**Mathematical Surveys and Monographs**

Volume: 119;
2005;
234 pp;
Hardcover

MSC: Primary 35; 76; 37; 86;
Secondary 46; 20

Print ISBN: 978-0-8218-3693-4

Product Code: SURV/119

List Price: $80.00

Individual Member Price: $64.00

**Electronic ISBN: 978-1-4704-1346-0
Product Code: SURV/119.E**

List Price: $80.00

Individual Member Price: $64.00

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#### Supplemental Materials

# Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics

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*Tian Ma; Shouhong Wang*

This monograph presents a geometric theory for incompressible flow and its applications to fluid dynamics. The main objective is to study the stability and transitions of the structure of incompressible flows and its applications to fluid dynamics and geophysical fluid dynamics. The development of the theory and its applications goes well beyond its original motivation of the study of oceanic dynamics.

The authors present a substantial advance in the use of geometric and topological methods to analyze and classify incompressible fluid flows. The approach introduces genuinely innovative ideas to the study of the partial differential equations of fluid dynamics. One particularly useful development is a rigorous theory for boundary layer separation of incompressible fluids.

The study of incompressible flows has two major interconnected parts. The first is the development of a global geometric theory of divergence-free fields on general two-dimensional compact manifolds. The second is the study of the structure of velocity fields for two-dimensional incompressible fluid flows governed by the Navier-Stokes equations or the Euler equations.

Motivated by the study of problems in geophysical fluid dynamics, the program of research in this book seeks to develop a new mathematical theory, maintaining close links to physics along the way. In return, the theory is applied to physical problems, with more problems yet to be explored.

The material is suitable for researchers and advanced graduate students interested in nonlinear PDEs and fluid dynamics.

#### Table of Contents

# Table of Contents

## Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics

- Contents vii8 free
- Preface ix10 free
- Introduction 112 free
- Chapter 1. Structure Classification of Divergence-Free Vector Fields 1728
- Chapter 2. Structural Stability of Divergence-Free Vector Fields 5162
- 2.1. Structural Stability of Divergence-Free Vector Fields with Free Boundary Conditions 5162
- 2.2. Structural Stability for Divergence-Free Vector Fields with Dirichlet Boundary Conditions 6071
- 2.3. Two Dimensional Hamiltonian Structural Stability 7182
- 2.4. Block Structure of Hamiltonian Vector Fields 7586
- 2.5. Local Structural Stability 7788
- Notes for Chapter 2 8091

- Chapter 3. Block Stability of Divergence-Free Vector Fields on Manifolds with Nonzero Genus 8192
- Chapter 4. Structural Stability of Solutions of Navier-Stokes Equations 109120
- 4.1. Genericity of Stable Steady States 109120
- 4.2. Properties for Structurally Stable Solutions on the Reynolds Numbers 114125
- 4.3. Asymptotic Hamiltonian Structural Stability 117128
- 4.4. Asymptotic Block Stability 123134
- 4.5. Periodic Structure of Solutions of the Navier-Stokes Equations 127138
- 4.6. Structure of Solutions of the Rayleigh-Benard Convection 142153
- Notes for Chapter 4 155166

- Chapter 5. Structural Bifurcation for One-Parameter Families of Divergence-Free Vector Fields 157168
- 5.1. Necessary Conditions for Structural Bifurcation 157168
- 5.2. Structural Bifurcation for Flows with No-Normal Flow Boundary Conditions 160171
- 5.3. Structural Bifurcation for Flows with Dirichlet Boundary Conditions 167178
- 5.4. Boundary Layer Separations of Incompressible Flows I 177188
- 5.5. Boundary Layer Separations of Incompressible Flows II 181192
- 5.6. Structural Bifurcation near Interior Singular Points 187198
- 5.7. Genericity of Structural Bifurcations 198209
- Notes for Chapter 5 201212

- Chapter 6. Two Examples 203214
- Bibliography 229240
- Index 233244

#### Readership

Advanced graduate students and research mathematicians interested in nonlinear PDEs and fluid dynamics.