Infinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2001
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge texts in applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 461 S. |
| ISBN: | 0521635640 |
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| 245 | 1 | 0 | |a Infinite-dimensional dynamical systems |b an introduction to dissipative parabolic PDEs and the theory of global attractors |c James C. Robinson |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2001 | |
| 300 | |a XVII, 461 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge texts in applied mathematics | |
| 650 | 4 | |a Attracteurs (Mathématiques) | |
| 650 | 7 | |a Dynamische systemen |2 gtt | |
| 650 | 7 | |a Partiële differentiaalvergelijkingen |2 gtt | |
| 650 | 4 | |a Équations différentielles paraboliques | |
| 650 | 4 | |a Attractors (Mathematics) | |
| 650 | 4 | |a Differential equations, Parabolic | |
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Datensatz im Suchindex
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| adam_text | Contents
Preface page xv
Introduction 1
Parti Functional Analysis 9
1 Banach and Hilbert Spaces 11
1.1 Banach Spaces and Some General Topology 11
1.2 The Euclidean Space Km 12
1.3 The Spaces C and C Y of Continuous Functions 14
1.3.1 Mollification and Approximation by Smooth Functions 18
1.4 The Lp Spaces of Lebesgue Integrable Functions 20
1.4.1 Lebesgue Integration 20
1.4.2 The Lebesgue Spaces LP(Q) with 1 p oo 22
1.4.3 The Lebesgue Space L°°(fi) 28
1.4.4 The Spaces L^itt) of Locally Integrable Functions 30
1.4.5 The lp Sequence Spaces, 1 p oo 32
1.5 Hilbert Spaces 33
1.5.1 The Orthogonal Projection onto a Linear Subspace 35
1.5.2 Bases in Hilbert Spaces 36
1.5.3 Noncompactness of the Unit Ball 38
Exercises 39
Notes 41
2 Ordinary Differential Equations 42
2.1 Existence and Uniqueness A Fixed Point Method 43
2.1.1 The Contraction Mapping Theorem 44
2.1.2 Local Existence for Lipschitz / 45
vii
viii Contents
2.2 Global Existence 48
2.3 Existence but No Uniqueness An Approximation Method 49
2.3.1 The Arzela Ascoli Theorem 49
2.3.2 Local Existence for Continuous / 51
2.4 Differential Inequalities 53
2.5 Continuous Dependence on Initial Conditions 56
2.6 Conclusion 59
Exercises 59
Notes 61
3 Linear Operators 62
3.1 Bounded Linear Operators on Banach Spaces 62
3.2 Domain, Range, Kernel, and the Inverse Operator 65
3.3 The Baire Category Theorem 66
3.4 Compact Operators 68
3.5 Compact Symmetric Operators on Hilbert Spaces 72
3.6 Obtaining an Eigenbasis from a Compact Symmetric Operator 74
3.7 Unbounded Operators 79
3.8 Extensions and Closable Operators 80
3.9 Spectral Theory for Unbounded Symmetric Operators 81
3.10 Positive Operators and Their Fractional Powers 83
Exercises 85
Notes 87
4 Dual Spaces 89
4.1 The Hahn Banach Theorem 89
4.2 Examples of Dual Spaces 93
4.2.1 The Dual Space of Lp, 1 p oo 93
4.2.2 The Dual Space of lp, 1 p oo 94
4.2.3 The Dual Spaces of L and L°° 96
4.2.4 The Dual Space of / and /°° 96
4.3 Dual Spaces of Hilbert Spaces 99
4.4 Reflexive Spaces 100
4.5 Notions of Weak Convergence 101
4.5.1 Weak Convergence 101
4.5.2 Weak * Convergence 104
4.6 The Alaoglu Weak * Compactness Theorem 105
Exercises 107
Notes 107
Contents ix
5 Sobolev Spaces 109
5.1 Generalised Notions of Derivatives 109
5.1.1 The Weak Derivative 109
5.1.2 The Distribution Derivative 111
5.2 General Sobolev Spaces 114
5.2.1 Sobolev Spaces and the Closure of Differential Operators 115
5.2.2 The Hilbert Space Hk(Q) 115
5.3 Outline of the Rest of the Chapter 119
5.4 C°°(Q)isDensemHk(Q) 120
5.5 An Extension Theorem 124
5.5.1 Extending Functions in Hk(R+) 125
5.5.2 Coordinate Changes 127
5.5.3 Straightening the Boundary 128
5.5.4 Extending Functions in Hk(Q) 129
5.6 Density of C°° (Q) in Hk (Q) 131
5.7 The Sobolev Embedding Theorem Hk,Cr, and U 132
5.7.1 Integrability of Functions in Sobolev Spaces 132
5.7.2 Sobolev Spaces and Spaces of Continuous Functions 139
5.7.3 The Sobolev Embedding Theorem 142
5.8 A Compactness Theorem 143
5.9 Boundary Values 145
5.10 Sobolev Spaces of Periodic Functions 149
Exercises 152
Notes 153
Part II Existence and Uniqueness Theory 157
6 The Laplacian 159
6.1 Classical, Strong, and Weak Solutions 160
6.2 Weak Solutions of Poisson s Equation 160
6.3 Higher Regularity for the Laplacian I:
Periodic Boundary Conditions 164
6.4 Higher Regularity for the Laplacian II:
Dirichlet Boundary Conditions 168
6.4.1 A Heuristic Estimate 168
6.4.2 Difference Quotients 170
6.4.3 Interior Regularity Result 172
6.5 Boundary Regularity for the Laplacian 175
6.5.1 Regularity up to a Flat Boundary 175
6.5.2 Regularity up to a C2 Boundary 180
x Contents
6.5.3 H2k(Q) and Domains of A* 183
Exercises 185
Notes 187
7 Weak Solutions of Linear Parabolic Equations 188
7.1 Banach Space Valued Function Spaces 188
7.2 Weak Solutions of Parabolic Equations 194
7.3 The Galerkin Method: Truncated Eigenfunction Expansions 197
7.4 Weak Solutions 200
7.4.1 The Galerkin Approximations 201
7.4.2 Uniform Bounds on un in Various Spaces 202
7.4.3 Extraction of an Appropriate Subsequence 203
7.4.4 Properties of the Weak Solution 205
7.4.5 Uniqueness and Continuous Dependence on Initial Conditions 206
7.5 Strong Solutions 206
7.6 Higher Regularity: Spatial and Temporal 209
Exercises 210
Notes 212
8 Nonlinear Reaction Diffusion Equations 213
8.1 Results to Deal with the Nonlinear Term 214
8.1.1 A Compactness Theorem 214
8.1.2 A Weak Version of the Dominated Convergence Theorem 217
8.2 The Basis for the Galerkin Expansion 219
8.3 Weak Solutions 221
8.3.1 A Semidynamical System on L2(£2) 226
8.4 Strong Solutions 227
Exercises 231
Notes 232
9 The Navier Stokes Equations: Existence and Uniqueness 234
9.1 The Stokes Operator 235
9.2 The Weak Form of the Navier Stokes Equation 239
9.3 Properties of the Trilinear Form 241
9.4 Existence of Weak Solutions 244
9.5 Unique Weak Solutions in Two Dimensions 250
9.6 Existence of Strong Solutions in Two Dimensions 252
9.7 Uniqueness of 3D Strong Solutions 255
9.8 Dynamical Systems Generated by the 2D Equations 256
Contents xi
Exercises 256
Notes 257
Part III Finite Dimensional Global Attractors 259
10 The Global Attractor: Existence and General Properties 261
10.1 Semigroups 261
10.2 Dissipation 262
10.3 Limit Sets and Attractors 265
10.3.1 Limit Sets 265
10.3.2 The Global Attractor 266
10.4 A Theorem for the Existence of Global Attractors 269
10.5 An Example The Lorenz Equations 271
10.6 Structure of the Attractor 272
10.6.1 Gradient Systems and Lyapunov Functions 274
10.7 How the Attractor Determines the Asymptotic Dynamics 276
10.8 Continuity Properties of the Attractor 278
10.8.1 Upper Semicontinuity 278
10.8.2 Lower Semicontinuity 279
10.9 Conclusion 280
Exercises 281
Notes 282
11 The Global Attractor for Reaction Diffusion Equations 285
11.1 Absorbing Sets and the Attractor 285
11.1.1 An Absorbing Set in L2 286
11.1.2 An Absorbing Set in //0 287
11.1.3 The Global Attractor 290
11.2 Regularity Results 290
11.2.1 A Bound in L00 290
11.2.2 A Bound in H2(Q) 293
11.2.3 Further Regularity 295
11.3 Injectivity on A 296
11.4 A Lyapunov Functional 299
11.5 The Chaffee Infante Equation 301
11.5.1 Stationary Points 301
11.5.2 Bifurcations around the Zero State 304
Exercises 306
Notes 307
xii Contents
12 The Global Attractor for the Navier Stokes Equations 309
12.1 2D Navier Stokes Equations 309
12.1.1 An Absorbing Set in L2 310
12.1.2 An Absorbing Set in H1 311
12.1.3 An Absorbing Set in H2 313
12.1.4 Comparison of the Attractors in H and V
and Further Regularity Results 315
12.1.5 Injectivity on the Attractor 316
12.2 The 3D Navier Stokes Equations 317
12.2.1 An Absorbing Set in V 318
12.2.2 An Absorbing Set in D(A) and a Global Attractor 322
12.3 Conclusion 323
Exercises 323
Notes 324
13 Finite Dimensional Attractors: Theory and Examples 325
13.1 Measures of Dimension 326
13.1.1 The Fractal Dimension 326
13.1.2 The Hausdorff Dimension 330
13.1.3 Hausdorff versus Fractal Dimension 334
13.2 Bounding the Attractor Dimension Dynamically 336
13.3 Example I: The Reaction Diffusion Equation 343
13.3.1 Uniform Differentiability 343
13.3.2 A Bound on the Attractor Dimension 346
13.4 Example II: The 2D Navier Stokes Equations 347
13.4.1 Uniform Differentiability 347
13.4.2 A Bound on the Attractor Dimension 349
13.5 Physical Interpretation of the Attractor Dimension 350
13.6 Conclusion 352
Exercises 352
Notes 354
Part IV Finite Dimensional Dynamics 357
14 The Squeezing Property: Determining Modes 359
14.1 The Squeezing Property 359
14.2 An Approximate Manifold Structure for A 360
14.3 Determining Modes 363
14.4 The Squeezing Property for Reaction Diffusion Equations 365
14.5 The 2D Navier Stokes Equations 369
Contents xiii
14.5.1 Checking the Squeezing Property 369
14.5.2 Approximate Inertial Manifolds 371
14.6 Finite Dimensional Exponential Attractors 374
14.7 Conclusion 379
Exercises 379
Notes 383
15 The Strong Squeezing Property: Inertial Manifolds 385
15.1 Inertial Manifolds and Slaving 385
15.2 A Geometric Existence Proof 388
15.2.1 The Strong Squeezing Property 388
15.2.2 The Existence Proof 391
15.3 Finding Conditions for the Strong Squeezing Property 394
15.4 Inertial Manifolds for Reaction Diffusion Equations 396
15.4.1 Preparing the Equation 396
15.4.2 Checking the Spectral Gap Condition 399
15.4.3 Extensions to Other Domains and Higher Dimensions 400
15.5 More General Conditions for the Strong Squeezing Property 401
15.5.1 Inertial Manifolds and the Navier Stokes Equations 401
15.6 Conclusion 403
Exercises 403
Notes 404
16 A Direct Approach 406
16.1 Parametrising the Attractor 407
16.1.1 Experimental Measurements as Parameters 411
16.2 An Extension Theorem 411
16.3 Embedding the Dynamics Without Uniqueness 412
16.3.1 Continuity of F on A for the Scalar
Reaction Diffusion Equation 414
16.3.2 Continuity of F on A for the 2D Navier Stokes Equations 415
16.4 A Discrete Time Utopian Theorem 415
16.4.1 The Topology of Global Attractors 416
16.4.2 The within e Discrete Utopian Theorem 419
16.5 Conclusion 422
Exercises 422
Notes 424
17 The Kuramoto Sivashinsky Equation 426
17.1 Preliminaries 426
xiv Contents
17.2 Existence and Uniqueness of Solutions 428
17.3 Absorbing Sets and the Global Attractor 429
17.4 The Attractor is Finite Dimensional 431
17.5 Inertial Manifolds 432
Notes 433
Appendix A Sobolev Spaces of Periodic Functions 435
A.I The Sobolev Embedding Theorem Hs, Cr, and Lp 435
A.I.I Conditions for HS(Q) C C°(Q) 435
A. 1.2 Integrability Properties of Functions in Hs 436
A.2 Rellich Kondrachov Compactness Theorem 437
Appendix B Bounding the Fractal Dimension Using the Decay
of Volume Elements 439
References 445
Index 453
|
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| id | DE-604.BV025258433 |
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| indexdate | 2024-07-09T22:29:51Z |
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| isbn | 0521635640 |
| language | English |
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| spelling | Robinson, James Cooper Verfasser aut Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors James C. Robinson 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2001 XVII, 461 S. txt rdacontent n rdamedia nc rdacarrier Cambridge texts in applied mathematics Attracteurs (Mathématiques) Dynamische systemen gtt Partiële differentiaalvergelijkingen gtt Équations différentielles paraboliques Attractors (Mathematics) Differential equations, Parabolic Attraktor (DE-588)4140563-8 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Dissipatives System (DE-588)4209641-8 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s Dissipatives System (DE-588)4209641-8 s Attraktor (DE-588)4140563-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019894843&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Robinson, James Cooper Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors Attracteurs (Mathématiques) Dynamische systemen gtt Partiële differentiaalvergelijkingen gtt Équations différentielles paraboliques Attractors (Mathematics) Differential equations, Parabolic Attraktor (DE-588)4140563-8 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Dissipatives System (DE-588)4209641-8 gnd |
| subject_GND | (DE-588)4140563-8 (DE-588)4173245-5 (DE-588)4209641-8 |
| title | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors |
| title_auth | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors |
| title_exact_search | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors |
| title_full | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors James C. Robinson |
| title_fullStr | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors James C. Robinson |
| title_full_unstemmed | Infinite-dimensional dynamical systems an introduction to dissipative parabolic PDEs and the theory of global attractors James C. Robinson |
| title_short | Infinite-dimensional dynamical systems |
| title_sort | infinite dimensional dynamical systems an introduction to dissipative parabolic pdes and the theory of global attractors |
| title_sub | an introduction to dissipative parabolic PDEs and the theory of global attractors |
| topic | Attracteurs (Mathématiques) Dynamische systemen gtt Partiële differentiaalvergelijkingen gtt Équations différentielles paraboliques Attractors (Mathematics) Differential equations, Parabolic Attraktor (DE-588)4140563-8 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd Dissipatives System (DE-588)4209641-8 gnd |
| topic_facet | Attracteurs (Mathématiques) Dynamische systemen Partiële differentiaalvergelijkingen Équations différentielles paraboliques Attractors (Mathematics) Differential equations, Parabolic Attraktor Parabolische Differentialgleichung Dissipatives System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019894843&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT robinsonjamescooper infinitedimensionaldynamicalsystemsanintroductiontodissipativeparabolicpdesandthetheoryofglobalattractors |