An Introduction to Semiflows
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Summary
This book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models.
The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial manifolds.
Table of Contents
DYNAMICAL PROCESSES
Introduction
Ordinary Differential Equations
Attracting Sets
Iterated Sequences
Lorenz' Equations
Duffing's Equation
Summary
ATTRACTORS OF SEMIFLOWS
Distance and Semidistance
Discrete and Continuous Semiflows
Invariant Sets
Attractors
Dissipativity
Absorbing Sets and Attractors
Attractors via a-Contractions
Fractal Dimension
A Priori Estimates
ATTRACTORS FOR SEMILINEAR EVOLUTION EQUATIONS
PDEEs as Dynamical Systems
Functional Framework
The Parabolic Problem
The Hyperbolic Problem
Regularity
Upper Semicontinuity of the Global Attractors
EXPONENTIAL ATTRACTORS
Introduction
The Discrete Squeezing Property
The Parabolic Problem
The Hyperbolic Problem
Proof of Theorem 4.4
Concluding Remarks
INERTIAL MANIFOLDS
Introduction
Definitions and Comparisons
Geometric Assumptions on the Semiflow
Strong Squeezing Property and Inertial Manifolds
A Modification
Inertial Manifolds for Evolution Equations
Applications
Semilinear Evolution Equations in One Space Dimension
EXAMPLES
Cahn-Hilliard Equations
Beam and von Kármán Equation
Navier-Stokes Equations
Maxwell's Equations
A NON-EXISTENCE RESULT FOR INERTIAL MANIFOLDS
The Initial-Boundary Value Problem
Overview of the Argument
The Linearized Problem
Inertial Manifolds for the Linearized Problem
C1 Linearization Equivalence
Perturbations of the Nonlinear Flow
Asymptotic Properties of the Perturbed Flow
The Non-Existence Result
Proof of Proposition 7.17
The C1 Linearization Equivalence Theorems.
APPENDIX: SELECTED RESULTS FROM ANALYSIS
A.1 Ordinary Differential Equations
A.2 Linear Spaces and their Duals
A.3 Semigroups of Linear Operators
A.4 Lebesgue Spaces
A.5 Sobolev Spaces of Scalar Valued Functions
A.6 Sobolev Spaces of Vector Valued Functions
A.7 The Spaces H(div,W) and H(curl,W)
A.8 Almost Periodic Functions
BIBLIOGRAPHY
INDEX
NOMENCLATURE
Editorial Reviews
"I am convinced that the authors have made this mathematically-demanding area as accessible as possible to the newcomer in the field. The pictures and 'simple ODE applications' help the reader gain intuition. This book does a fine job of presenting difficult material on applications to readers without in-depth knowledge of partial differential equations in a painless and enjoyable way."
- Hal Smith, Arizona State University
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