Note: Intro -- Euler Equations, Navier-Stokes Equations and Turbulence -- Peter Constantin -- 1 Introduction -- 2 Euler Equations -- 3 An Infinite Energy Blow Up Example -- 4 Navier-Stokes Equations -- 5 Approximations -- 6 The QG Equation -- 7 Dissipation and Spectra -- References -- CKN Theory of Singularities of Weak Solutions of the Navier-Stokes Equations -- Giovanni Gallavotti -- 1 Leray's Solutions and Energy -- 2 Kinematic Inequalities -- 3 Pseudo Navier Stokes Velocity -- Pressure Pairs. Scaling Operators -- 4 The Theorems of Scheffer and of Caffarelli--Kohn--Nirenberg -- 5 Fractal Dimension of Singularitiesof the Navier--Stokes Equation, d=3 -- 5.1 Dimension and Measure of Hausdorff -- 5.2 Hausdorff Dimension of Singular Times in the Navier--Stokes Solutions (d=3) -- 5.3 Hausdorff Dimension in Space--Time of the Solutions of NS, (d=3) -- 6 Problems. The Dimensional Bounds of the CKN Theory -- References -- Approximation of Weak Limits and Related Problems -- Alexandre V. Kazhikhov -- 1 Strong Approximation of Weak Limits by Averagings -- 1.1 Notations and Basic Notions from Orlicz Function Spaces Theory -- 1.2 Strong Approximation of Weak Limits -- Step 1. Simple example. -- Step 2. One-dimensional case, Steklov averaging. -- Step 3. The general case. -- Remark 1.1 -- 1.3 Applications to Navier-Stokes Equations -- 2 Transport Equations in Orlicz Spaces -- 2.1 Statement of Problem -- 2.2 Existence and Uniqueness Theorems -- 2.3 Gronwall-type Inequality and Osgood Uniqueness Theorem -- 2.4 Conclusive Remarks -- 3 Some Remarks on Compensated Compactness Theory -- 3.1 Introduction -- 3.2 Classical Compactness (Aubin-Simon Theorem) -- 3.3 Compensated Compactness -- ``div-curl" Lemma -- 3.4 Compensated Compactness-theorem of L. Tartar -- 3.5 Generalizations and Examples -- References -- Oscillating Patterns in Some Nonlinear Evolution Equations. , Yves Meyer -- 1 Introduction -- 2 A Model Case: the Nonlinear Heat Equation -- 3 Navier-Stokes Equations -- 4 The L2-theory is Unstable -- 5 T. Kato's Theorem -- 6 The Kato Theorem Revisited by Marco Cannone -- 7 The Kato Theory with Lorentz Spaces -- 8 Vortex Filaments and a Theorem by Y. Giga and T. Miyakawa -- 9 Vortex Patches -- 10 The H. Koch & -- D. Tataru Theorem -- 11 Localized Velocity Fields -- 12 Large Time Behavior of Solutions to the Navier-Stokes Equations -- 13 Improved Gagliardo-Nirenberg Inequalities -- 14 The Space BV of Functions with Bounded Variation in the Plane -- 15 Gagliardo-Nirenberg Inequalities and BV -- 16 Improved Poincaré Inequalities -- 17 A Direct Proof of Theorem 15.3 -- 18 Littlewood-Paley Analysis -- 19 Littlewood-Paley Analysis and Wavelet Analysis -- References -- Asymptotic Analysis of Fluid Equations -- Seiji Ukai -- 1 Introduction -- 2 Schemes for Establishing Asymptotic Relations -- 2.1 From Newton Equation to Boltzmann Equation: Boltzmann--Grad Limit -- Newton Equation -- Hard Sphere Gas -- Liouville Equation -- BBGKY Hierarchy -- Boltzmann Hierarchy -- Boltzmann Equation -- Collision Operator Q -- 2.2 From Boltzmann Equation to Fluid Equations --Multi-Scale Analysis -- The Case (,)=(0,0): Compressible Euler Equation (C.E.) -- The Case > -- 0, =0 -- The Case 0, > -- 0 -- 3 Abstract Cauchy-Kovalevskaya Theorem -- 3.1 Example 1: Pseudo Differential Equation -- 3.2 Example 2: Local Solutions of the Boltzmann Equation -- 4 The Boltzmann-Grad Limit -- 4.1 Integral Equations -- 4.2 Local Solutions and Uniform Estimates -- 4.3 Lanford's Theorem -- 5 Fluid Dynamical Limits -- 5.1 Preliminary -- 5.2 Main Theorems -- 5.3 Proof of Theorem 5.1 -- 5.4 Proof of Theorems 5.2 and 5.3 -- References.

Note: Intro -- Preface -- Contents -- Complex Fluids and Lagrangian Particles -- 1 Introduction -- 2 Lagrangian Particles: Small Data -- 3 Lagrangian Particles: Uniqueness -- 4 Large Data -- References -- Ergodicity Results for the Stochastic Navier-Stokes Equations: An Introduction -- 1 Introduction -- 2 Preliminaries -- 2.1 A Short Introduction to the Stochastic Navier-Stokes Equations -- 2.1.1 Notations -- 2.1.2 The Linear Equation -- 2.1.3 Existence and Uniqueness of Solutions for (6) -- 2.2 Transition Semi-group and Invariant Measures -- 2.3 Uniqueness of the Invariant Measure and Ergodicity: General Results -- 3 The Two Dimensional Case -- 3.1 The Strictly Dissipative Case -- 3.1.1 A Simple Example -- 3.1.2 The Navier-Stokes Equations with Large Viscosity -- 3.2 The Non Degenerate Case -- 3.2.1 A Simple Example -- 3.2.2 Use of Lyapunov Functionals -- 3.2.3 Another Proof of Ergodicity Based on More Analytical Arguments -- 3.3 Degenerate Noise: Case of a Large Number of Excited Modes -- 3.3.1 A Simplified Setting -- 3.3.2 The Navier-Stokes Equation -- 3.3.3 Another Methods Based on a Generalization of Doob Theorem -- 3.4 Very Degenerate Noise -- 4 The Three Dimensional Case -- 4.1 The Kolmogorov Equation -- 4.1.1 A Priori Estimates -- 4.1.2 Passage to the Limit -- 4.2 Markov Solutions and Transition Semi-group -- 4.3 Ergodicity -- 4.4 Further Results -- References -- Steady-State Navier-Stokes Problem Past a Rotating Body: Geometric-Functional Properties and Related Questions -- 1 Introduction -- 2 Review of Some Basic Results in Nonlinear Analysis -- 2.1 Operators in Banach Spaces -- 2.1.1 Basic Definitions -- 2.1.2 Continuous, Bounded and Closed Operators: Linear Operators -- 2.1.3 Operators of Class Ck -- 2.1.4 Compact Operators -- 2.1.5 Proper Operators -- 2.1.6 Fredholm Operators -- 2.1.7 Inverse Mapping and Implicit Function Theorems. , 2.2 Sard-Smale Theorem and Its Relevant Consequences -- 2.2.1 Fredholm Maps: Sard-Smale Theorem -- 2.2.2 Mod 2 Degree for C2 Proper Fredholm Maps of Index 0 -- 2.2.3 Parametrized Sard-Smale Theorem -- 3 Structure of the Set of Steady-State Solutions Past a Rotating Obstacle -- 3.1 The Navier-Stokes Problem in Banach Spaces -- 3.1.1 Preliminary Considerations -- 3.1.2 The Space X(Ω) and Its Relevant Properties -- 3.1.3 The Generalized Oseen Operator in X(Ω) -- 3.1.4 Suitable Extensions of the Boundary Data -- 3.1.5 The Nonlinear Oseen Operator in X(Ω) -- 3.2 Relevant Properties of the Operator N -- 3.3 Structure of the Solution Manifold -- 3.3.1 Control by a Finite Number of Parameters -- 4 Some Results on Steady Bifurcation of Solutions to the Navier-Stokes Problem Past a Rotating Obstacle -- 4.1 Review of Elementary Bifurcation Theory in Banach Spaces -- 4.1.1 Bifurcation Points of Equations in Banach Spaces -- 4.1.2 A Sufficient Condition for the Existence of a Bifurcation Point -- 4.2 Application to Flow in an Exterior Domain: Convection in an Unbounded Porous Medium -- 4.3 On Steady Bifurcation of Solutions to the Navier-Stokes Problem Past a Rotating Obstacle -- References -- Analysis of Generalized Newtonian Fluids -- 1 Theoretical and Numerical Analysis of Steady Problems for Generalized Newtonian Fluids -- 1.1 Introduction -- 1.2 Notation and the Extra Stress Tensor S -- 1.2.1 Notation and Function Spaces -- 1.2.2 Basic Properties of the Extra Stress Tensor -- 1.3 Existence of Weak Solutions -- 1.4 Finite Element Approximation of p-Stokes Systems -- 1.4.1 Interpolation in Orlicz-Sobolev Spaces -- 1.4.2 Error Estimates -- References -- Selected Topics of Local Regularity Theory for Navier-StokesEquations -- 1 Introduction -- 2 Stokes System -- 3 ε-Regularity Theory -- 4 Why Suitable Weak Solution? -- 5 Mild Bounded Ancient Solutions. , 6 Bounded Ancient Solutions -- 7 Liouville Type Theorems -- 7.1 LPS Quantities -- 7.2 2D Case -- 7.3 Axially Symmetric Case with No Swirl -- 7.4 Axially Symmetric Case -- 8 Axially Symmetric Suitable Weak Solutions -- 9 Backward Uniqueness for Navier-Stokes Equations -- 10 Appendix -- 10.1 Carleman-Type Inequalities -- 10.2 Unique Continuation Across Spatial Boundaries -- 10.3 Backward Uniqueness for Heat Operator in Half Space -- References.