Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Math. Soc.
2006
|
| Series: | Zurich lectures in advanced mathematics
|
| Subjects: | |
| Online Access: | Table of contents only Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [88]-92) and index |
| Physical Description: | IX, 92 S. Ill. |
| ISBN: | 3037190213 9783037190210 |
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MARC
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| 100 | 1 | |a Kuksin, Sergej B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |c Sergei B. Kuksin |
| 264 | 1 | |a Zürich |b European Math. Soc. |c 2006 | |
| 300 | |a IX, 92 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Zurich lectures in advanced mathematics | |
| 500 | |a Includes bibliographical references (p. [88]-92) and index | ||
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Navier-Stokes equations | |
| 650 | 4 | |a Hydrodynamics | |
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| 776 | 0 | 8 | |i Erscheint auch als |a Kuksin, Sergej B. |t Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |n Online-Ausgabe |z 978-3-03719-521-5 |w (DE-604)BV036752996 |
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Record in the Search Index
| _version_ | 1804137959326220288 |
|---|---|
| adam_text | Contents
0
Introduction
.................................
vii
1
Function spaces
............................... 1
1.1
Function spaces for functions of
z
.................. 1
1.2
Functions of
t
and
x
......................... 3
2
The deterministic 2D Navier-Stokes Equation
.............. 5
2.1
Leray decomposition
......................... 5
2.2
Properties of the nonlinearity
В
................... 8
2.3
The existence and uniqueness theorem
............... 10
2.4
Improving the smoothness of solutions
............... 14
2.5
The NS semigroup
.......................... 18
2.6
Singular forces
............................ 19
2.7
Some hydrodynamical terminology
................. 22
3
Random kick-forces
............................. 24
3.1
Ingredients for the constructions
.................. 24
3.2
The kicked NSE
............................ 25
3.3
Stationary measures
......................... 27
3.4
More estimates
............................ 28
4
White-forced equations
........................... 30
4.1
White in time forces
......................... 30
4.2
The white-forced 2D NSE
...................... 31
4.3
Estimates for solutions
........................ 33
4.4
Stationary measures
......................... 36
4.5
High-frequency random kicks
.................... 37
5
Preliminaries from measure theory
.................... 39
5.1
Weak convergence of measures and Lipschitz-dual distance
. ... 39
5.2
Variational distance
......................... 40
5.3
Coupling
................................ 41
5.4
Kantorovich functionals
....................... 42
6
Uniqueness of a stationary measure: kick-forces
............. 43
6.1
The main lemma
........................... 43
6.2
Weak solution of
(6.1)........................ 45
6.3
The theorem
............................. 46
6.4
Corollaries from the theorem
.................... 50
6.5 3D
XSE with small random kicks
.................. 51
6.6
Stationary measures and random attractors
............ 52
6.7
Appendix: Summary of the proof of Theorem
6.4......... 53
vi
Contents
7
Uniqueness of a stationary measure: white-forces
............ 56
7.1
The main theorem
.......................... 56
7.2
Stationary measures for equation, perturbed
by high frequency kicks
....................... 58
8
Ergodicity and the strong law of large numbers
............. 60
9
The martingale approximation and CLT
................. 63
10
The Eulerian limit
............................. 66
10.1
White-forces, proportional to the square-root
of the viscosity
............................ 66
10.2
One negative result
.......................... 71
10.3
Other scalings
............................. 73
10.4
Discussion
............................... 74
10.5
Kicked equations
........................... 75
11
Balance relations for the white-forced NSE
............... 77
11.1
The balance relations
......................... 77
11.2
The co-area form of the balance relations
............. 80
12
Comments
.................................. 83
Bibliography
................................... 88
Index
....................................... 93
|
| adam_txt |
Contents
0
Introduction
.
vii
1
Function spaces
. 1
1.1
Function spaces for functions of
z
. 1
1.2
Functions of
t
and
x
. 3
2
The deterministic 2D Navier-Stokes Equation
. 5
2.1
Leray decomposition
. 5
2.2
Properties of the nonlinearity
В
. 8
2.3
The existence and uniqueness theorem
. 10
2.4
Improving the smoothness of solutions
. 14
2.5
The NS semigroup
. 18
2.6
Singular forces
. 19
2.7
Some hydrodynamical terminology
. 22
3
Random kick-forces
. 24
3.1
Ingredients for the constructions
. 24
3.2
The kicked NSE
. 25
3.3
Stationary measures
. 27
3.4
More estimates
. 28
4
White-forced equations
. 30
4.1
White in time forces
. 30
4.2
The white-forced 2D NSE
. 31
4.3
Estimates for solutions
. 33
4.4
Stationary measures
. 36
4.5
High-frequency random kicks
. 37
5
Preliminaries from measure theory
. 39
5.1
Weak convergence of measures and Lipschitz-dual distance
. . 39
5.2
Variational distance
. 40
5.3
Coupling
. 41
5.4
Kantorovich functionals
. 42
6
Uniqueness of a stationary measure: kick-forces
. 43
6.1
The main lemma
. 43
6.2
Weak solution of
(6.1). 45
6.3
The theorem
. 46
6.4
Corollaries from the theorem
. 50
6.5 3D
XSE with small random kicks
. 51
6.6
Stationary measures and random attractors
. 52
6.7
Appendix: Summary of the proof of Theorem
6.4. 53
vi
Contents
7
Uniqueness of a stationary measure: white-forces
. 56
7.1
The main theorem
. 56
7.2
Stationary measures for equation, perturbed
by high frequency kicks
. 58
8
Ergodicity and the strong law of large numbers
. 60
9
The martingale approximation and CLT
. 63
10
The Eulerian limit
. 66
10.1
White-forces, proportional to the square-root
of the viscosity
. 66
10.2
One negative result
. 71
10.3
Other scalings
. 73
10.4
Discussion
. 74
10.5
Kicked equations
. 75
11
Balance relations for the white-forced NSE
. 77
11.1
The balance relations
. 77
11.2
The co-area form of the balance relations
. 80
12
Comments
. 83
Bibliography
. 88
Index
. 93 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:49:17Z |
| indexdate | 2024-07-09T21:20:35Z |
| institution | BVB |
| isbn | 3037190213 9783037190210 |
| language | English |
| lccn | 2007423032 |
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| physical | IX, 92 S. Ill. |
| publishDate | 2006 |
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| publisher | European Math. Soc. |
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| series2 | Zurich lectures in advanced mathematics |
| spelling | Kuksin, Sergej B. Verfasser aut Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Sergei B. Kuksin Zürich European Math. Soc. 2006 IX, 92 S. Ill. txt rdacontent n rdamedia nc rdacarrier Zurich lectures in advanced mathematics Includes bibliographical references (p. [88]-92) and index Differential equations, Partial Differentiable dynamical systems Navier-Stokes equations Hydrodynamics Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Navier-Stokes-Gleichung (DE-588)4041456-5 gnd rswk-swf Navier-Stokes-Gleichung (DE-588)4041456-5 s Strömungsmechanik (DE-588)4077970-1 s DE-604 Erscheint auch als Kuksin, Sergej B. Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Online-Ausgabe 978-3-03719-521-5 (DE-604)BV036752996 http://www.loc.gov/catdir/toc/fy0713/2007423032.html Table of contents only Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698416&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kuksin, Sergej B. Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Differential equations, Partial Differentiable dynamical systems Navier-Stokes equations Hydrodynamics Strömungsmechanik (DE-588)4077970-1 gnd Navier-Stokes-Gleichung (DE-588)4041456-5 gnd |
| subject_GND | (DE-588)4077970-1 (DE-588)4041456-5 |
| title | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |
| title_auth | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |
| title_exact_search | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |
| title_exact_search_txtP | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |
| title_full | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Sergei B. Kuksin |
| title_fullStr | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Sergei B. Kuksin |
| title_full_unstemmed | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Sergei B. Kuksin |
| title_short | Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions |
| title_sort | randomly forced nonlinear pdes and statistical hydrodynamics in 2 space dimensions |
| topic | Differential equations, Partial Differentiable dynamical systems Navier-Stokes equations Hydrodynamics Strömungsmechanik (DE-588)4077970-1 gnd Navier-Stokes-Gleichung (DE-588)4041456-5 gnd |
| topic_facet | Differential equations, Partial Differentiable dynamical systems Navier-Stokes equations Hydrodynamics Strömungsmechanik Navier-Stokes-Gleichung |
| url | http://www.loc.gov/catdir/toc/fy0713/2007423032.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698416&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kuksinsergejb randomlyforcednonlinearpdesandstatisticalhydrodynamicsin2spacedimensions |