Geometric theory of incompressible flows with applications to fluid dynamics / Tian Ma, Shouhong Wang.

Format:

Book

Author/Creator:

Ma, Tian, 1956- author.

Wang, Shouhong, 1962- author.

Series:

Mathematical surveys and monographs ; volume 119.

Mathematical surveys and monographs ; volume 119

Language:

English

Subjects (All):

Global analysis (Mathematics).

Vector fields.

Differential equations, Partial.

Manifolds.

Fluid dynamics.

Geophysics.

Physical Description:

1 online resource (248 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2005]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""Introduction""; ""0.1. Representation of Fluid Flows""; ""0.2. Motivation and Main Objectives""; ""0.3. The User's Guide""; ""Notes for Introduction""; ""Chapter 1. Structure Classification of Divergence-Free Vector Fields""; ""1.1. Limit Set Theorem""; ""1.2. Poincare-Hopf Index Theorem on Manifolds with Boundaries""; ""1.3. Structural Classification""; ""1.4. Topological Classification""; ""Notes for Chapter 1""; ""Chapter 2. Structural Stability of Divergence-Free Vector Fields""

""2.1. Structural Stability of Divergence-Free Vector Fields with Free Boundary Conditions""""2.2. Structural Stability for Divergence-Free Vector Fields with Dirichlet Boundary Conditions""; ""2.3. Two Dimensional Hamiltonian Structural Stability""; ""2.4. Block Structure of Hamiltonian Vector Fields""; ""2.5. Local Structural Stability""; ""Notes for Chapter 2""; ""Chapter 3. Block Stability of Divergence-Free Vector Fields on Manifolds with Nonzero Genus""; ""3.1. Instability on Manifolds with Nonzero Genus""; ""3.2. Block Structure and Block Stability""

""3.3. Structural Evolution of the Taylor Vortices""""Notes for Chapter 3""; ""Chapter 4. Structural Stability of Solutions of Navier-Stokes Equations""; ""4.1. Genericity of Stable Steady States""; ""4.2. Properties for Structurally Stable Solutions on the Reynolds Numbers""; ""4.3. Asymptotic Hamiltonian Structural Stability""; ""4.4. Asymptotic Block Stability""; ""4.5. Periodic Structure of Solutions of the Navier-Stokes Equations""; ""4.6. Structure of Solutions of the Rayleigh-Benard Convection""; ""Notes for Chapter 4""

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

""Chapter 6. Two Examples""""6.1. Fluid Flow Maps and Double-Gyre Ocean Circulation""; ""6.2. Boundary Layer Separation on Driven Cavity Flow""; ""Notes for Chapter 6""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""K""; ""L""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""V""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 229-232) and index.

Description based on print version record.

ISBN:

1-4704-1346-9