Hardcover ISBN:  9780821836934 
Product Code:  SURV/119 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Electronic ISBN:  9781470413460 
Product Code:  SURV/119.E 
List Price:  $80.00 
MAA Member Price:  $72.00 
AMS Member Price:  $64.00 

Book DetailsMathematical Surveys and MonographsVolume: 119; 2005; 234 ppMSC: Primary 35; 76; 37; 86; Secondary 46; 20;
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 divergencefree fields on general twodimensional compact manifolds. The second is the study of the structure of velocity fields for twodimensional incompressible fluid flows governed by the NavierStokes 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.ReadershipAdvanced graduate students and research mathematicians interested in nonlinear PDEs and fluid dynamics.

Table of Contents

Chapters

1. Structure classification of divergencefree vector fields

2. Structural stability of divergencefree vector fields

3. Block stability of divergencefree vector fields on manifolds with nonzero genus

4. Structural stability of solutions of NavierStokes equations

5. Structural bifurcation for oneparameter families of divergencefree vector fields

6. Two examples


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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 divergencefree fields on general twodimensional compact manifolds. The second is the study of the structure of velocity fields for twodimensional incompressible fluid flows governed by the NavierStokes 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.
Advanced graduate students and research mathematicians interested in nonlinear PDEs and fluid dynamics.

Chapters

1. Structure classification of divergencefree vector fields

2. Structural stability of divergencefree vector fields

3. Block stability of divergencefree vector fields on manifolds with nonzero genus

4. Structural stability of solutions of NavierStokes equations

5. Structural bifurcation for oneparameter families of divergencefree vector fields

6. Two examples