HI CAMBRIDGE n S P UNIVERSITY PRESS

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1 Infinite-Dimensional Dynamical Systems An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors JAMES C. ROBINSON University of Warwick HI CAMBRIDGE n S P UNIVERSITY PRESS

2 Preface page xv Introduction 1 Parti Functional Analysis 9 1 Banach and Hilbert Spaces Banach Spaces and Some General Topology The Euclidean Space W" The Spaces C r and C' y of Continuous Functions Mollification and Approximation by Smooth Functions The L p Spaces of Lebesgue Integrable Functions Lebesgue Integration The Lebesgue Spaces L P (Q.) with 1 < p < oo The Lebesgue Space! 00 ( 2) The Spaces L c ( 2) of Locally Integrable Functions The l p Sequence Spaces, 1 < p < oo Hilbert Spaces The Orthogonal Projection onto a Linear Subspace Bases in Hilbert Spaces Noncompactness of the Unit Ball Ordinary Differential Equations Existence and Uniqueness - A Fixed-Point Method The Contraction Mapping Theorem Local Existence for Lipschitz / 45 vn

3 viii Global Existence Existence but No Uniqueness - An Approximation Method The Arzela-Ascoli Theorem Local Existence for Continuous / Differential Inequalities Continuous Dependence on Initial Conditions Conclusion Linear Operators Bounded Linear Operators on Banach Spaces Domain, Range, Kernel, and the Inverse Operator The Baire Category Theorem Compact Operators Compact Symmetric Operators on Hilbert Spaces Obtaining an Eigenbasis from a Compact Symmetric Operator Unbounded Operators Extensions and Closable Operators ', Spectral Theory for Unbounded Symmetric Operators Positive Operators and Their Fractional Powers Dual Spaces The Hahn-Banach Theorem Examples of Dual Spaces The Dual Space of L p, 1 < p < oo The Dual Space of l p, 1 < p < oo The Dual Spaces of L 1 and L 00 The Dual Space of I 1 and / Dual Spaces of Hilbert Spaces Reflexive Spaces Notions of Weak Convergence Weak Convergence Weak-* Convergence The Alaoglu Weak-* Compactness Theorem

4 ' Sobolev Spaces Generalised Notions of Derivatives The Weak Derivative The Distribution Derivative General Sobolev Spaces Sobolev Spaces and the Closure of Differential Operators The Hilbert Space H k (Q) Outline of the Rest of the Chapter C^^isDenseintf^ft) An Extension Theorem Extending Functions in //*(R+) Coordinate Changes Straightening the Boundary Extending Functions in H k (Q) Density of C (^) in H k ( 2) The Sobolev Embedding Theorem - H k, C, and L p Integrability of Functions in Sobolev Spaces Sobolev Spaces and Spaces of Continuous Functions The Sobolev Embedding Theorem A Compactness Theorem Boundary Values Sobolev Spaces of Periodic Functions ix Part II Existence and Uniqueness Theory The Laplacian Classical, Strong, and Weak Solutions Weak Solutions of Poisson's Equation Hisher Reeularitv for the Lanlacian I: Periodic Boundary Conditions Higher Regularity for the Laplacian II: Dirichlet Boundary Conditions A Heuristic Estimate Difference Quotients Interior Regularity Result Boundary Regularity for the Laplacian Regularity up to a Flat Boundary Regularity up to a C 2 Boundary 180

5 x H lk (Q.) and Domains of A k Weak Solutions of Linear Parabolic Equations Banach-Space Valued Function Spaces Weak Solutions of Parabolic Equations The Galerkin Method: Truncated Eigenfunction Expansions Weak Solutions The Galerkin Approximations, Uniform Bounds on u n in Various Spaces Extraction of an Appropriate Subsequence Properties of the Weak Solution Uniqueness and Continuous Dependence on Initial Conditions Strong Solutions Higher Regularity: Spatial and Temporal ' Nonlinear Reaction-Diffusion Equations Results to Deal with the Nonlinear Term A Compactness Theorem ^ A Weak Version of the Dominated Convergence Theorem The Basis for the Galerkin Expansion Weak Solutions A Semidynamical System on L 2 (Sl) Strong Solutions The Navier-Stokes Equations: Existence and Uniqueness The Stokes Operator The Weak Form of the Navier-Stokes Equation Properties of the Trilinear Form Existence of Weak Solutions Unique Weak Solutions in Two Dimensions Existence of Strong Solutions in Two Dimensions Uniqueness of 3D Strong Solutions Dynamical Systems Generated by the 2D Equations 256

6 xi Part III Finite-Dimensional Global Attractors The Global Attractor: Existence and General Properties Semigroups Dissipation Limit Sets and Attractors Limit Sets > The Global Attractor A Theorem for the Existence of Global Attractors An Example - The Lorenz Equations Structure of the Attractor Gradient Systems and Lyapunov Functions How the Attractor Determines the Asymptotic Dynamics Continuity Properties of the Attractor Upper Semicontinuity Lower Semicontinuity Conclusion The Global Attractor for Reaction-Diffusion Equations Absorbing Sets and the Attractor An Absorbing Set in L 2 An Absorbing Set in H$ The Global Attractor Regularity Results A Bound in L A Bound in// 2 ( 2) Further Regularity Injectivity on A A Lyapunov Functional The Chaffee-Infante Equation Stationary Points Bifurcations around the Zero State

7 xii 12 The Global Attractor for the Navier-Stokes Equations D Navier-Stokes Equations An Absorbing Set in L An Absorbing Set in H An Absorbing Set in H Comparison of the Attractors in H and V and Further Regularity Results Injectivity on the Attractor The 3D Navier-Stokes Equations An Absorbing Set in V An Absorbing Set in D(A) and a Global Attractor Conclusion Finite-Dimensional Attractors: Theory and Examples Measures of Dimension The "Fractal" Dimension The Hausdorff Dimension ; Hausdorff versus Fractal Dimension Bounding the Attractor Dimension Dynamically Example I: The Reaction-Diffusion Equation Uniform Differentiability A Bound on the Attractor Dimension Example II: The 2D Navier-Stokes Equations Uniform Differentiability A Bound on the Attractor Dimension Physical Interpretation of the Attractor Dimension Conclusion Part IV Finite-Dimensional Dynamics The Squeezing Property: Determining Modes The Squeezing Property An Approximate Manifold Structure for A Determining Modes The Squeezing Property for Reaction-Diffusion Equations The 2D Navier-Stokes Equations 369

8 Checking the Squeezing Property Approximate Inertial Manifolds Finite-Dimensional Exponential Attractors Conclusion The Strong Squeezing Property: Inertial Manifolds Inertial Manifolds and "Slaving" A Geometric Existence Proof The Strong Squeezing Property The Existence Proof Finding Conditions for the Strong Squeezing Property Inertial Manifolds for Reaction Diffusion Equations Preparing the Equation Checking the Spectral Gap Condition Extensions to Other Domains and Higher Dimensions More General Conditions for the Strong Squeezing Property Inertial Manifolds and the Navier-Stokes Equations Conclusion A Direct Approach Parametrising the Attractor Experimental Measurements as Parameters An Extension Theorem Embedding the Dynamics Without Uniqueness Continuity of F on A for the Scalar Reaction-Diffusion Equation Continuity of F on A for the 2D Navier-Stokes Equations A Discrete-Time Utopian Theorem The Topology of Global Attractors The "within e" Discrete Utopian Theorem ; Conclusion The Kuramoto-Sivashinsky Equation Preliminaries ^ 426 xiii

9 xiv \1.2 Existence and Uniqueness of Solutions Absorbing Sets and the Global Attractor The Attractor is Finite-Dimensional Inertial Manifolds Appendix A Sobolev Spaces of Periodic Functions 435 A. 1 The Sobolev Embedding Theorem - H s, C, and L p 435 A.1.1 Conditions for H s (Q) c C (Q) 435 A. 1.2 Integrability Properties of Functions in H s 436 A.2 Rellich-Kondrachov Compactness Theorem 437 Appendix B Bounding the Fractal Dimension Using the Decay of Volume Elements 439 References 445 Index 453