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An Introduction to Semiflows

An Introduction to Semiflows by Albert J. Milani
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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
CRC Press; October 2004
403 pages; ISBN 9781420035117
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Title: An Introduction to Semiflows
Author: Albert J. Milani; Norbert J. Koksch
 
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