An Introduction to Semiflows 1st Edition by Albert Milani, Norbert Koksch 0367454289 9780367454289

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ISBN 10: 0367454289

ISBN 13: 9780367454289

Author: Albert J. Milani, Norbert J. Koksch

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.

An Introduction to Semiflows 1st Table of contents:

1 Dynamical Processes

1.1 Introduction

1.2 Ordinary Differential Equations

1.2.1 Well-Posedness

1.2.2 Regular and Chaotic Systems

1.2.3 Dependence on Parameters

1.2.4 Autonomous Equations

1.3 Attracting Sets

1.4 Iterated Sequences

1.4.1 Poincaré Maps

1.4.2 Bernoulli’s Sequences

1.4.3 Tent Maps

1.4.4 Logistic Maps

1.5 Lorenz’ Equations

1.5.1 The Differential System

1.5.2 Equilibrium Points

1.6 Duffing’s Equation

1.6.1 The General Model

1.6.2 A Linearized Model

1.7 Summary

2 Attractors of Semiflows

2.1 Distance and Semidistance

2.2 Discrete and Continuous Semiflows

2.2.1 Types of Semiflows

2.2.2 Example: Lorenz’ Equations

2.3 Invariant Sets

2.3.1 Orbits

2.3.2 Limit Sets

2.3.3 Stability of Stationary Points

2.3.4 Invariance of Orbits and ω-Limit Sets

2.4 Attractors

2.4.1 Attracting Sets

2.4.2 Global Attractors

2.4.3 Compactness

2.5 Dissipativity

2.6 Absorbing Sets and Attractors

2.6.1 Attractors of Compact Semiflows

2.6.2 A Generalization

2.7 Attractors via α-Contractions

2.7.1 Measuring Noncompactness

2.7.2 A Route to α-Contractions

2.8 Fractal Dimension

2.9 A Priori Estimates

2.9.1 Integral and Differential Inequalities

2.9.2 Exponential Inequality

3 Attractors for Semilinear Evolution Equations

3.1 PDEEs as Dynamical Systems

3.1.1 The Model IBV Problems

3.1.2 Construction of the Attractors

3.2 Functional Framework

3.2.1 Function Spaces

3.2.2 Orthogonal Bases

3.2.3 Finite Dimensional Subspaces

3.3 The Parabolic Problem

3.3.1 Step 1: The Solution Operator

3.3.2 Step 2: Absorbing Sets

3.3.3 Step 3: Compactness of the Solution Operator

3.3.4 Step 4: Conclusion

3.3.5 Backward Uniqueness

3.4 The Hyperbolic Problem

3.4.1 Step 1: The Solution Operator

3.4.2 Step 2: Absorbing Sets

3.4.3 Step 3: Compactness of the Solution Operator

3.4.4 Step 4: Conclusion

3.4.5 Attractors via α-Contractions

3.5 Regularity

3.6 Upper Semicontinuity of the Global Attractors

4 Exponential Attractors

4.1 Introduction

4.2 The Discrete Squeezing Property

4.2.1 Orthogonal Projections

4.2.2 Squeezing Properties

4.2.3 Squeezing Properties and Exponential Attractors

4.2.4 Proof of Theorem 4.5

4.3 The Parabolic Problem

4.3.1 Step 1: Absorbing Sets in X1

4.3.2 Step 2: The Discrete Squeezing Property

4.4 The Hyperbolic Problem

4.4.1 Step 1: Absorbing Sets in X1

4.4.2 Step 2: The Discrete Squeezing Property

4.5 Proof of Theorem 4.4

4.5.1 Outline

4.5.2 The Cone Property

4.5.3 The Basic Covering Step

4.5.4 The First and Second Iterates

4.5.5 The General Iterate

4.5.6 Conclusion

4.6 Concluding Remarks

5 Inertial Manifolds

5.1 Introduction

5.2 Definitions and Comparisons

5.2.1 Lipschitz Manifolds and Inertial Manifolds

5.2.2 Inertial Manifolds and Exponential Attractors

5.2.3 Methods of Construction of the Inertial Manifold

5.3 Geometric Assumptions on the Semiflow

5.3.1 The Cone Invariance Property

5.3.2 Decay and Squeezing Properties

5.3.3 Consequences of the Decay Property

5.4 Strong Squeezing Property and Inertial Manifolds

5.4.1 Surjectivity and Uniform Boundedness

5.4.2 Construction of the Inertial Manifold

5.5 A Modification

5.5.1 The Modified Strong Squeezing Property

5.5.2 Consequences of the Modified Strong Squeezing Property

5.5.3 Construction of the Inertial Manifold, 2

5.5.4 Comparison of the Squeezing Properties

5.6 Inertial Manifolds for Evolution Equations

5.6.1 The Evolution Problem

5.6.2 The Spectral Gap Condition

5.6.3 The Strong Squeezing Properties

5.6.4 Uniform Boundedness and Surjectivity

5.7 Applications

5.7.1 Semilinear Heat Equations

5.7.2 Semilinear Wave Equations

5.8 Semilinear Evolution Equations in One Space Dimension

5.8.1 The Parabolic Problem

5.8.2 Absorbing Sets

5.8.3 Adjusting the Nonlinearity

5.8.4 The Inertial Manifold

5.8.5 The Hyperbolic Perturbation

5.8.6 Concluding Remarks

6 Examples

6.1 Cahn-Hilliard Equations

6.1.1 Introduction

6.1.2 The Cahn-Hilliard Semiflows

6.1.3 Absorbing Sets

6.1.4 The Global Attractor

6.1.5 The Exponential Attractor

6.1.6 The Inertial Manifold

6.2 Beam and von Kármán Equation

6.2.1 Functional Framework and Notations

6.2.2 The Beam Equation Semiflow

6.2.3 Absorbing Sets

6.2.4 The Global Attractor

6.2.5 The Exponential Attractor

6.2.6 Inertial Manifold

6.2.7 von Kármán Equations

6.3 Navier-Stokes Equations

6.3.1 The Equations and their Functional Framework

6.3.2 The 2-Dimensional Navier-Stokes Semiflow

6.3.3 Absorbing Sets and Attractor

6.3.4 The Exponential Attractor

6.4 Maxwell’s Equations

6.4.1 The Equations and their Functional Framework

6.4.2 The Quasi-Stationary Maxwell Semiflow

6.4.3 Absorbing Sets and Attractors

7 A Nonexistence Result for Inertial Manifolds

7.1 The Initial-Boundary Value Problem

7.2 Overview of the Argument

7.3 The Linearized Problem

7.4 Inertial Manifolds for the Linearized Problem

7.5 C1 Linearization Equivalence

7.6 Perturbations of the Nonlinear Flow

7.7 Asymptotic Properties of the Perturbed Flow

7.8 The Nonexistence Result

7.9 Proof of Proposition 7.17

7.10 The C1 Linearization Equivalence Theorems

7.10.1 Equivalence for a Single Operator

7.10.2 Equivalence for Groups of Operators

Appendix: Selected Results from Analysis

A.1 Ordinary Differential Equations

A.1.1 Classical Solutions

A.1.2 Generalized Solutions

A.1.3 Stability for Autonomous Systems

A.2 Linear Spaces and their Duals

A.2.1 Orthonormal Bases in Hilbert spaces

A.2.2 Dual Spaces and the Hahn-Banach Theorem

A.2.3 Linear Operators in Banach Spaces

A.2.4 Adjoint of a Bounded Operator

A.2.5 Adjoint of an Unbounded Operator

A.2.6 Gelfand Triples of Hilbert Spaces

A.2.7 Linear Operators in Gelfand Triples

A.2.8 Eigenvalues of Compact Operators

A.2.9 Fractional Powers of Positive Operators

A.2.10 Interpolation Spaces

A.2.11 Differential Calculus in Banach Spaces

A.3 Semigroups of Linear Operators

A.3.1 General Results

A.3.2 Applications to PDEs

A.4 Lebesgue Spaces

A.4.1 The Spaces Lp(Ω)

A.4.2 Inequalities

A.4.3 Other Properties of the Spaces Lp(Ω)

A.5 Sobolev Spaces of Scalar Valued Functions

A.5.1 Distributions in Ω

A.5.2 The Spaces Hm(Ω), m ∈ ℕ

A.5.3 The Spaces Hs(Ω), s ∈ ℝ≥0

A.5.4 The Spaces Hs0(Ω), s ∈ ℝ≥0, and Hs(Ω), s ∈ ℝ<0

A.5.5 The Laplace Operator

A.6 Sobolev Spaces of Vector Valued Functions

A.6.1 Lebesgue and Sobolev Spaces

A.6.2 The Intermediate Derivatives Theorem

A.7 The Spaces H(div, Ω) and H (curl, Ω)

A.7.1 Notations

A.7.2 The Space H(div, Ω)

A.7.3 The Space H(curl, Ω)

A.7.4 Relations between H(div, Ω) and H(curl, Ω)

A.8 Almost Periodic Functions

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