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Mathematics of Two-Dimensional Turbulence

Book Description

This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier–Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) – proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication Page
  5. Contents
  6. Preface
  7. 1 - Preliminaries
    1. 1.1 Function Spaces
    2. 1.2 Basic Facts from Measure Theory
    3. 1.3 Markov Processes and Random Dynamical Systems
    4. Notes and Comments
  8. 2 - Two-Dimensional Navier–Stokes Equations
    1. 2.1 Cauchy Problem for the Deterministic System
    2. 2.2 Stochastic Navier–Stokes Equations
    3. 2.3 Navier–Stokes Equations Perturbed by a Random Kick Force
    4. 2.4 Navier–Stokes Equations Perturbed by Spatially Regular White Noise
    5. 2.5 Existence of a Stationary Distribution
    6. 2.6 Appendix: Some Technical Proofs
    7. Notes and Comments
  9. 3 - Uniqueness of Stationary Measure and Mixing
    1. 3.1 Three Results on Uniqueness and Mixing
    2. 3.2 Dissipative RDS with Bounded Kicks
    3. 3.3 Navier–Stokes System Perturbed by White Noise
    4. 3.4 Navier–Stokes System with Unbounded Kicks
    5. 3.5 Further Results and Generalisations
    6. 3.6 Appendix: Some Technical Proofs
    7. 3.7 Relevance of the Results for Physics
    8. Notes and Comments
  10. 4 - Ergodicity and Limiting Theorems
    1. 4.1 Ergodic Theorems
    2. 4.2 Random Attractors and Stationary Distributions
    3. 4.3 Dependence of a Stationary Measure on the Random Force
    4. 4.4 Relevance of the Results for Physics
    5. Notes and Comments
  11. 5 - Inviscid Limit
    1. 5.1 Balance Relations
    2. 5.2 Limiting Measures
    3. 5.3 Relevance of the Results for Physics
    4. Notes and Comments
  12. 6 - Miscellanies
    1. 6.1 3D Navier–Stokes System in Thin Domains
    2. 6.2 Ergodicity and Markov Selection
    3. 6.3 Navier–Stokes Equations with very Degenerate Noise
  13. Appendix
    1. A.1 Monotone Class Theorem
    2. A.2 Standard Measurable Spaces
    3. A.3 Projection Theorem
    4. A.4 Gaussian Random Variables
    5. A.5 Weak convergence of Random Measures
    6. A.6 The Gelfand Triple and Yosida Approximation
    7. A.7 Itô Formula in Hilbert Spaces
    8. A.8 Local Time for Continuous Itô Processes
    9. A.9 Krylov’s Estimate
    10. A.10 Girsanov’s Theorem
    11. A.11 Martingales, Submartingales, and Supermartingales
    12. A.12 Limit Theorems For Discrete-Time Martingales
    13. A.13 Martingale Approximation for Markov Processes
    14. A.14 Generalised Poincaré Inequality
    15. A.15 Functions in Sobolev Spaces with a Discrete Essential Range
  14. Solutions to Selected Exercises
  15. Notation and Conventions
  16. References
  17. Index