The Kolmogorov-Obukhov theory of turbulence : a mathematical theory of turbulence

Cover......Page 1
The Kolmogorov-Obukhov 
Theory of Turbulence......Page 4
Preface......Page 8
Contents......Page 10
1.1 Introduction to Turbulence......Page 12
1.2 The Navier–Stokes Equation for Fluid Flow......Page 15
1.2.1 Energy and Dissipation......Page 16
1.3 Laminar Versus Turbulent Flow......Page 17
1.4 Two Examples of Fluid Instability Creating Large Noise......Page 19
1.4.1 Stability......Page 21
1.5 The Central Limit Theorem and the Large Deviation Principle, in Probability Theory......Page 26
1.5.1 Cramér's Theorem......Page 28
1.5.2 Stochastic Processes and Time Change......Page 29
1.6 Poisson Processes and Brownian Motion......Page 31
1.6.1 Finite-Dimensional Brownian Motion......Page 35
1.6.2 The Wiener Process......Page 37
1.7 The Noise in Fully Developed Turbulence......Page 39
1.7.1 The Generic Noise......Page 42
1.8 The Stochastic Navier–Stokes Equation for Fully Developed Turbulence......Page 43
2.1 Ito Processes and Ito's Calculus......Page 46
2.2 The Generator of an Ito Diffusion and Kolmogorov's Equation......Page 48
2.2.2 Girsanov's Theorem and Cameron–Martin......Page 50
2.3 Jumps and Lévy Processes......Page 51
2.4 Spectral Theory for the Operator K......Page 53
2.5 The Feynman–Kac Formula and the Log-Poissonian Processes......Page 57
2.6 The Kolmogorov–Obukhov–She–Leveque Theory......Page 59
2.7 Estimates of the Structure Functions......Page 60
2.8 The Solution of the Stochastic Linearized Navier–Stokes Equation......Page 64
3.1 The Invariant Measure of the Stochastic Navier–Stokes Equation......Page 66
3.1.1 The Invariant Measure of Turbulence......Page 68
3.2 The Invariant Measure for the Velocity Differences......Page 70
3.3 The Differential Equation for the Probability Density Function......Page 72
3.4 The PDF for the Turbulent Velocity Differences......Page 73
3.5 Comparison with Simulations and Experiments......Page 77
3.6 Description of Simulations and Experiments......Page 80
3.7 The Invariant Measure of the Stochastic Vorticity Equation......Page 81
3.7.1 The Invariant Measure of Turbulent Vorticity......Page 84
4.1 Leray's Theory......Page 85
4.2 The A Priori Estimate of the Turbulent Solutions......Page 88
4.3 Existence Theory of the Stochastic Navier–Stokes Equation......Page 95
Appendix A  The Bound for a Swirling Flow......Page 99
Appendix B  Detailed Estimates of S2 and S3......Page 107
Appendix C  The Generalized Hyperbolic Distributions......Page 110
Reference......Page 112
Index......Page 116