Cover

Half-title

Title

Copyright

Dedication

Contents

Preface

1 The equations of motion

1.1 Introduction

1.2 Euler's equations for an incompressible fluid

1.3 Energy, body forces, vorticity, and enstrophy

1.4 Viscosity, the stress tensor, and the Navier-Stokes equations

1.5 Thermal convection and the Boussinesq equations

1.6 References and further reading

Exercises

2 Dimensionless parameters and stability

2.1 Dimensionless parameters

2.2 Linear and nonlinear stability, differential inequalities

2.3 References and further reading

Exercises

3 Turbulence

3.1 Introduction

3.2 Statistical turbulence theory and the closure problem

3.3 Spectra, Kolmogorov's scaling theory, and turbulent length scales

3.4 References and further reading

Exercises

4 Degrees of freedom, dynamical systems, and attractors

4.1 Introduction

4.2 Dynamical systems, attractors, and their dimension

4.3 The Lorenz system

4.4 References and further reading

Exercises

5 On the existence, uniqueness, and regularity of solutions

5.1 Introduction

5.2 Existence and uniqueness for ODEs

5.3 Galerkin approximations and weak solutions of the Navier-Stokes equations

5.4 Uniqueness and the regularity problem

5.5 References and further reading

Exercises

6 Ladder results for the Navier-Stokes equations

6.1 Introduction

6.2 The Navier-Stokes ladder theorem

6.3 A natural definition of a length scale

6.4 The dynamical wavenumbers KN, r

6.5 Estimates for the Navier-Stokes equations

6.5.1 Estimates for F0

6.5.2 Estimates for and

6.5.3 Estimates for lim t→∞ F1, , and

6.6 A ladder for the thermal convection equations

6.7 References and further reading

Exercises

7 Regularity and length scales for the 2d and 3d Navier-Stokes equations

7.1 Introduction

7.2 A global attractor and length scales in the 2d case

7.2.1 A global attractor

7.2.2 Length scales in the 2d Navier-Stokes equations

7.3 3d Navier-Stokes regularity?

7.3.1 Problems with 3d Navier-Stokes regularity

7.3.2 A Bound on in 3d

7.3.3 Bounds on <||u||∞> and <||Du||∞1/2>

7.4 The Kolmogorov length and intermittency

7.5 Singularities and the Euler equations

7.6 References and further reading

Exercises

8 Exponential decay of the Fourier power spectrum

8.1 Introduction

8.2 A differential inequality for ||ext|v|Vu||22

8.3 A bound on ||ext|v|Vu||22

8.4 Decay of the Fourier spectrum

8.5 References and further reading

Exercises

9 The attractor dimension for the Navier-Stokes equations

9.1 Introduction

9.2 The 2d attractor dimension estimate

9.3 The 3d attractor dimension estimate

9.4 References and further reading

Exercises

10 Energy dissipation rate estimates for boundary-driven flows

10.1 Introduction

10.2 Boundary-driven shear flow

10.3 Thermal convection in a horizontal plane

10.4 Discussion

10.5 References and further reading

Exercises

Appendix A Inequalities

References

Index