics

Translated by

Constantin I. Chueshov

from the Russian edition ( « A CTA » , 1999)

You can ORDER

ORDER this book

while visiting the website

of «A CTA » Scientific Publishing House

http://www.acta.com.ua

Translation edited by

Maryna B. Khorolska

www.acta.com.ua/en/

I. D. Chueshov

Introduction

to the Theory of Infinite-Dimensional

Introduction

to the Theory of Infinite-Dimensional

Dissipative Systems

A CTA 2002

UDC 517

2000 Mathematics Subject Classification:

primary 37L05; secondary 37L30, 37L25.

This book provides an exhaustive introduction to the scope

of main ideas and methods of the theory of infinite-dimen-

sional dissipative dynamical systems which has been rapidly

developing in recent years. In the examples systems genera-

ted by nonlinear partial differential equations arising in the

different problems of modern mechanics of continua are con-

sidered. The main goal of the book is to help the reader to

master the basic strategies used in the study of infinite-di-

mensional dissipative systems and to qualify him/her for an

independent scientific research in the given branch. Experts

in nonlinear dynamics will find many fundamental facts in the

convenient and practical form in this book.

The core of the book is composed of the courses given by

the author at the Department of Mechanics and Mathematics

at Kharkov University during a number of years. This book

contains a large number of exercises which make the main

text more complete. It is sufficient to know the fundamentals

of functional analysis and ordinary differential equations to

read the book.

Translated by Constantin I. Chueshov

from the Russian edition ( « A CTA » , 1999)

Translation edited by Maryna B. Khorolska

A CTA Scientific Publishing House

Kharkiv, Ukraine

E-mail: we@acta.com.ua

Свідоцтво ДК №179

ISBN 966-7021-20-3 (series)

ISBN 966-7021-64-5

Contents

. . . . Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 1 .

e Theory

of Infinite-Dimensional Dynamical Sy

Basic Concepts of

Basic Concepts of th

tth

the Theory

e Theory

of Infinite-Dimensional Dynamical Sy

of Infinite-Dimensional Dynamical Syst

sst

stems

ems

. . . . § 1 Notion of Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . 11

. . . . § 2 Trajectories and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . 17

. . . . § 3 Definition of Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

. . . . § 4 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . . 24

. . . . § 5 Theorems on Existence of Global Attractor . . . . . . . . . . . . . . 28

. . . . § 6 On the Structure of Global Attractor . . . . . . . . . . . . . . . . . . . 34

. . . . § 7 Stability Properties of Attractor and Reduction Principle . . 45

. . . . § 8 Finite Dimensionality of Invariant Sets . . . . . . . . . . . . . . . . . 52

. . . . § 9 Existence and Properties of Attractors of a Class

of Infinite-Dimensional Dissipative Systems . . . . . . . . . . . . . 61

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 2 .

Long-Time Behaviour of Solutions

to a Class of Semilinear Parabolic Equations

Long-Time Behaviour of Solutions

to a Class of Semilinear Parabolic Equations

. . . . § 1 Positive Operators with Discrete Spectrum . . . . . . . . . . . . . . 77

. . . . § 2 Semilinear Parabolic Equations in Hilbert Space . . . . . . . . . . 85

. . . . § 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

. . . . § 4 Existence Conditions and Properties of Global Attractor . . 101

. . . . § 5 Systems with Lyapunov Function . . . . . . . . . . . . . . . . . . . . . 108

. . . . § 6 Explicitly Solvable Model of Nonlinear Diffusion . . . . . . . . . 118

. . . . § 7 Simplified Model of Appearance of Turbulence in Fluid . . . 130

. . . . § 8 On Retarded Semilinear Parabolic Equations . . . . . . . . . . . 138

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Contents

Chapter 3 .

Inertial Manifolds

. . . . § 1 Basic Equation and Concept of Inertial Manifold . . . . . . . . 149

. . . . § 2 Integral Equation for Determination of Inertial Manifold . . 155

. . . . § 3 Existence and Properties of Inertial Manifolds . . . . . . . . . . 161

. . . . § 4 Continuous Dependence of Inertial Manifold

on Problem Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

. . . . § 5 Examples and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

. . . . § 6 Approximate Inertial Manifolds

for Semilinear Parabolic Equations . . . . . . . . . . . . . . . . . . . 182

. . . . § 7 Inertial Manifold for Second Order in Time Equations . . . . 189

. . . . § 8 Approximate Inertial Manifolds for Second Order

in Time Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

. . . . § 9 Idea of Nonlinear Galerkin Method . . . . . . . . . . . . . . . . . . . 209

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Chapter 4 .

The Problem on Nonlinear

Oscillations of a Plate in a Supersonic Gas Flow

The Problem on Nonlinear

Oscillations of a Plate in a Supersonic Gas Flow

. . . . § 1 Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

. . . . § 2 Auxiliary Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

. . . . § 3 Theorem on the Existence and Uniqueness of Solutions . . 232

. . . . § 4 Smoothness of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

. . . . § 5 Dissipativity and Asymptotic Compactness . . . . . . . . . . . . . 246

. . . . § 6 Global Attractor and Inertial Sets . . . . . . . . . . . . . . . . . . . . 254

. . . . § 7 Conditions of Regularity of Attractor . . . . . . . . . . . . . . . . . . 261

. . . . § 8 On Singular Limit in the Problem

of Oscillations of a Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

. . . . § 9 On Inertial and Approximate Inertial Manifolds . . . . . . . . . 276

. . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

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