The Navier-Stokes equations: an elementary functional analytic approach
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2001
|
| Series: | Birkhäuser advanced texts
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 355 - 363 Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | X, 367 S. 26 cm |
| ISBN: | 3764365455 9783034805506 |
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MARC
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| 100 | 1 | |a Sohr, Hermann |e Verfasser |0 (DE-588)106421379 |4 aut | |
| 245 | 1 | 0 | |a The Navier-Stokes equations |b an elementary functional analytic approach |c Hermann Sohr |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2001 | |
| 300 | |a X, 367 S. |b 26 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Birkhäuser advanced texts | |
| 500 | |a Literaturverzeichnis Seite 355 - 363 | ||
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Navier-Stokes, Équations de | |
| 650 | 7 | |a Navier-Stokes-vergelijkingen |2 gtt | |
| 650 | 4 | |a Navier-Stokes equations | |
| 650 | 0 | 7 | |a Navier-Stokes-Gleichung |0 (DE-588)4041456-5 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-009461206 | ||
Record in the Search Index
| _version_ | 1804128664616435712 |
|---|---|
| adam_text | Contents
Preface
..................................... ix
I Introduction
1
Basic notations
.............................. 1
1.1
The equations of Navier-Stokes
.................. 1
1.2
Further notations
.......................... 3
1.3
Linearized equations
........................ 6
2
Description of the functional analytic approach
............ 7
2.1
The role of the Stokes operator A
................. 7
2.2
The stationary linearized case
................... 10
2.3
The stationary nonlinear case
................... 11
2.4
The nonstationary linearized case
................. 13
2.5
The full nonlinear case
....................... 16
3
Function spaces
............................. 22
3.1
Smooth functions
.......................... 22
3.2
Smoothness properties of the boundary
ΘΩ
........... 25
3.3
¿«-spaces
.............................. 27
3.4
The boundary spaces Lfl{du)
................... 30
3.5
Distributions
............................ 34
3.6
Sobolev spaces
........................... 38
II Preliminary Results
1
Embedding properties and related facts
................ 43
1.1
Poincaré
inequalities
........................ 43
1.2
Traces and Green s formula
.................... 47
1.3
Embedding properties
....................... 52
1.4
Decomposition of domains
..................... 55
1.5
Compact embeddings
........................ 56
1.6
Representation of functionals
................... 61
1.7
Mollification method
........................ 64
yj Contents
2
The operators V and
div
........................ 67
2.1
Solvability of
div v
=
g
and Vp =
ƒ................ 67
2.2
A criterion for gradients
...................... 72
2.3
Regularity results on
div
υ
=
g
.................. 78
2.4
Further results on the equation
div v
=
g
............. 79
2.5
Helmholtz decomposition in L2-spaces
.............. 81
3
Elementary functional analytic properties
............... 89
3.1
Basic facts on Banach spaces
................... 89
3.2
Basic facts on Hilbert spaces
................... 93
3.3
The Laplace operator
Δ
......................100
3.4
Resolvent and Yosida approximation
...............104
III The Stationary Navier-Stokes Equations
1
Weak solutions of the Stokes equations
................107
1.1
The notion of weak solutions
...................107
1.2
Embedding properties of Wffiu)
.................110
1.3
Existence of weak solutions
....................112
1.4
The nonhomogeneous case
div u
=
g
...............114
1.5
Regularity properties of weak solutions
..............116
2
The Stokes operator A
.........................127
2.1
Definition and properties
......................127
2.2
The square root A? of A
......................132
2.3
The Stokes operator A in Rn
...................135
2.4
Embedding properties of D(Aa)
..................141
2.5
Completion of the space D{Aa)
..................146
2.6
The operator
A~iPåiv
......................153
3
The stationary Navier-Stokes equations
................157
3.1
Weak solutions
...........................157
3.2
The nonlinear term
и
■
Vu.....................
159
3.3
The associated pressure
ρ
.....................163
3.4
Existence of weak solutions in bounded domains
........165
3.5
Existence of weak solutions in unbounded domains
.......168
3.6
Regularity properties for the stationary nonlinear system
.... 173
3.7
Some uniqueness results
......................178
Contents
vii
IV The Linearized Nonstationary Theory
1
Preliminaries for the time dependent linear theory
..........185
1.1
The nonstationary Stokes system
.................185
1.2
Basic spaces for the time dependent theory
...........186
1.3
The vector valued operator
J¿
...................191
1.4
Time dependent gradients Vp
...................198
1.5
A special solution class of the homogeneous system
.......203
1.6
The inhomogeneous evolution equation u
4-
Au
= ƒ......212
2
Theory of weak solutions in the linearized case
............219
2.1
Weak solutions
...........................219
2.2
Equivalent formulation and approximation
............221
2.3
Energy equality and strong continuity
..............225
2.4
Representation formula for weak solutions
............230
2.5
Basic estimates of weak solutions
.................237
2.6
Associated pressure of weak solutions
...............246
2.7
Regularity properties of weak solutions
..............253
V The Full Nonlinear Navier-Stokes Equations
1
Weak solutions
..............................261
1.1
Definition of weak solutions
.................... 261
1.2
Properties of the nonlinear term
и
-Vu..............
265
1.3
Integral equation for weak solutions and weak continuity
.... 270
1.4
Energy equality and strong continuity
.............. 272
1.5
Serrin s uniqueness condition
................... 276
1.6
Integrability properties of weak solutions in space and time,
the scale of Serrin s quantity
....................282
1.7
Associated pressure of weak solutions
...............295
1.8
Regularity properties of weak solutions
..............296
2
Approximation of the Navier-Stokes equations
............305
2.1
Approximate Navier-Stokes system
................ 305
2.2
Properties of approximate weak solutions
............ 307
2.3
Regularity properties of approximate weak solutions
...... 311
2.4
Smooth solutions of the Navier-Stokes equations
with slightly modified forces
..................312
2.5
Existence of approximate weak solutions
.............315
2.6
Uniform norm bounds of approximate weak solutions
......318
viii Contents
3
Existence
of weak solutions of the Navier-Stokes system
....... 320
3.1
Main result
............................. 320
3.2
Preliminary compactness results
................. 323
3.3
Proof of Theorem
3.1.1....................... 329
3.4
Weighted energy inequalities and time decay
........... 334
3.5
Exponential decay for domains for which the
Poincaré
inequality holds
..................... 336
3.6
Generalized energy inequality
................... 339
4
Strong solutions of the Navier-Stokes system
............. 343
4.1
The notion of strong solutions
................... 343
4.2
Existence results
.......................... 344
Bibliography
.................................. 355
Index
...................................... 365
|
| any_adam_object | 1 |
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.353 |
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| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
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| id | DE-604.BV013832931 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:52:50Z |
| institution | BVB |
| isbn | 3764365455 9783034805506 |
| language | English |
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| physical | X, 367 S. 26 cm |
| publishDate | 2001 |
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| publisher | Birkhäuser |
| record_format | marc |
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| spelling | Sohr, Hermann Verfasser (DE-588)106421379 aut The Navier-Stokes equations an elementary functional analytic approach Hermann Sohr Basel [u.a.] Birkhäuser 2001 X, 367 S. 26 cm txt rdacontent n rdamedia nc rdacarrier Birkhäuser advanced texts Literaturverzeichnis Seite 355 - 363 Hier auch später erschienene, unveränderte Nachdrucke Navier-Stokes, Équations de Navier-Stokes-vergelijkingen gtt Navier-Stokes equations Navier-Stokes-Gleichung (DE-588)4041456-5 gnd rswk-swf Navier-Stokes-Gleichung (DE-588)4041456-5 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009461206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sohr, Hermann The Navier-Stokes equations an elementary functional analytic approach Navier-Stokes, Équations de Navier-Stokes-vergelijkingen gtt Navier-Stokes equations Navier-Stokes-Gleichung (DE-588)4041456-5 gnd |
| subject_GND | (DE-588)4041456-5 |
| title | The Navier-Stokes equations an elementary functional analytic approach |
| title_auth | The Navier-Stokes equations an elementary functional analytic approach |
| title_exact_search | The Navier-Stokes equations an elementary functional analytic approach |
| title_full | The Navier-Stokes equations an elementary functional analytic approach Hermann Sohr |
| title_fullStr | The Navier-Stokes equations an elementary functional analytic approach Hermann Sohr |
| title_full_unstemmed | The Navier-Stokes equations an elementary functional analytic approach Hermann Sohr |
| title_short | The Navier-Stokes equations |
| title_sort | the navier stokes equations an elementary functional analytic approach |
| title_sub | an elementary functional analytic approach |
| topic | Navier-Stokes, Équations de Navier-Stokes-vergelijkingen gtt Navier-Stokes equations Navier-Stokes-Gleichung (DE-588)4041456-5 gnd |
| topic_facet | Navier-Stokes, Équations de Navier-Stokes-vergelijkingen Navier-Stokes equations Navier-Stokes-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009461206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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