Handbook of Mathematical Fluid Dynamics (eBook)

Front cover 1
Handbook of Mathematical Fluid Dynamics 4
Copyright page 5
Contents of the Handbooks 6
Preface 8
List of Contributors 10
Contents 12
Chapter 1. On the Contact Topology and Geometry of Ideal Fluids 14
1. Ideal fluids on Riemannian manifolds 16
2. The geometry of steady solutions 20
3. Basic contact topology 23
4. Contact structures and steady Euler fields in 3-d 27
5. Knots and links in three-dimensional flows 29
6. Instability 37
7. Concluding unscientific postscript 45
References 47
Chapter 2. Shock Reflection in Gas Dynamics 52
Definitions 54
1. Models for gas dynamics 56
2. Multi-dimensional shocks 66
3. Reflection along a planar wall 90
4. Reflection at a wedge 103
5. Reflection at a wedge: Qualitative aspects 114
6. Regular Reflection at a wedge: Quantitative aspects 118
Acknowledgement 132
References 132
Chapter 3. The Mathematical Theory of the Incompressible Limit in Fluid Dynamics 136
1. Introduction 138
2. Scaling and formal limit 138
3. Remarks on the early development of the theory of the incompressible limit 149
4. Justification of the incompressible limit for periodic barotropic flow 150
5. Survey of further results 164
6. Some open problems 168
References 168
Chapter 4. Local Regularity Theory of the Navier-Stokes EquationsLocal Regularity Theory Navier-Stokes Equations 172
1. Introduction 174
2. Local regularity theory for the Stokes system 175
3. Interior case: Ladyzhenskaya-Prodi-Serrin condition 180
4. Boundary case: Ladyzhenskaya-Prodi-Serrin condition 186
5. Interior case: epsilon-regularity theory 189
6. Boundary case: epsilon-regularity theory 195
7. Local L3,-solutions 203
8. Other local regularity results 207
Acknowledgement 210
References 211
Chapter 5. On the influence of the Earth's Rotation on Geophysical Flows 214
Introduction 216
1. Modelling geophysical flows 217
2. A simplified model for midlatitudes 247
3. Taking into account spatial variations at midlatitudes 275
4. The tropics 292
Acknowledgement 340
References 340
Chapter 6. The Foundations of Oceanic Dynamics and Climate Modeling 344
1. Introduction 346
2. Oceanic flows 349
3. Solution concepts 366
4. Skew product dynamics 394
5. Thin domain dynamics 407
6. Climate modeling 408
7. Concluding remarks 411
Acknowledgements 415
References 415
Chapter 7. Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities 420
1. Introductory remarks 422
2. Models and their physical attributes 431
3. Mathematical analysis of the models 439
4. Special solutions 450
References 455
Chapter 8. The Navier-Stokes System in Domains with Cylindrical Outlets to Infinity. Leray's Problem 458
Introduction 460
1. Chapter I. Function spaces and auxiliary results 465
2. Chapter II. Poiseuille type flows 480
3. Chapter III. Steady problems 541
4. Chapter IV. Time-dependent problems 579
Acknowledgments 656
References 656
Chapter 9. Periodic Homogenization Problems in Incompressible Fluid Equations 662
1. Introduction 664
2. Fluid flows in porous media 665
3. Advection and diffusion of passive scalars 689
4. Convection effects 702
Acknowledgments 709
References 709
Author Index 712
Subject Index 720