| Preface | p. xi |
| The equations of motion | p. 1 |
| Introduction | p. 1 |
| Euler's equations for an incompressible fluid | p. 1 |
| Energy, body forces, vorticity, and enstrophy | p. 7 |
| Viscosity, the stress tensor, and the Navier-Stokes equations | p. 12 |
| Thermal convection and the Boussinesq equations | p. 18 |
| References and further reading | p. 21 |
| Exercises | p. 22 |
| Dimensionless parameters and stability | p. 23 |
| Dimensionless parameters | p. 23 |
| Linear and nonlinear stability, differential inequalities | p. 29 |
| References and further reading | p. 38 |
| Exercises | p. 38 |
| Turbulence | p. 40 |
| Introduction | p. 40 |
| Statistical turbulence theory and the closure problem | p. 40 |
| Spectra, Kolmogorov's scaling theory, and turbulent length scales | p. 49 |
| References and further reading | p. 59 |
| Exercises | p. 60 |
| Degrees of freedom, dynamical systems, and attractors | p. 61 |
| Introduction | p. 61 |
| Dynamical systems, attractors, and their dimension | p. 62 |
| The Lorenz system | p. 74 |
| References and further reading | p. 86 |
| Exercises | p. 87 |
| On the existence, uniqueness, and regularity of solutions | p. 88 |
| Introduction | p. 88 |
| Existence and uniqueness for ODEs | p. 89 |
| Galerkin approximations and weak solutions of the Navier-Stokes equations | p. 96 |
| Uniqueness and the regularity problem | p. 104 |
| References and further reading | p. 113 |
| Exercises | p. 113 |
| Ladder results for the Navier-Stokes equations | p. 114 |
| Introduction | p. 114 |
| The Navier-Stokes ladder theorem | p. 117 |
| A natural definition of a length scale | p. 125 |
| The dynamical wavenumbers k[subscript N,r] | p. 127 |
| Estimates for the Navier-Stokes equations | p. 128 |
| Estimates for F[subscript 0] | p. 129 |
| Estimates for [left angle bracket]F[subscript 1 right angle bracket] and [left angle bracket]k[superscript 2 subscript 1,0 right angle bracket] | p. 130 |
| Estimates for lim[subscript t[right arrow infinity]F[subscript 1], [left angle bracket]F[subscript 2 right angle bracket], and [left angle bracket]k[superscript 2 subscript 2,1 right angle bracket] | p. 131 |
| A ladder for the thermal convection equations | p. 132 |
| References and further reading | p. 134 |
| Exercises | p. 134 |
| Regularity and length scales for the 2d and 3d Navier-Stokes equations | p. 137 |
| Introduction | p. 137 |
| A global attractor and length scales in the 2d case | p. 138 |
| A global attractor | p. 139 |
| Length scales in the 2d Navier-Stokes equations | p. 139 |
| 3d Navier-Stokes regularity? | p. 144 |
| Problems with 3d Navier-Stokes regularity | p. 144 |
| A Bound on [left angle bracket]k[subscript N,1 right angle bracket] in 3d | p. 146 |
| Bounds on [left angle bracket double vertical line]u[double vertical line subscript [infinity right angle bracket] and [left angle bracket double vertical line]Du[double vertical line superscript 1/2 subscript [infinity right angle bracket] | p. 148 |
| The Kolmogorov length and intermittency | p. 149 |
| Singularities and the Euler equations | p. 152 |
| References and further reading | p. 155 |
| Exercises | p. 155 |
| Exponential decay of the Fourier power spectrum | p. 157 |
| Introduction | p. 157 |
| A differential inequality for [double vertical line]e[superscript [alpha]t [down triangle, open] down triangle, open]u[double vertical line superscript 2 subscript 2] | p. 157 |
| A bound on [double vertical line]e[superscript [alpha]t [down triangle, open] down triangle, open]u[double vertical line superscript 2 subscript 2] | p. 163 |
| Decay of the Fourier spectrum | p. 165 |
| References and further reading | p. 167 |
| Exercises | p. 167 |
| The attractor dimension for the Navier-Stokes equations | p. 169 |
| Introduction | p. 169 |
| The 2d attractor dimension estimate | p. 170 |
| The 3d attractor dimension estimate | p. 177 |
| References and further reading | p. 179 |
| Exercises | p. 180 |
| Energy dissipation rate estimates for boundary-driven flows | p. 181 |
| Introduction | p. 181 |
| Boundary-driven shear flow | p. 182 |
| Thermal convection in a horizontal plane | p. 192 |
| Discussion | p. 197 |
| References and further reading | p. 203 |
| Exercises | p. 204 |
| Inequalities | p. 205 |
| References | p. 209 |
| Index | p. 213 |
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