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Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics
Tian Ma, Sichuan University, Chengdu, China, and Shouhong Wang, Indiana University, Bloomington, IN

Mathematical Surveys and Monographs
2005; 234 pp; hardcover
Volume: 119
ISBN-10: 0-8218-3693-5
ISBN-13: 978-0-8218-3693-4
List Price: US$80
Member Price: US$64
Order Code: SURV/119
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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 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.


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

Table of Contents

  • Introduction
  • Structure classification of divergence-free vector fields
  • Structural stability of divergence-free vector fields
  • Block stability of divergence-free vector fields on manifolds with nonzero genus
  • Structural stability of solutions of Navier-Stokes equations
  • Structural bifurcation for one-parameter families of divergence-free vector fields
  • Two examples
  • Bibliography
  • Index
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