Mathematical foundation of turbulent viscous flows: lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Series: | Lecture notes in mathematics
1871 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | IX, 252 S. Ill. |
| ISBN: | 3540285865 9783540285861 |
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| 245 | 1 | 0 | |a Mathematical foundation of turbulent viscous flows |b lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |c P. Constantin ... Ed.: M. Cannone ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2005 | |
| 300 | |a IX, 252 S. |b Ill. | ||
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Record in the Search Index
| _version_ | 1804135103865028608 |
|---|---|
| adam_text | Contents
Euler
Equations, Navier-Stokes Equations and Turbulence
Peter
Constantin
................................................ 1
1
Introduction
.................................................. 1
2
Euler
Equations
............................................... 2
3
An Infinite Energy Blow Up Example
............................ 10
4
Navier-Stokes Equations
........................................ 16
5
Approximations
............................................... 24
6
The QG Equation
............................................. 26
7
Dissipation and Spectra
........................................ 30
References
...................................................... 39
CKN Theory of Singularities of Weak Solutions
of the Navier-Stokes Equations
Giovanni Gallavotti
.............................................. 45
1
Leray s Solutions and Energy
................................... 47
2
Kinematic Inequalities
......................................... 49
3
Pseudo
Navier
Stokes Velocity
-
Pressure Pairs. Scaling Operators
... 50
4
The Theorems of Scheffer and of Caffarelli-Kohn-Nirenberg
........ 54
5
Fractal Dimension of Singularities
of the Navier-Stokes Equation,
d
= 3 ............................ 58
5.1
Dimension and Measure of Hausdorff
........................ 58
5.2
Hausdorff Dimension of Singular Times in the Navier-Stokes
Solutions (d
= 3).......................................... 59
5.3
Hausdorff Dimension in Space-Time of the Solutions of NS,
(d
= 3)................................................... 61
6
Problems. The Dimensional Bounds of the CKN Theory
............ 63
References
...................................................... 73
Approximation of Weak Limits and Related Problems
Alexandre
V. Kazhikhov
.......................................... 75
1
Strong Approximation of Weak Limits by Averagings
.............. 76
VIII Contents
1.1
Notations and Basic Notions
from Orlicz Function Spaces Theory
76
1.2
Strong Approximation of Weak Limits
....................... 79
Step
1.
Simple example
................................. 79
Step
2.
One-dimensional case, Steklov averaging
........... 80
Step
3.
The general case
................................ 81
Remark
1.1........................................... 82
1.3
Applications to Navier-Stokes Equations
..................... 82
2
Transport Equations in Orlicz Spaces
............................ 83
2.1
Statement of Problem
...................................... 83
2.2
Existence and Uniqueness Theorems
......................... 85
2.3
Gronwall-type Inequality and Osgood Uniqueness Theorem
..... 86
2.4
Conclusive Remarks
....................................... 88
3
Some Remarks on Compensated Compactness Theory
.............. 89
3.1
Introduction
.............................................. 89
3.2
Classical Compactness
(Aubin-Simon
Theorem)
............... 91
3.3
Compensated Compactness
-
div-curl Lemma
............... 92
3.4
Compensated Compactness-theorem of L. Tartar
.............. 94
3.5
Generalizations and Examples
.............................. 96
References
...................................................... 98
Oscillating Patterns in Some Nonlinear Evolution Equations
Yves Meyer
.....................................................101
1
Introduction
..................................................101
2
A Model Case: the Nonlinear Heat Equation
......................103
3
Navier-Stokes Equations
........................................113
4
The L2-theory is Unstable
......................................118
5
T. Kato s Theorem
............................................124
6
The
Kato
Theorem Revisited by Marco
Cannone
..................127
7
The
Kato
Theory with
Lorentz
Spaces
...........................132
8
Vortex Filaments and a Theorem by Y.
Giga
and T. Miyakawa
......136
9
Vortex Patches
................................................142
10
The H. Koch
k D.
Tataru
Theorem
..............................144
11
Localized Velocity Fields
.......................................146
12
Large Time Behavior of Solutions to the Navier-Stokes Equations
... 151
13
Improved Gagliardo-Nirenberg Inequalities
........................154
14
The Space BV of Functions with Bounded Variation in the Plane
... 157
15
Gagliardo-Nirenberg Inequalities and BV
.........................161
16
Improved
Poincaré
Inequalities
..................................166
17
A Direct Proof of Theorem
15.3.................................170
18
Littlewood-Paley Analysis
......................................172
19
Littlewood-Paley Analysis and Wavelet Analysis
...................178
References
...................... .........182
Contents
IX
Asymptotic Analysis of Fluid Equations
Seiji Ukai
.......................................................189
1
Introduction
..................................................189
2
Schemes for Establishing Asymptotic Relations
....................194
2.1
From Newton Equation to Boltzmann Equation:
Boltzmann-Grad Limit
....................................194
Newton Equation
-
Hard Sphere Gas
........................195
Liouville Equation
.........................................196
BBGKY Hierarchy
........................................196
Boltzmann Hierarchy
......................................198
Boltzmann Equation
.......................................198
Collision Operator
Q
......................................200
2.2
From Boltzmann Equation to Fluid Equations
-
Multi-Scale Analysis
.......................................202
The Case
(α, β)
= (0,0):
Compressible
Euler
Equation (C.E.)
... 204
The Case a
> 0,
β
= 0.....................................205
The Case a
> 0,
β
> 0.....................................205
3
Abstract Cauchy-Kovalevskaya Theorem
.........................212
3.1
Example
1:
Pseudo
Differential Equation
.....................216
3.2
Example
2:
Local Solutions of the Boltzmann Equation
........219
4
The Boltzmann-Grad Limit
.....................................223
4.1
Integral Equations
.........................................223
4.2
Local Solutions and Uniform Estimates
......................224
4.3
Lanford s Theorem
........................................229
5
Fluid Dynamical Limits
........................................232
5.1
Preliminary
...............................................233
5.2
Main Theorems
...........................................235
5.3
Proof of Theorem
5.1......................................238
5.4
Proof of Theorems
5.2
and
5.3..............................243
References
......................................................248
|
| adam_txt |
Contents
Euler
Equations, Navier-Stokes Equations and Turbulence
Peter
Constantin
. 1
1
Introduction
. 1
2
Euler
Equations
. 2
3
An Infinite Energy Blow Up Example
. 10
4
Navier-Stokes Equations
. 16
5
Approximations
. 24
6
The QG Equation
. 26
7
Dissipation and Spectra
. 30
References
. 39
CKN Theory of Singularities of Weak Solutions
of the Navier-Stokes Equations
Giovanni Gallavotti
. 45
1
Leray's Solutions and Energy
. 47
2
Kinematic Inequalities
. 49
3
Pseudo
Navier
Stokes Velocity
-
Pressure Pairs. Scaling Operators
. 50
4
The Theorems of Scheffer and of Caffarelli-Kohn-Nirenberg
. 54
5
Fractal Dimension of Singularities
of the Navier-Stokes Equation,
d
= 3 . 58
5.1
Dimension and Measure of Hausdorff
. 58
5.2
Hausdorff Dimension of Singular Times in the Navier-Stokes
Solutions (d
= 3). 59
5.3
Hausdorff Dimension in Space-Time of the Solutions of NS,
(d
= 3). 61
6
Problems. The Dimensional Bounds of the CKN Theory
. 63
References
. 73
Approximation of Weak Limits and Related Problems
Alexandre
V. Kazhikhov
. 75
1
Strong Approximation of Weak Limits by Averagings
. 76
VIII Contents
1.1
Notations and Basic Notions
from Orlicz Function Spaces Theory
76
1.2
Strong Approximation of Weak Limits
. 79
Step
1.
Simple example
. 79
Step
2.
One-dimensional case, Steklov averaging
. 80
Step
3.
The general case
. 81
Remark
1.1. 82
1.3
Applications to Navier-Stokes Equations
. 82
2
Transport Equations in Orlicz Spaces
. 83
2.1
Statement of Problem
. 83
2.2
Existence and Uniqueness Theorems
. 85
2.3
Gronwall-type Inequality and Osgood Uniqueness Theorem
. 86
2.4
Conclusive Remarks
. 88
3
Some Remarks on Compensated Compactness Theory
. 89
3.1
Introduction
. 89
3.2
Classical Compactness
(Aubin-Simon
Theorem)
. 91
3.3
Compensated Compactness
-
"div-curl" Lemma
. 92
3.4
Compensated Compactness-theorem of L. Tartar
. 94
3.5
Generalizations and Examples
. 96
References
. 98
Oscillating Patterns in Some Nonlinear Evolution Equations
Yves Meyer
.101
1
Introduction
.101
2
A Model Case: the Nonlinear Heat Equation
.103
3
Navier-Stokes Equations
.113
4
The L2-theory is Unstable
.118
5
T. Kato's Theorem
.124
6
The
Kato
Theorem Revisited by Marco
Cannone
.127
7
The
Kato
Theory with
Lorentz
Spaces
.132
8
Vortex Filaments and a Theorem by Y.
Giga
and T. Miyakawa
.136
9
Vortex Patches
.142
10
The H. Koch
k D.
Tataru
Theorem
.144
11
Localized Velocity Fields
.146
12
Large Time Behavior of Solutions to the Navier-Stokes Equations
. 151
13
Improved Gagliardo-Nirenberg Inequalities
.154
14
The Space BV of Functions with Bounded Variation in the Plane
. 157
15
Gagliardo-Nirenberg Inequalities and BV
.161
16
Improved
Poincaré
Inequalities
.166
17
A Direct Proof of Theorem
15.3.170
18
Littlewood-Paley Analysis
.172
19
Littlewood-Paley Analysis and Wavelet Analysis
.178
References
. .182
Contents
IX
Asymptotic Analysis of Fluid Equations
Seiji Ukai
.189
1
Introduction
.189
2
Schemes for Establishing Asymptotic Relations
.194
2.1
From Newton Equation to Boltzmann Equation:
Boltzmann-Grad Limit
.194
Newton Equation
-
Hard Sphere Gas
.195
Liouville Equation
.196
BBGKY Hierarchy
.196
Boltzmann Hierarchy
.198
Boltzmann Equation
.198
Collision Operator
Q
.200
2.2
From Boltzmann Equation to Fluid Equations
-
Multi-Scale Analysis
.202
The Case
(α, β)
= (0,0):
Compressible
Euler
Equation (C.E.)
. 204
The Case a
> 0,
β
= 0.205
The Case a
> 0,
β
> 0.205
3
Abstract Cauchy-Kovalevskaya Theorem
.212
3.1
Example
1:
Pseudo
Differential Equation
.216
3.2
Example
2:
Local Solutions of the Boltzmann Equation
.219
4
The Boltzmann-Grad Limit
.223
4.1
Integral Equations
.223
4.2
Local Solutions and Uniform Estimates
.224
4.3
Lanford's Theorem
.229
5
Fluid Dynamical Limits
.232
5.1
Preliminary
.233
5.2
Main Theorems
.235
5.3
Proof of Theorem
5.1.238
5.4
Proof of Theorems
5.2
and
5.3.243
References
.248 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T13:53:36Z |
| indexdate | 2024-07-09T20:35:11Z |
| institution | BVB |
| isbn | 3540285865 9783540285861 |
| language | English |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 P. Constantin ... Ed.: M. Cannone ... Berlin [u.a.] Springer 2005 IX, 252 S. Ill. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1871 Turbulente Strömung (DE-588)4117265-6 gnd rswk-swf Viskose Strömung (DE-588)4226965-9 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2003 Martina Franca gnd-content Turbulente Strömung (DE-588)4117265-6 s Viskose Strömung (DE-588)4226965-9 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Constantin, Peter 1951- Sonstige (DE-588)1034846574 oth Cannone, Marco Sonstige oth Lecture notes in mathematics 1871 (DE-604)BV000676446 1871 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2687067&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung TU Muenchen application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014624566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 Lecture notes in mathematics Turbulente Strömung (DE-588)4117265-6 gnd Viskose Strömung (DE-588)4226965-9 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4117265-6 (DE-588)4226965-9 (DE-588)4114528-8 (DE-588)1071861417 |
| title | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |
| title_auth | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |
| title_exact_search | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |
| title_exact_search_txtP | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |
| title_full | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 P. Constantin ... Ed.: M. Cannone ... |
| title_fullStr | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 P. Constantin ... Ed.: M. Cannone ... |
| title_full_unstemmed | Mathematical foundation of turbulent viscous flows lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 P. Constantin ... Ed.: M. Cannone ... |
| title_short | Mathematical foundation of turbulent viscous flows |
| title_sort | mathematical foundation of turbulent viscous flows lectures given at the c i m e summer school held in martina franca italy september 1 5 2003 |
| title_sub | lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 1 - 5, 2003 |
| topic | Turbulente Strömung (DE-588)4117265-6 gnd Viskose Strömung (DE-588)4226965-9 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Turbulente Strömung Viskose Strömung Mathematisches Modell Konferenzschrift 2003 Martina Franca |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2687067&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014624566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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