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 / Edition 1

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Five leading specialists reflect on different and complementary approaches to fundamental questions in the study of the Fluid Mechanics and Gas Dynamics equations. Constantin presents the Euler equations of ideal incompressible fluids and discusses the blow-up problem for the Navier-Stokes equations of viscous fluids, describing some of the major mathematical questions of turbulence theory. These questions are connected to the Caffarelli-Kohn-Nirenberg theory of singularities for the incompressible Navier-Stokes equations that is explained in Gallavotti's lectures. Kazhikhov introduces the theory of strong approximation of weak limits via the method of averaging, applied to Navier-Stokes equations. Y. Meyer focuses on several nonlinear evolution equations - in particular Navier-Stokes - and some related unexpected cancellation properties, either imposed on the initial condition, or satisfied by the solution itself, whenever it is localized in space or in time variable. Ukai presents the asymptotic analysis theory of fluid equations. He discusses the Cauchy-Kovalevskaya technique for the Boltzmann-Grad limit of the Newtonian equation, the multi-scale analysis, giving the compressible and incompressible limits of the Boltzmann equation, and the analysis of their initial layers.

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Table of Contents

Euler Equations, Navier-Stokes Equations and Turbulence   Peter Constantin     1
Introduction     1
Euler Equations     2
An Infinite Energy Blow Up Example     10
Navier-Stokes Equations     16
Approximations     24
The QG Equation     26
Dissipation and Spectra     30
References     39
CKN Theory of Singularities of Weak Solutions of the Navier-Stokes Equations   Giovanni Gallavotti     45
Leray's Solutions and Energy     47
Kinematic Inequalities     49
Pseudo Navier Stokes Velocity - Pressure Pairs. Scaling Operators     50
The Theorems of Scheffer and of Caffarelli-Kohn-Nirenberg     54
Fractal Dimension of Singularities of the Navier-Stokes Equation, d = 3     58
Dimension and Measure of Hausdorff     58
Hausdorff Dimension of Singular Times in the Navier-Stokes Solutions (d = 3)     59
Hausdorff Dimension in Space-Time of the Solutions of NS, (d = 3)     61
Problems. The Dimensional Bounds of the CKN Theory     63
References     73
Approximation of Weak Limits and Related Problems   Alexandre V. Kazhikhov     75
Strong Approximation of Weak Limits byAveragings     76
Notations and Basic Notions from Orlicz Function Spaces Theory     76
Strong Approximation of Weak Limits     79
Simple example     79
One-dimensional case, Steklov averaging     80
The general case     81
Remark 1.1     82
Applications to Navier-Stokes Equations     82
Transport Equations in Orlicz Spaces     83
Statement of Problem     83
Existence and Uniqueness Theorems     85
Gronwall-type Inequality and Osgood Uniqueness Theorem     86
Conclusive Remarks     88
Some Remarks on Compensated Compactness Theory     89
Introduction     89
Classical Compactness (Aubin-Simon Theorem)     91
Compensated Compactness - "div-curl" Lemma     92
Compensated Compactness-theorem of L. Tartar     94
Generalizations and Examples     96
References     98
Oscillating Patterns in Some Nonlinear Evolution Equations   Yves Meyer     101
Introduction     101
A Model Case: the Nonlinear Heat Equation     103
Navier-Stokes Equations     113
The L[superscript 2]-theory is Unstable     118
T. Kato's Theorem     124
The Kato Theorem Revisited   Marco Cannone     127
The Kato Theory with Lorentz Spaces     132
Vortex Filaments and a Theorem   Y. Giga   T. Miyakawa     136
Vortex Patches     142
The H. Koch & D. Tataru Theorem     144
Localized Velocity Fields     146
Large Time Behavior of Solutions to the Navier-Stokes Equations     151
Improved Gagliardo-Nirenberg Inequalities     154
The Space BV of Functions with Bounded Variation in the Plane     157
Gagliardo-Nirenberg Inequalities and BV     161
Improved Poincare Inequalities     166
A Direct Proof of Theorem 15.3     170
Littlewood-Paley Analysis     172
Littlewood-Paley Analysis and Wavelet Analysis     178
References     182
Asymptotic Analysis of Fluid Equations   Seiji Ukai     189
Introduction     189
Schemes for Establishing Asymptotic Relations     194
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
From Boltzmann Equation to Fluid Equations - Multi-Scale Analysis     202
The Case ([alpha],[Beta]) = (0,0): Compressible Euler Equation (C.E.)     204
The Case [alpha] > 0, [Beta] = 0     205
The Case [alpha] [greater than or equal] 0,[Beta] > 0     205
Abstract Cauchy-Kovalevskaya Theorem     212
Example 1: Pseudo Differential Equation     216
Example 2: Local Solutions of the Boltzmann Equation     219
The Boltzmann-Grad Limit     223
Integral Equations     223
Local Solutions and Uniform Estimates     224
Lanford's Theorem     229
Fluid Dynamical Limits     232
Preliminary     233
Main Theorems     235
Proof of Theorem 5.1     238
Proof of Theorems 5.2 and 5.3     243
References     248
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