Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors

by James C. Robinson

• ISBN13: 9780521632041
• ISBN10: 0521632048
• Format: Hardcover
• Copyright: 2001-04-23
• Publisher: Cambridge University Press

Summary

This book develops the theory of global attractors for a class of parabolic PDEs which includes reaction-diffusion equations and the Navier-Stokes equations, two examples that are treated in detail. A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson's equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systems of the title. Attention then switches to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular, the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves 'finite-dimensional'. The book is intended as a didactic text for first year graduates, and assumes only a basic knowledge of Banach and Hilbert spaces, and a working understanding of the Lebesgue integral.

Table of Contents

Preface	xv
Introduction	1	(8)
Part I Functional Analysis	9	(148)
Banach and Hilbert Spaces	11	(31)
Banach Spaces and Some General Topology	11	(1)
The Euclidean Space Rm	12	(2)
The Spaces Cr and Cry of Continuous Functions	14	(6)
Mollification and Approximation by Smooth Functions	18	(2)
The Lp Spaces of Lebesgue Integrable Functions	20	(13)
Lebesgue Integration	20	(2)
The Lebesgue Spaces Lp (Ω) with 1 ≤ p < ∞	22	(6)
The Lebesgue Space L∞ (Omega;)	28	(2)
The Spaces Lploc (Omega;) of Locally Integrable Functions	30	(2)
The lp Sequence Spaces, 1 ≤ p ≤ ∞	32	(1)
Hilbert Spaces	33	(9)
The Orthogonal Projection onto a Linear Subspace	35	(1)
Bases in Hilbert Spaces	36	(2)
Noncompactness of the Unit Ball	38	(1)
Exercises	39	(2)
Notes	41	(1)
Ordinary Differential Equations	42	(20)
Existence and Uniqueness - A Fixed-Point Method	43	(5)
The Contraction Mapping Theorem	44	(1)
Local Existence for Lipschitz f	45	(3)
Global Existence	48	(1)
Existence but No Uniqueness--An Approximation Method	49	(4)
The Arzela--Ascoli Theorem	49	(2)
Local Existence for Continuous f	51	(2)
Differential Inequalities	53	(3)
Continuous Dependence on Initial Conditions	56	(3)
Conclusion	59	(3)
Exercises	59	(2)
Notes	61	(1)
Linear Operators	62	(27)
Bounded Linear Operators on Banach Spaces	62	(3)
Domain, Range, Kernel, and the Inverse Operator	65	(1)
The Baire Category Theorem	66	(2)
Compact Operators	68	(4)
Compact Symmetric Operators on Hilbert Spaces	72	(2)
Obtaining an Eigenbasis from a Compact Symmetric Operator	74	(5)
Unbounded Operators	79	(1)
Extensions and Closable Operators	80	(1)
Spectral Theory for Unbounded Symmetric Operators	81	(2)
Positive Operators and Their Fractional Powers	83	(6)
Exercises	85	(2)
Notes	87	(2)
Dual Spaces	89	(20)
The Hahn--Banach Theorem	89	(4)
Examples of Dual Spaces	93	(6)
The Dual Space of Lp, 1 < p < ∞	93	(1)
The Dual Space of lp, 1 < p < ∞	94	(2)
The Dual Spaces of L1 and L∞	96	(1)
The Dual Space of l1 and l∞	96	(3)
Dual Spaces of Hilbert Spaces	99	(1)
Reflexive Spaces	100	(1)
Notions of Weak Convergence	101	(4)
Weak Convergence	101	(3)
Weak-* Convergence	104	(1)
The Alaoglu Weak-* Compactness Theorem	105	(4)
Exercises	107	(1)
Notes	107	(2)
Sobolev Spaces	109	(48)
Generalised Nations of Derivatives	109	(5)
The Weak Derivative	109	(2)
The Distribution Derivative	111	(3)
General Sobolev Spaces	114	(5)
Sobolev Spaces and the Closure of Differential Operators	115	(1)
The Hilbert Space Hk (Omega;)	115	(4)
Outline of the Rest of the Chapter	119	(1)
C∞ (Omega;) is Dense in Hk (Omega;)	120	(4)
An Extension Theorem	124	(7)
Extending Functions in Hk (R+m)	125	(2)
Coordinate Changes	127	(1)
Straightening the Boundary	128	(1)
Extending Functions in Hk (Omega;)	129	(2)
Density of C∞ (Omega;) in Hk (Omega;)	131	(1)
The Sobolev Embedding Theroem--Hk, Cr, and Lp	132	(11)
Integrability of Functions in Sobolev Spaces	132	(7)
Sobolev Spaces and Spaces of Continuous Functions	139	(3)
The Sobolev Embedding Theorem	142	(1)
A Compactness Theorem	143	(2)
Boundary Values	145	(4)
Sobolev Spaces of Periodic Functions	149	(8)
Exercises	152	(1)
Notes	153	(4)
Part II Existence and Uniqueness Theory	157	(102)
The Laplacian	159	(29)
Classical, Strong, and Weak Solutions	160	(1)
Weak Solutions of Poisson's Equation	160	(4)
Higher Regularity for the Laplacian I: Periodic Boundary Conditions	164	(4)
Higher Regularity for the Laplacian II: Dirichlet Boundary Conditions	168	(7)
A Heuristic Estimate	168	(2)
Difference Quotients	170	(2)
Interior Regularity Result	172	(3)
Boundary Regularity for the Laplacian	175	(13)
Regularity up to a Flat Boundary	175	(5)
Regularity up to a C2 Boundary	180	(3)
H2k (Omega;) and Domains of Ak	183	(2)
Exercises	185	(2)
Notes	187	(1)
Weak Solutions of Linear Parabolic Equations	188	(25)
Banach-Space Valued Function Spaces	188	(6)
Weak Solutions of Parabolic Equations	194	(3)
The Galerkin Method: Truncated Eigenfunction Expansions	197	(3)
Weak Solutions	200	(6)
The Galerkin Approximations	201	(1)
Uniform Bounds on un in Various Spaces	202	(1)
Extraction of an Appropriate Subsequence	203	(2)
Properties of the Weak Solution	205	(1)
Uniqueness and Continuous Dependence on Initial Conditions	206	(1)
Strong Solutions	206	(3)
Higher Regularity: Spatial and Temporal	209	(4)
Exercises	210	(2)
Notes	212	(1)
Nonlinear Reaction--Diffusion Equations	213	(21)
Results to Deal with the Nonlinear Term	214	(5)
A Compactness Theorem	214	(3)
A Weak Version of the Dominated Convergence Theorem	217	(2)
The Basis for the Galerkin Expansion	219	(2)
Weak Solutions	221	(6)
A Semidynamical System on L2 (Omega;)	226	(1)
Strong Solutions	227	(7)
Exercises	231	(1)
Notes	232	(2)
The Navier--Stokes Equations: Existence and Uniqueness	234	(25)
The Stokes Operator	235	(4)
The Weak Form of the Navier--Stokes Equation	239	(2)
Properties of the Trilinear Form	241	(3)
Existence of Weak Solutions	244	(6)
Unique Weak Solutions in Two Dimensions	250	(2)
Existence of Strong Solutions in Two Dimensions	252	(3)
Uniqueness of 3D Strong Solutions	255	(1)
Dynamical Systems Generated by the 2D Equations	256	(3)
Exercises	256	(1)
Notes	257	(2)
Part III Finite-Dimensional Global Attractors	259	(98)
The Global Attractor: Existence and General Properties	261	(24)
Semigroups	261	(1)
Dissipation	262	(3)
Limit Sets and Attractors	265	(4)
Limit Sets	265	(1)
The Global Attractor	266	(3)
A Theorem for the Existence of Global Attractors	269	(2)
An Example--The Lorenz Equations	271	(1)
Structure of the Attractor	272	(4)
Gradient Systems and Lyapunov Functions	274	(2)
How the Attractor Determines the Asymptotic Dynamics	276	(2)
Continuity Properties of the Attractor	278	(2)
Upper Semicontinuity	278	(1)
Lower Semicontinuity	279	(1)
Conclusion	280	(5)
Exercises	281	(1)
Notes	282	(3)
The Global Attractor for Reaction--Diffusion Equations	285	(24)
Absorbing Sets and the Attractor	285	(5)
An Absorbing Set in L2	286	(1)
An Absorbing Set in H10	287	(3)
The Global Attractor	290	(1)
Regularity Results	290	(6)
A Bound in L∞	290	(3)
A Bound in H2 (Omega;)	293	(2)
Further Regularity	295	(1)
Injectivity on A	296	(3)
A Lyapunov Functional	299	(2)
The Chaffee--Infante Equation	301	(8)
Stationary Points	301	(3)
Bifurcations around the Zero State	304	(2)
Exercises	306	(1)
Notes	307	(2)
The Global Attractor for the Navier--Stokes Equations	309	(16)
2D Navier--Stokes Equations	309	(8)
An Absorbing Set in L2	310	(1)
An Absorbing Set in H1	311	(2)
An Absorbing Set in H2	313	(2)
Comparison of the Attractors in H and V and Further Regularity Results	315	(1)
Injectivity on the Attractor	316	(1)
The 3D Navier--Stokes Equations	317	(6)
An Absorbing Set in V	318	(4)
An Absorbing Set in D(A) and a Global Attractor	322	(1)
Conclusion	323	(2)
Exercises	323	(1)
Notes	324	(1)
Finite-Dimensional Attractors: Theory and Examples	325	(32)
Measures of Dimension	326	(10)
The ``Fractal'' Dimension	326	(4)
The Hausdorff Dimension	330	(4)
Hausdorff versus Fractal Dimension	334	(2)
Bounding the Attractor Dimension Dynamically	336	(7)
Example I: The Reaction--Diffusion Equation	343	(4)
Uniform Differentiability	343	(3)
A Bound on the Attractor Dimension	346	(1)
Example II: The 2D Navier--Stokes Equations	347	(3)
Uniform Differentiability	347	(2)
A Bound on the Attractor Dimension	349	(1)
Physical Interpretation of the Attractor Dimension	350	(2)
Conclusion	352	(5)
Exercises	352	(2)
Notes	354	(3)
Part IV Finite-Dimensional Dynamics	357	(78)
The Squeezing Property: Determining Modes	359	(26)
The Squeezing Property	359	(1)
An Approximate Manifold Structure for A	360	(3)
Determining Modes	363	(2)
The Squeezing Property for Reaction--Diffusion Equations	365	(4)
The 2D Navier--Stokes Equations	369	(5)
Checking the Squeezing Property	369	(2)
Approximate Inertial Manifolds	371	(3)
Finite-Dimensional Exponential Attractors	374	(5)
Conclusion	379	(6)
Exercises	379	(4)
Notes	383	(2)
The Strong Squeezing Property: Inertial Manifolds	385	(21)
Inertial Manifolds and ``Slaving''	385	(3)
A Geometric Existence Proof	388	(6)
The Strong Squeezing Property	388	(3)
The Existence Proof	391	(3)
Finding Conditions for the Strong Squeezing Property	394	(2)
Inertial Manifolds for Reaction--Diffusion Equations	396	(5)
Preparing the Equation	396	(3)
Checking the Spectral Gap Condition	399	(1)
Extensions to Other Domains and Higher Dimensions	400	(1)
More General Conditions for the Strong Squeezing Property	401	(2)
Inertial Manifolds and the Navier--Stokes Equations	401	(2)
Conclusion	403	(3)
Exercises	403	(1)
Notes	404	(2)
A Direct Approach	406	(20)
Parametrising the Attractor	407	(4)
Experimental Measurements as Parameters	411	(1)
An Extension Theorem	411	(1)
Embedding the Dynamics Without Uniqueness	412	(3)
Continuity of F on A for the Scalar Reaction--Diffusion Equation	414	(1)
Continuity of F on A for the 2D Navier--Stokes Equations	415	(1)
A Discrete-Time Utopian Theorem	415	(7)
The Topology of Global Attractors	416	(3)
The ``within &epsis;'' Discrete Utopian Theorem	419	(3)
Conclusion	422	(4)
Exercises	422	(2)
Notes	424	(2)
The Kuramoto--Sivashinsky Equation	426	(9)
Preliminaries	426	(2)
Existence and Uniqueness of Solutions	428	(1)
Absorbing Sets and the Global Attractor	429	(2)
The Attractor is Finite-Dimensional	431	(1)
Inertial Manifolds	432	(3)
Notes	433	(2)
Appendix A Sobolev Spaces of Periodic Functions	435	(4)
A.1 The Sobolev Embedding Theorem--Hs, Cr, and Lp	435	(2)
A.1.1 Conditions for Hs (Q) ⊂ C0 (Q)	435	(1)
A.1.2 Integrability Properties of Functions in Hs	436	(1)
A.2 Rellich--Kondrachov Compactness Theorem	437	(2)
Appendix B Bounding the Fractal Dimension Using the Decay of Volume Elements	439	(6)
References	445	(8)
Index	453