Infinite-dimensional dynamical systems in mechanics and physics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2014
|
| Edition: | [softcover repr. of the hardcover 2. ed. 1998] |
| Series: | Applied mathematical sciences
68 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 648 S. Ill., graph. Darst. |
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MARC
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| 100 | 1 | |a Temam, Roger |d 1940- |e Verfasser |0 (DE-588)1024798135 |4 aut | |
| 245 | 1 | 0 | |a Infinite-dimensional dynamical systems in mechanics and physics |
| 250 | |a [softcover repr. of the hardcover 2. ed. 1998] | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2014 | |
| 300 | |a XXI, 648 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 68 | |
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| 650 | 0 | 7 | |a Unendlichdimensionales System |0 (DE-588)4207956-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differenzierbares dynamisches System |0 (DE-588)4137931-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Evolutionsgleichung |0 (DE-588)4129061-6 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
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Record in the Search Index
| _version_ | 1804152918075506688 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
vii
Preface to the First Edition
ix
GENERAL INTRODUCTION.
The User s Guide
1
Introduction
1
1.
Mechanism and Description of Chaos. The Finite-Dimensional Case
2
2.
Mechanism and Description of Chaos. The Infinite-Dimensional Case
6
3.
The Global Attractor. Reduction to Finite Dimension
10
4.
Remarks on the Computational Aspect
12
5.
The User s Guide
13
CHAPTER I
General Results and Concepts on Invariant Sets and Attractors
15
Introduction
15
1.
Semigroups, Invariant Sets, and Attractors
16
1
.1.
Semigroups of Operators
16
1.2.
Functional Invariant Sets
18
1.3.
Absorbing Sets and Attractors
20
1.4.
A Remark on the Stability of the Attractors
28
2.
Examples in Ordinary Differential Equations
29
2.1.
The Pendulum
29
2.2.
The
Minea
System
32
2.3.
The
Lorenz
Model
34
3.
Fractal Interpolation and Attractors
36
3.1.
The General Framework
37
3.2.
The Interpolation Process
38
3.3.
Proof of Theorem
3.1 40
xvi Contents
CHAPTER II
Elements
of Functional Analysis
43
Introduction
43
1.
Function Spaces
43
1.1.
Definition of the Spaces. Notations
43
1.2.
Properties of Sobolev Spaces
45
1.3.
Other Sobolev Spaces
49
1.4.
Further Properties of Sobolev Spaces
51
2.
Linear Operators
53
2,
Í.
Bilinear Forms and Linear Operators
54
2.2.
Concrete Examples of Linear Operators
58
3.
Linear Evolution Equations of the First Order in Time
68
3.1.
Hypotheses
68
3.2.
A Result of Existence and Uniqueness
70
3.3.
Regularity Results
71
3.4.
Time-Dependent Operators
74
4.
Linear Evolution Equations of the Second Order in Time
76
4.1.
The Evolution Problem
76
4.2.
Another Result
79
4.3.
Time-Dependent Operators
80
CHAPTER III
Attractors of the Dissipative Evolution Equation of the First Order
in Time: Reaction-Diffusion Equations. Fluid Mechanics and
Pattern Formation Equations
82
Introduction
82
1.
Reaction-Diffusion Equations
83
1.1.
Equations with a Polynomial Nonlinearity
84
1.2.
Equations with an Invariant Region
93
2.
Navier-Stokes Equations
(« = 2) 104
2.1.
The Equations and Their Mathematical Setting
105
2.2.
Absorbing Sets and Attractors
109
2.3.
Proof of Theorem
2.1
ИЗ
3.
Other Equations in Fluid Mechanics
115
3.1.
Abstract Equation. General Results
115
3.2.
Fluid Driven by Its Boundary
118
3.3.
Magnetohydrodynamics
(MHD)
123
3.4.
Geophysical Flows (Flows on a Manifold)
127
3.5.
Thermohydraulics
í
33
4.
Some Pattern Formation Equations
141
4.1.
The Kuramoto-Sivashinsky Equation
141
4.2.
The Cahn-Hilliard Equation
151
5. Semilinear
Equations
162
5.1.
The Equations. The Semigroup
162
5.2.
Absorbing Sets and Attractors
167
5.3.
Proof of Theorem
5.2 170
Contents xvii
6.
Backward Uniqueness
171
6.1.
An Abstract Result
172
6.2.
Applications
175
CHAPTER IV
Attractors of Dissipative Wave Equations
179
Introduction
179
1.
Linear Equations: Summary and Additional Results
180
1.1.
The General Framework
181
1.2.
Exponential Decay
183
1.3.
Bounded Solutions on the Real Line
186
2.
The Sine-Gordon Equation
188
2.1.
The Equation and Its Mathematical Setting
189
2.2.
Absorbing Sets and Attractors
191
2.3.
Other Boundary Conditions
196
3.
A Nonlinear Wave Equation of Relativistic Quantum Mechanics
202
3.1.
The Equation and Its Mathematical Setting
202
3.2.
Absorbing Sets and Attractors
206
4.
An Abstract Wave Equation
212
4.1.
The Abstract Equation. The Group of Operators
212
4.2.
Absorbing Sets and Attractors
215
4.3.
Examples
220
4.4.
Proof of Theorem
4.1
(Sketch)
224
5.
The Ginzburg-Landau Equation
226
5.1.
The Equations and Its Mathematical Setting
227
5.2.
Absorbing Sets and Attractors
230
6.
Weakly Dissipative Equations. I. The Nonlinear Schrodinger Equation
234
6.1.
The Nonlinear Schrodinger Equation
235
6.2.
Existence and Uniqueness of Solution. Absorbing Sets
236
6.3.
Decomposition of the Semigroup
239
6.4.
Comparison of
z
and
Z
for Large Times
250
6.5.
Application to the Attractor. The Main Result
252
6.6.
Determining Modes
254
7.
Weakly Dissipative Equations II. The Korteweg-De
Vries
Equation
256
7.1.
The Equation and its Mathematical Setting
257
7.2.
Absorbing Sets and Attractors
260
7.3.
Regularity of the Attractor
269
7.4.
Proof of the Results in Section
7.1 272
7.5.
Proof of Proposition
7.2 290
8.
Unbounded Case: The Lack of Compactness
306
8.1.
Preliminaries
307
8.2.
The Global Attractor
312
9.
Regularity of Attractors
316
9.1.
A Preliminary Result
317
9.2.
Example of Partial Regularity
322
9.3.
Example of
<žř°°
Regularity
324
10.
Stability of Attractors
329
xviii Contents
CHAPTER V
Lyapunov
Exponents and Dimension of Attractors
335
Introduction
335
1. Linear and Multilinear Algebra
336
1.1.
Exterior Product of Hubert Spaces
336
1.2.
Multilinear Operators and Exterior Products
340
1.3.
Image of a Ball by a Linear Operator
347
2.
Lyapunov Exponents and Lyapunov Numbers
355
2.1.
Distortion of Volumes Produced by the Semigroup
355
2.2.
Definition of the Lyapunov Exponents and Lyapunov Numbers
357
2.3.
Evolution of the Volume Element and Its Exponential Decay:
The Abstract Framework
362
3.
Hausdorff and Fractal Dimensions of Attractors
365
3.1.
Hausdorff and Fractal Dimensions
365
3.2.
Covering Lemmas
367
3.3.
The Main Results
368
3.4.
Application to Evolution Equations
377
CHAPTER VI
Explicit Bounds on the Number of Degrees of Freedom and the
Dimension of Attractors of Some Physical Systems
380
Introduction
380
1.
The
Lorenz Attractor 381
2.
Reaction-Diffusion Equations
385
2.1.
Equations with a Polynomial Nonlinearity
386
2.2.
Equations with an Invariant Region
392
3.
Navier-Stokes Equations (n
= 2) 397
3.1.
General Boundary Conditions
398
3.2.
Improvements for the Space-Periodic Case
404
4.
Other Equations in Fluid Mechanics
412
4.1.
The Linearized Equations (The Abstract Framework)
412
4.2.
Fluid Driven by Its Boundary
413
4.3.
Magnetohydrodynamics
420
4.4.
Flows on a Manifold
425
4.5.
Thermohydraulics
430
5.
Pattern Formation Equations
434
5.1.
The Kuramoto-Sivashinsky Equation
435
5.2.
The Cahn-Hilliard Equations
441
6.
Dissipative Wave Equations
446
6.1.
The Linearized Equation
447
6.2.
Dimension of the Attractor
450
6.3.
Sine-Gordon Equations
453
6.4.
Some Lemmas
454
7.
The Ginzburg-Landau Equation
456
7.1.
The Linearized Equation
456
7.2.
Dimension of the Attractor
457
8.
Differentiability of the Semigroup
46
í
Contents xix
CHAPTER
VII
Non-
Well-Posed Problems, Unstable Manifolds, Lyapunov
Functions, and Lower Bounds on Dimensions
465
Introduction
465
PART A: Non-Well-Posed Problems
466
1.
Dissipativity and Well Posedness
466
1.1.
General Definitions
466
1.2.
The Class of Problems Studied
467
1.3.
The Main Result
471
2.
Estimate of Dimension for
Non-
Well-Posed Problems:
Examples in Fluid Dynamics
475
2.1.
The Equations and Their Linearization
476
2.2.
Estimate of the Dimension of X
411
2.3.
The Three-Dimensional Navier-Stokes Equations
479
PART B: Unstable Manifolds, Lyapunov Functions, and Lower
Bounds on Dimensions
482
3.
Stable and Unstable Manifolds
482
3.1.
Structure of a Mapping in the Neighborhood of a Fixed Point
483
3.2.
Application to Attractors
485
3.3.
Unstable Manifold of a Compact Invariant Set
489
4.
The Attractor of a Semigroup with a Lyapunov Function
490
4.1.
A General Result
490
4.2.
Additional Results
492
4.3.
Examples
495
5.
Lower Bounds on Dimensions of Attractors: An Example
496
CHAPTER
VIII
The Cone and Squeezing Properties.
Inerţial
Manifolds
498
Introduction
498
1.
The Cone Property
499
1.1.
The Cone Property
499
1.2.
Generalizations
502
1.3.
The Squeezing Property
504
2.
Construction of an
Inerţial
Manifold: Description of the Method
505
2.1.
Inerţial
Manifolds: The Method of Construction
505
2.2.
The Initial and Prepared Equations
506
2.3.
The Mapping ZT
509
3.
Existence of an Inertia! Manifold
512
3.1.
The Result of Existence
513
3.2.
First Properties of 3~
514
3.3.
Utilization of the Cone Property
516
3.4.
Proof of Theorem
3.1
(End)
522
3.5.
Another Form of Theorem
3.1 525
4.
Examples
526
4.1.
Example
1:
The Kuramoto-Sivashinsky Equation
526
xx Contents
4.2.
Example
2:
Approximate Inertia! Manifolds for the
Navier-Stokes Equations
528
4.3.
Example
3:
Reaction—Diffusion Equations
530
4.4.
Example
4:
The Ginzburg-Landau Equation
531
5.
Approximation and Stability of the Inertia! Manifold with
Respect to Perturbations
532
CHAPTER IX
Inerţial
Manifolds and Slow Manifolds. The Non-Self-Adjoint Case
536
Introduction
536
1.
The Functional Setting
537
1.1.
Notations and Hypotheses
537
1.2.
Construction of the
Inerţial
Manifold
539
2.
The Main Result (Lipschitz Case)
541
2.
Í.
Existence of Inertia! Manifolds
541
2.2.
Properties of
ЗГ
542
2.3.
Smoothness Property of
Φ (Φ
is
^1)
548
2.4.
Proof of Theorem
2.
1
550
3.
Complements and
Applications
553
3.1.
The Locally Lipschitz Case
553
3.2.
Dimension of the Inertia! Manifold
555
4.
Inerţial
Manifolds and Slow Manifolds
559
4.1.
The Motivation
559
4.2.
The Abstract Equation
560
4.3.
An Equation of Navier-Stokes Type
562
CHAPTER X
Approximation of Attractors and
Inerţial
Manifolds.
Convergent Families of Approximate inertia! Manifolds
565
Introduction
565
1.
Construction of the Manifolds
566
1.1.
Approximation of the Differential Equation
566
1.2.
The Approximate Manifolds
569
2.
Approximation of Attractors
571
2.1.
Properties of
-3~¿
571
2.2.
Distance to the Attractor
573
2.3.
The Main Result
576
3.
Convergent Families of Approximate
Inerţial
Manifolds
578
3.1.
Properties of
&¿
579
3.2.
Distance to the Exact
Inerţial
Manifold
581
3.3.
Convergence to the Exact Inertia! Manifold
583
APPENDIX
Collective Sobolev Inequalities
585
Introduction
585
1.
Notations and Hypotheses
586
1.
í
.
The Operator
U
586
í.
2.
The
Schrödinger-Type
Operators
588
Contents xxi
2.
Spectral
Estimates for
Schrödinger-Type
Operators
590
2.1.
The Birman-Schwinger Inequality
590
2.2.
The Spectral Estimate
593
3.
Generalization of the Sobolev-Lieb-Thirring Inequality (I)
596
4.
Generalization of the Sobolev Lieb-Thirring Inequality (II)
602
4.1.
The Space-Periodic Case
603
4.2.
The General Case
605
4.3.
Proof of Theorem
4.1 607
5.
Examples
610
Bibliography
613
Index
645
|
| any_adam_object | 1 |
| author | Temam, Roger 1940- |
| author_GND | (DE-588)1024798135 |
| author_facet | Temam, Roger 1940- |
| author_role | aut |
| author_sort | Temam, Roger 1940- |
| author_variant | r t rt |
| building | Verbundindex |
| bvnumber | BV042319902 |
| ctrlnum | (OCoLC)903231457 (DE-599)BVBBV042319902 |
| edition | [softcover repr. of the hardcover 2. ed. 1998] |
| format | Book |
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| id | DE-604.BV042319902 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:18:20Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027756853 |
| oclc_num | 903231457 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XXI, 648 S. Ill., graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Temam, Roger 1940- Verfasser (DE-588)1024798135 aut Infinite-dimensional dynamical systems in mechanics and physics [softcover repr. of the hardcover 2. ed. 1998] New York [u.a.] Springer 2014 XXI, 648 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 68 Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Unendlichdimensionales System (DE-588)4207956-1 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Evolutionsgleichung (DE-588)4129061-6 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 s Evolutionsgleichung (DE-588)4129061-6 s DE-604 Unendlichdimensionales System (DE-588)4207956-1 s Dynamisches System (DE-588)4013396-5 s Randwertproblem (DE-588)4048395-2 s 1\p DE-604 Differenzierbares dynamisches System (DE-588)4137931-7 s 2\p DE-604 Applied mathematical sciences 68 (DE-604)BV000005274 68 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027756853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Temam, Roger 1940- Infinite-dimensional dynamical systems in mechanics and physics Applied mathematical sciences Nichtlineares dynamisches System (DE-588)4126142-2 gnd Unendlichdimensionales System (DE-588)4207956-1 gnd Dynamisches System (DE-588)4013396-5 gnd Randwertproblem (DE-588)4048395-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Evolutionsgleichung (DE-588)4129061-6 gnd |
| subject_GND | (DE-588)4126142-2 (DE-588)4207956-1 (DE-588)4013396-5 (DE-588)4048395-2 (DE-588)4137931-7 (DE-588)4129061-6 |
| title | Infinite-dimensional dynamical systems in mechanics and physics |
| title_auth | Infinite-dimensional dynamical systems in mechanics and physics |
| title_exact_search | Infinite-dimensional dynamical systems in mechanics and physics |
| title_full | Infinite-dimensional dynamical systems in mechanics and physics |
| title_fullStr | Infinite-dimensional dynamical systems in mechanics and physics |
| title_full_unstemmed | Infinite-dimensional dynamical systems in mechanics and physics |
| title_short | Infinite-dimensional dynamical systems in mechanics and physics |
| title_sort | infinite dimensional dynamical systems in mechanics and physics |
| topic | Nichtlineares dynamisches System (DE-588)4126142-2 gnd Unendlichdimensionales System (DE-588)4207956-1 gnd Dynamisches System (DE-588)4013396-5 gnd Randwertproblem (DE-588)4048395-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Evolutionsgleichung (DE-588)4129061-6 gnd |
| topic_facet | Nichtlineares dynamisches System Unendlichdimensionales System Dynamisches System Randwertproblem Differenzierbares dynamisches System Evolutionsgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027756853&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT temamroger infinitedimensionaldynamicalsystemsinmechanicsandphysics |