​​​​​​​Turbulence is a major problem facing modern societies. It makes airline passengers return to their seats and fasten their seatbelts but it also creates drag on the aircraft that causes it to use more fuel and create more pollution. The same applies to cars, ships and the space shuttle. The mathematical theory of turbulence has been an unsolved problems for 500 years and the development of the statistical theory of the Navier-Stokes equations describes turbulent flow has been an open problem. The Kolmogorov-Obukhov Theory of Turbulence develops a statistical theory of turbulence from the stochastic Navier-Stokes equation and the physical theory, that was proposed by Kolmogorov and Obukhov in 1941. The statistical theory of turbulence shows that the noise in developed turbulence is a general form which can be used to present a mathematical model for the stochastic Navier-Stokes equation. The statistical theory of the stochastic Navier-Stokes equation is developed in a pedagogical manner and shown to imply the Kolmogorov-Obukhov statistical theory. This book looks at a new mathematical theory in turbulence which may lead to many new developments in vorticity and Lagrangian turbulence. But even more importantly it may produce a systematic way of improving direct Navier-Stokes simulations and lead to a major jump in the technology both preventing and utilizing turbulence.Table of ContentsCoverThe Kolmogorov-Obukhov Theory of Turbulence - A Mathematical Theory of TurbulenceISBN 9781461462613 ISBN 9781461462620PrefaceContentsThe Mathematical Formulation of Fully Developed Turbulence1.1 Introduction to Turbulence1.2 The Navier-Stokes Equation for Fluid Flow1.2.1 Energy and Dissipation1.3 Laminar Versus Turbulent Flow1.4 Two Examples of Fluid Instability Creating Large Noise1.4.1 Stability1.5 The Central Limit Theorem and the Large Deviation Principle, in Probability Theory1.5.1 Cramer'� s Theorem1.5.2 Stochastic Processes and Time Change1.6 Poisson Processes and Brownian Motion1.6.1 Finite-Dimensional Brownian Motion1.6.2 The Wiener Process1.7 The Noise in Fully Developed Turbulence1.7.1 The Generic Noise1.8 The Stochastic Navier-Stokes Equation for Fully Developed TurbulenceProbability and the Statistical Theory of Turbulence2.1 Ito Processes and Ito's Calculus2.2 The Generator of an Ito Diffusion and Kolmogorov's Equation2.2.1 The Feynman-Kac Formula2.2.2 Girsanov's Theorem and Cameron-Martin2.3 Jumps and Levy� Processes2.4 Spectral Theory for the Operator K2.5 The Feynman-Kac Formula and the Log-Poissonian Processes2.6 The Kolmogorov-Obukhov-She-Leveque Theory2.7 Estimates of the Structure Functions2.8 The Solution of the Stochastic Linearized Navier-Stokes EquationThe Invariant Measure and the Probability Density Function3.1 The Invariant Measure of the Stochastic Navier-Stokes Equation3.1.1 The Invariant Measure of Turbulence3.2 The Invariant Measure for the Velocity Differences3.3 The Differential Equation for the Probability Density Function3.4 The PDF for the Turbulent Velocity Differences3.5 Comparison with Simulations and Experiments3.6 Description of Simulations and Experiments3.7 The Invariant Measure of the Stochastic Vorticity Equation3.7.1 The Invariant Measure of Turbulent VorticityExistence Theory of Swirling Flow4.1 Leray's Theory4.2 The A Priori Estimate of the Turbulent Solutions4.3 Existence Theory of the Stochastic Navier-Stokes EquationThe Bound for a Swirling FlowDetailed Estimates of S2 and S3The Generalized Hyperbolic DistributionsReferencesIndex