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

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Overview

"This book develops the theory of global attractors for a class of parabolic PDEs that 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 that generate the infinite-dimensional dynamical systems of the title. Attention then turns to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space that 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 graduate students and assumes only a basic knowledge of elementary functional analysis."--BOOK JACKET.
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Editorial Reviews

From the Publisher
"The book is written clearly and concisely. It is well structured, and the material is presented in a rigorous, coherent fashion...[it] constitutes an excellent resource for researchers and advanced graduate students in applied mathematics, dynamical systems, nonlinear dynamics, and computational mechanics. Its acquisition by libraries is strongly recommended." Applied Mechanics Reviews
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Product Details

Table of Contents

Preface
Introduction 1
Pt. I Functional Analysis 9
1 Banach and Hilbert Spaces 11
2 Ordinary Differential Equations 42
3 Linear Operators 62
4 Dual Spaces 89
5 Sobolev Spaces 109
Pt. II Existence and Uniqueness Theory 157
6 The Laplacian 159
7 Weak Solutions of Linear Parabolic Equations 188
8 Nonlinear Reaction-Diffusion Equations 213
9 The Navier-Stokes Equations: Existence and Uniqueness 234
Pt. III Finite-Dimensional Global Attractors 259
10 The Global Attractor: Existence and General Properties 261
11 The Global Attractor for Reaction-Diffusion Equations 285
12 The Global Attractor for the Navier-Stokes Equations 309
13 Finite-Dimensional Attractors: Theory and Examples 325
Pt. IV Finite-Dimensional Dynamics 357
14 The Squeezing Property: Determining Modes 359
15 The Strong Squeezing Property: Inertial Manifolds 385
16 A Direct Approach 406
17 The Kuramoto-Sivashinsky Equation 426
App. A Sobolev Spaces of Periodic Functions 435
App. B Bounding the Fractal Dimension Using the Decay of Volume Elements 439
References 445
Index 453
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