Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Transcription

1 Roger Temam Infinite-Dimensional Dynamical Systems in Mechanics and Physics Second Edition With 13 Illustrations Springer

2 Contents Preface to the Second Edition Preface to the First Edition vii ix GENERAL INTRODUCTION. The User's Guide 1 Introduction 1 1. Mechanism and Description of Chaos. The Finite-Dimensional Case 2 2. Mechanism and Description of Chaos. The Infinite-Dimensional Case 6 3. The Global Attractor. Reduction to Finite Dimension Remarks on the Computational Aspect The User's Guide 13 CHAPTER I General Results and Concepts on Invariant Sets and Attractors 15 Introduction Semigroups, Invariant Sets, and Attractors Semigroups of Operators Functional Invariant Sets Absorbing Sets and Attractors A Remark on the Stability of the Attractors Examples in Ordinary Differential Equations The Pendulum The Minea System The Lorenz Model Fractal Interpolation and Attractors The General Framework The Interpolation Process Proof of Theorem

3 xvi Contents CHAPTER II Elements of Functional Analysis 43 Introduction Function Spaces Definition of the Spaces. Notations Properties of Sobolev Spaces Other Sobolev Spaces Further Properties of Sobolev Spaces Linear Operators Bilinear Forms and Linear Operators "Concrete" Examples of Linear Operators Linear Evolution Equations of the First Order in Time Hypotheses A Result of Existence and Uniqueness Regularity Results Time-Dependent Operators Linear Evolution Equations of the Second Order in Time The Evolution Problem Another Result Time-Dependent Operators 80 CHAPTER III Attractors of the Dissipative Evolution Equation of the First Order in Time: Reaction-Diffusion Equations. Fluid Mechanics and Pattern Formation Equations 82 Introduction Reaction-Diffusion Equations Equations with a Polynomial Nonlinearity Equations with an Invariant Region Navier-Stokes Equations (n = 2) The Equations and Their Mathematical Setting Absorbing Sets and Attractors Proof of Theorem Other Equations in Fluid Mechanics Abstract Equation. General Results Fluid Driven by Its Boundary Magnetohydrodynamics (MHD) Geophysical Flows (Flows on a Manifold) Thermohydraulics Some Pattern Formation Equations The Kuramoto-Sivashinsky Equation The Cahn-Hilliard Equation Semilinear Equations The Equations. The Semigroup Absorbing Sets and Attractors Proof of Theorem

4 Contents xvii 6. Backward Uniqueness An Abstract Result Applications 175 CHAPTER IV Attractors of Dissipative Wave Equations 179 Introduction Linear Equations: Summary and Additional Results The General Framework Exponential Decay Bounded Solutions on the Real Line The Sine-Gordon Equation The Equation and Its Mathematical Setting Absorbing Sets and Attractors Other Boundary Conditions A Nonlinear Wave Equation of Relativistic Quantum Mechanics The Equation and Its Mathematical Setting Absorbing Sets and Attractors An Abstract Wave Equation The Abstract Equation. The Group of Operators Absorbing Sets and Attractors Examples Proof of Theorem 4.1 (Sketch) The Ginzburg-Landau Equation The Equations and Its Mathematical Setting Absorbing Sets and Attractors Weakly Dissipative Equations. I. The Nonlinear Schrodinger Equation The Nonlinear Schrodinger Equation Existence and Uniqueness of Solution. Absorbing Sets Decomposition of the Semigroup Comparison of z and Z for Large Times Application to the Attractor. The Main Result Determining Modes Weakly Dissipative Equations II. The Korteweg-De Vries Equation The Equation and its Mathematical Setting Absorbing Sets and Attractors Regularity of the Attractor Proof of the Results in Section Proof of Proposition Unbounded Case: The Lack of Compactness Preliminaries The Global Attractor Regularity of Attractors A Preliminary Result Example of Partial Regularity Example of # e0 Regularity Stability of Attractors 329

5 xviii " Contents CHAPTER V Lyapunov Exponents and Dimension of Attractors 335 Introduction Linear and Multilinear Algebra Exterior Product of Hilbert Spaces Multilinear Operators and Exterior Products Image of a Ball by a Linear Operator Lyapunov Exponents and Lyapunov Numbers Distortion of Volumes Produced by the Semigroup Definition of the Lyapunov Exponents and Lyapunov Numbers Evolution of the Volume Element and Its Exponential Decay: The Abstract Framework Hausdorff and Fractal Dimensions of Attractors Hausdorff and Fractal Dimensions Covering Lemmas The Main Results Application to Evolution Equations 377 CHAPTER VI Explicit Bounds on the Number of Degrees of Freedom and the Dimension of Attractors of Some Physical Systems 380 Introduction The Lorenz Attractor Reaction-Diffusion Equations Equations with a Polynomial Nonlinearity Equations with an Invariant Region Navier-Stokes Equations (n = 2) General Boundary Conditions Improvements for the Space-Periodic Case Other Equations in Fluid Mechanics The Linearized Equations (The Abstract Framework) Fluid Driven by Its Boundary Magnetohydrodynamics Flows on a Manifold Thermohydraulics Pattern Formation Equations The Kuramoto-Sivashinsky Equation The Cahn-Hilliard Equations Dissipative Wave Equations The Linearized Equation Dimension of the Attractor Sine-Gordon Equations Some Lemmas The Ginzburg-Landau Equation The Linearized Equation Dimension of the Attractor Differentiability of the Semigroup 461

6 Contents xix CHAPTER VII Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions 465 Introduction 465 PART A: NON-WELL-POSED PROBLEMS Dissipativity and Well Posedness General Definitions The Class of Problems Studied The Main Result Estimate of Dimension for Non-Well-Posed Problems: Examples in Fluid Dynamics The Equations and Their Linearization Estimate of the Dimension of X All 2.3. The Three-Dimensional Navier-Stokes Equations 479 PART B: UNSTABLE MANIFOLDS, LYAPUNOV FUNCTIONS, AND LOWER BOUNDS ON DIMENSIONS Stable and Unstable Manifolds Structure of a Mapping in the Neighborhood of a Fixed Point Application to Attractors Unstable Manifold of a Compact Invariant Set The Attractor of a Semigroup with a Lyapunov Function A General Result Additional Results Examples Lower Bounds on Dimensions of Attractors: An Example 496 CHAPTER VIII The Cone and Squeezing Properties. Inertial Manifolds 498 Introduction The Cone Property The Cone Property Generalizations The Squeezing Property Construction of an Inertial Manifold: Description of the Method Inertial Manifolds: The Method of Construction The Initial and Prepared Equations The Mapping 3~ Existence of an Inertial Manifold 512 '3.1. The Result of Existence First Properties of 2T Utilization of the Cone Property Proof of Theorem 3.1 (End) Another Form of Theorem Examples Example 1: The Kuramoto-Sivashinsky Equation 526

7 xx Contents 4.2. Example 2: Approximate Inertial Manifolds for the Navier-Stokes Equations Example 3: Reaction-Diffusion Equations Example 4: The Ginzburg-Landau Equation Approximation and Stability of the Inertial Manifold with Respect to Perturbations 532 CHAPTER IX Inertial Manifolds and Slow Manifolds. The Non-Self-Adjoint Case 536 Introduction The Functional Setting Notations and Hypotheses Construction of the Inertial Manifold The Main Result (Lipschitz Case) Existence of Inertial Manifolds Properties of ST Smoothness Property of <D(<D is "g 71 ) Proof of Theorem Complements and Applications The Locally Lipschitz Case Dimension of the Inertial Manifold Inertial Manifolds and Slow Manifolds The Motivation The Abstract Equation An Equation of Navier-Stokes Type 562 CHAPTER X Approximation of Attractors and Inertial Manifolds. Convergent Families of Approximate Inertial Manifolds 565 Introduction Construction of the Manifolds Approximation of the Differential Equation The Approximate Manifolds Approximation of Attractors Properties of 3T Distance to the Attractor The Main Result Convergent Families of Approximate Inertial Manifolds Properties of &~j Distance to the Exact Inertial Manifold Convergence to the Exact Inertial Manifold 583 APPENDIX Collective Sobolev Inequalities 585 Introduction Notations and Hypotheses The Operator U The Schrodinger-Type Operators 588

8 Contents xxi 2. Spectral Estimates for Schrodinger-Type Operators The Birman-Schwinger Inequality The Spectral Estimate Generalization of the Sobolev-Lieb-Thirring Inequality (I) Generalization of the Sobolev-Lieb-Thirring Inequality (II) The Space-Periodic Case The General Case Proof of Theorem Examples 610 Bibliography 613 Index 645