An Introduction to Semiflows

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ISBN: 9781584884583 | 1584884584
Cover: Hardcover
Copyright: 10/14/2004

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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.

Dynamical Processes	p. 1
Introduction	p. 1
Ordinary Differential Equations	p. 4
Well-Posedness	p. 6
Regular and Chaotic Systems	p. 8
Dependence on Parameters	p. 10
Autonomous Equations	p. 11
Attracting Sets	p. 16
Iterated Sequences	p. 20
Poincare Maps	p. 22
Bernoulli's Sequences	p. 24
Tent Maps	p. 27
Logistic Maps	p. 30
Lorenz' Equations	p. 32
The Differential System	p. 33
Equilibrium Points	p. 33
Duffing's Equation	p. 36
The General Model	p. 36
A Linearized Model	p. 38
Summary	p. 40
Attractors of Semiflows	p. 41
Distance and Semidistance	p. 41
Discrete and Continuous Semiflows	p. 42
Types of Semiflows	p. 42
Example: Lorenz' Equations	p. 46
Invariant Sets	p. 49
Orbits	p. 50
Limit Sets	p. 52
Stability of Stationary Points	p. 55
Invariance of Orbits and [omega]-Limit Sets	p. 59
Attractors	p. 64
Attracting Sets	p. 64
Global Attractors	p. 65
Compactness	p. 67
Dissipativity	p. 68
Absorbing Sets and Attractors	p. 70
Attractors of Compact Semiflows	p. 70
A Generalization	p. 72
Attractors via [alpha]-Contractions	p. 73
Measuring Noncompactness	p. 74
A Route to [alpha]-Contractions	p. 79
Fractal Dimension	p. 81
A Priori Estimates	p. 83
Integral and Differential Inequalities	p. 84
Exponential Inequality	p. 86
Attractors for Semilinear Evolution Equations	p. 89
PDEEs as Dynamical Systems	p. 89
The Model IBV Problems	p. 89
Construction of the Attractors	p. 92
Functional Framework	p. 95
Function Spaces	p. 95
Orthogonal Bases	p. 97
Finite Dimensional Subspaces	p. 98
The Parabolic Problem	p. 99
Step 1: The Solution Operator	p. 100
Step 2: Absorbing Sets	p. 105
Step 3: Compactness of the Solution Operator	p. 106
Step 4: Conclusion	p. 107
Backward Uniqueness	p. 108
The Hyperbolic Problem	p. 111
Step 1: The Solution Operator	p. 112
Step 2: Absorbing Sets	p. 114
Step 3: Compactness of the Solution Operator	p. 116
Step 4: Conclusion	p. 121
Attractors via [alpha]-Contractions	p. 121
Regularity	p. 123
Upper Semicontinuity of the Global Attractors	p. 132
Exponential Attractors	p. 135
Introduction	p. 135
The Discrete Squeezing Property	p. 137
Orthogonal Projections	p. 137
Squeezing Properties	p. 138
Squeezing Properties and Exponential Attractors	p. 139
Proof of Theorem 4.5	p. 141
The Parabolic Problem	p. 143
Step 1: Absorbing Sets in x[subscript 1]	p. 143
Step 2: The Discrete Squeezing Property	p. 144
The Hyperbolic Problem	p. 147
Step 1: Absorbing Sets in x[subscript 1]	p. 147
Step 2: The Discrete Squeezing Property	p. 149
Proof of Theorem 4.4	p. 152
Outline	p. 152
The Cone Property	p. 153
The Basic Covering Step	p. 155
The First and Second Iterates	p. 159
The General Iterate	p. 161
Conclusion	p. 162
Concluding Remarks	p. 175
Inertial Manifolds	p. 177
Introduction	p. 177
Definitions and Comparisons	p. 179
Lipschitz Manifolds and Inertial Manifolds	p. 179
Inertial Manifolds and Exponential Attractors	p. 183
Methods of Construction of the Inertial Manifold	p. 185
Geometric Assumptions on the Semiflow	p. 189
The Cone Invariance Property	p. 189
Decay and Squeezing Properties	p. 191
Consequences of the Decay Property	p. 193
Strong Squeezing Property and Inertial Manifolds	p. 195
Surjectivity and Uniform Boundedness	p. 195
Construction of the Inertial Manifold	p. 197
A Modification	p. 201
The Modified Strong Squeezing Property	p. 201
Consequences of the Modified Strong Squeezing Property	p. 203
Construction of the Inertial Manifold, 2	p. 204
Comparison of the Squeezing Properties	p. 206
Inertial Manifolds for Evolution Equations	p. 208
The Evolution Problem	p. 208
The Spectral Gap Condition	p. 209
The Strong Squeezing Properties	p. 212
Uniform Boundedness and Surjectivity	p. 215
Applications	p. 218
Semilinear Heat Equations	p. 219
Semilinear Wave Equations	p. 220
Semilinear Evolution Equations in One Space Dimension	p. 229
The Parabolic Problem	p. 229
Absorbing Sets	p. 230
Adjusting the Nonlinearity	p. 234
The Inertial Manifold	p. 235
The Hyperbolic Perturbation	p. 238
Concluding Remarks	p. 239
Examples	p. 241
Cahn-Hilliard Equations	p. 241
Introduction	p. 242
The Cahn-Hilliard Semiflows	p. 244
Absorbing Sets	p. 247
The Global Attractor	p. 250
The Exponential Attractor	p. 251
The Inertial Manifold	p. 254
Beam and von Karman Equation	p. 261
Functional Framework and Notations	p. 261
The Beam Equation Semiflow	p. 262
Absorbing Sets	p. 263
The Global Attractor	p. 267
The Exponential Attractor	p. 269
Inertial Manifold	p. 271
von Karman Equations	p. 272
Navier-Stokes Equations	p. 272
The Equations and their Functional Framework	p. 272
The 2-Dimensional Navier-Stokes Semiflow	p. 275
Absorbing Sets and Attractor	p. 276
The Exponential Attractor	p. 278
Maxwell's Equations	p. 280
The Equations and their Functional Framework	p. 281
The Quasi-Stationary Maxwell Semiflow	p. 285
Absorbing Sets and Attractors	p. 288
A Nonexistence Result for Inertial Manifolds	p. 291
The Initial-Boundary Value Problem	p. 291
Overview of the Argument	p. 293
The Linearized Problem	p. 295
Inertial Manifolds for the Linearized Problem	p. 298
C[superscript 1] Linearization Equivalence	p. 303
Perturbations of the Nonlinear Flow	p. 304
Asymptotic Properties of the Perturbed Flow	p. 307
The Nonexistence Result	p. 309
Proof of Proposition 7.17	p. 310
The C[superscript 1] Linearization Equivalence Theorems	p. 316
Equivalence for a Single Operator	p. 316
Equivalence for Groups of Operators	p. 320
Selected Results from Analysis	p. 323
Ordinary Differential Equations	p. 323
Classical Solutions	p. 323
Generalized Solutions	p. 324
Stability for Autonomous Systems	p. 325
Linear Spaces and their Duals	p. 328
Orthonormal Bases in Hilbert spaces	p. 328
Dual Spaces and the Hahn-Banach Theorem	p. 329
Linear Operators in Banach Spaces	p. 330
Adjoint of a Bounded Operator	p. 333
Adjoint of an Unbounded Operator	p. 335
Gelfand Triples of Hilbert Spaces	p. 335
Linear Operators in Gelfand Triples	p. 336
Eigenvalues of Compact Operators	p. 338
Fractional Powers of Positive Operators	p. 340
Interpolation Spaces	p. 342
Differential Calculus in Banach Spaces	p. 344
Semigroups of Linear Operators	p. 345
General Results	p. 345
Applications to PDEs	p. 347
Lebesgue Spaces	p. 349
The Spaces L[superscript p] ([Omega])	p. 349
Inequalities	p. 350
Other Properties of the Spaces L[superscript p] ([Omega])	p. 351
Sobolev Spaces of Scalar Valued Functions	p. 352
Distributions in [Omega]	p. 353
The Spaces H[superscript m] ([Omega]), m [set membership] N	p. 353
The Spaces H[superscript s] ([Omega]), s [set membership] R[subscript greater than or equal 0]	p. 355
The Spaces H[superscript s subscript 0] ([Omega]), s [set membership] R[subscript greater than or equal 0], and H[superscript s] ([Omega]), s [set membership] R[subscript less than sign 0]	p. 357
The Laplace Operator	p. 358
Sobolev Spaces of Vector Valued Functions	p. 361
Lebesgue and Sobolev Spaces	p. 361
The Intermediate Derivatives Theorem	p. 362
The Spaces H(div, [Omega]) and H(curl, [Omega])	p. 363
Notations	p. 364
The Space H(div, [Omega])	p. 364
The Space H(curl, [Omega])	p. 365
Relations between H(div, [Omega]) and H(curl, [Omega])	p. 367
Almost Periodic Functions	p. 371
Bibliography	p. 373
Index	p. 381
Nomenclature	p. 385
Table of Contents provided by Rittenhouse. All Rights Reserved.

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