##
**Geometric theory of incompressible flows with applications to fluid dynamics.**
*(English)*
Zbl 1099.76002

Mathematical Surveys and Monographs 119. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3693-5/hbk). ix, 234 p. (2005).

This monograph presents in a rigorous manner a non-traditional, in many parts new, mathematical treatment of various topics related to the kinematics and dynamics of incompressible fluid flows that have not yet become standard knowledge of fluid mechanics. Globally viewed, the main topics which this monograph is dealing with are topological structure, structural stability and structural bifurcation of a flow. In the elapsed period of about ten years the authors published a series of papers on closely related topics. The most of them have been incorporated in this monograph. The mathematical approach in this book is entirely within the framework of modern dynamical systems theory, especially the theory of differentiable dynamical systems on compact manifolds, including the stability theory applied to fluid mechanics.

The monograph consists of a nicely organized introduction and six main chapters, followed by the bibliography and an index at the end. Each chapter begins with a short introduction explaining the subject and ends with short bibliographical notes. In sections and sub-sections in between the incisive definition-theorem-proof style is mostly used. Contents: 1. “Structure classification of divergence-free vector fields” (limit set theorem, Poincaré-Hopf index theorem on manifolds with boundaries, structural classification, topological classification); 2. “Structural stability of divergence-free vector fields” (structural stability of divergence-free vector fields with free boundary conditions, structural stability of divergence-free vector fields with Dirichlet boundary conditions, two-dimensional Hamiltonian structural stability, block structure of Hamiltonian vector fields, local structural stability); 3. “Block stability of divergence-free vector fields on manifolds with non-zero genus” (instability on manifolds with non-zero genus, block structure and block stability, structural stability of Taylor vortices); 4. “Structural stability of solutions of Navier-Stokes equations” (genericity of stable steady states, properties of structurally stable solutions depending on Reynolds number, asymptotic Hamiltonian structural stability, asymptotic block stability, periodic structure of solutions of Navier-Stokes equations, structure of solutions of Rayleigh-Bérnard convection equation); 5. “Structural bifurcations for one-parameter families of divergence-free vector fields” (necessary conditions for structural bifurcations, structural bifurcation for flows with non-normal flow boundary conditions, structural bifurcation for flows with Dirichlet boundary conditions, boundary layer separations of incompressible flows, boundary layer separations of incompressible flows, structural bifurcation near interior singular points, genericity of structural bifurcations); 6. “Two examples” (fluid flow maps and double-gyre ocean circulation, boundary layer separation in driven cavity flow).

The first three chapters cover the kinematics (specifically the topology) of two-dimensional incompressible flows. Moreover, here the authors develop a global geometric theory of divergence-free vector fields on general two-dimensional compact manifolds with or without boundaries, regardless the physical meaning of the vector fields, i.e., regardless of which evolution partial differential equation the considered vector field is a solution. However, many of the theorems proved in this book concern divergence-free vector fields defined on two-dimensional, orientable differentiable Riemannian manifolds with boundary (e.g., a planar region, a sub-manifold of a sphere or two-dimensional torus). On the boundary, with regard to fluid dynamics, either the no-penetration (in the case of vector fields originated from Euler equation) or the real-fluid boundary condition (in the case of vector fields originated from Navier-Stokes equations) is prescribed. The analysis of the topology and stability of a flow comprises the investigation of qualitative behavior of the instantaneous velocity vector field in a neighborhood of its singular (critical) points in the free flow and on the boundary, in which the magnitude of velocity (and the wall shear stress in the case of Dirichlet boundary conditions) vanishes. The topological structure of a flow is known when we know the type, number and position of its critical points. The singular point theory for vector fields forms a basis for the investigations throughout this book. Restricting to divergence-free vector fields, the authors present several original results as refinements or generalizations of known dynamical systems theorems, or by introducing new ideas like the block structure and block stability for incompressible flows on two-dimensional manifolds with non-zero genus.

The last three chapters deal with classification, stability and evolution of topological structures of specified vector fields considered as solutions of Euler and Navier-Stokes equations. Not all sections and sub-sections of these chapters could be further commented here in more detail. However, a particular mention deserves section 5.4 on the boundary layer separation (a structural bifurcation of index zero) of planar incompressible flows governed by full Navier-Stokes equations and initial and Dirichlet boundary conditions. A separation equation is presented and mathematically examined without resorting to Prandtl’s boundary layer equations. The proof links the point and time of separation with Reynolds number, external forcing, the curvature of the boundary, and the initial velocity field. The authors’ opinion that this theory will indeed be useful in future theoretical and engineering studies of boundary layer separation. However, it cannot be applied to determine the separation point a priori, as was wrongly stated in a paper by the authors (see reference [64] in the book). The results of two numerical simulations (chapter 6) produced by CFD-techniques confirm the validity of the global qualitative theory presented in this book.

The book is self-contained in the sense that almost every theorem or lemma is provided with a proof. However, various topics are presented in a rather brief manner only for experts in the field; for those, like this reviewer, whose main activity is not in dynamical systems theory, somewhat more prerequisite material would have been welcome for better understanding. The book is not free from typographical errors, and it would be easier readable when bold-face vector notation would have been used. Though this book is not for easy reading, it is a good contribution that presents material at the edge of the current research in the field.

The monograph consists of a nicely organized introduction and six main chapters, followed by the bibliography and an index at the end. Each chapter begins with a short introduction explaining the subject and ends with short bibliographical notes. In sections and sub-sections in between the incisive definition-theorem-proof style is mostly used. Contents: 1. “Structure classification of divergence-free vector fields” (limit set theorem, Poincaré-Hopf index theorem on manifolds with boundaries, structural classification, topological classification); 2. “Structural stability of divergence-free vector fields” (structural stability of divergence-free vector fields with free boundary conditions, structural stability of divergence-free vector fields with Dirichlet boundary conditions, two-dimensional Hamiltonian structural stability, block structure of Hamiltonian vector fields, local structural stability); 3. “Block stability of divergence-free vector fields on manifolds with non-zero genus” (instability on manifolds with non-zero genus, block structure and block stability, structural stability of Taylor vortices); 4. “Structural stability of solutions of Navier-Stokes equations” (genericity of stable steady states, properties of structurally stable solutions depending on Reynolds number, asymptotic Hamiltonian structural stability, asymptotic block stability, periodic structure of solutions of Navier-Stokes equations, structure of solutions of Rayleigh-Bérnard convection equation); 5. “Structural bifurcations for one-parameter families of divergence-free vector fields” (necessary conditions for structural bifurcations, structural bifurcation for flows with non-normal flow boundary conditions, structural bifurcation for flows with Dirichlet boundary conditions, boundary layer separations of incompressible flows, boundary layer separations of incompressible flows, structural bifurcation near interior singular points, genericity of structural bifurcations); 6. “Two examples” (fluid flow maps and double-gyre ocean circulation, boundary layer separation in driven cavity flow).

The first three chapters cover the kinematics (specifically the topology) of two-dimensional incompressible flows. Moreover, here the authors develop a global geometric theory of divergence-free vector fields on general two-dimensional compact manifolds with or without boundaries, regardless the physical meaning of the vector fields, i.e., regardless of which evolution partial differential equation the considered vector field is a solution. However, many of the theorems proved in this book concern divergence-free vector fields defined on two-dimensional, orientable differentiable Riemannian manifolds with boundary (e.g., a planar region, a sub-manifold of a sphere or two-dimensional torus). On the boundary, with regard to fluid dynamics, either the no-penetration (in the case of vector fields originated from Euler equation) or the real-fluid boundary condition (in the case of vector fields originated from Navier-Stokes equations) is prescribed. The analysis of the topology and stability of a flow comprises the investigation of qualitative behavior of the instantaneous velocity vector field in a neighborhood of its singular (critical) points in the free flow and on the boundary, in which the magnitude of velocity (and the wall shear stress in the case of Dirichlet boundary conditions) vanishes. The topological structure of a flow is known when we know the type, number and position of its critical points. The singular point theory for vector fields forms a basis for the investigations throughout this book. Restricting to divergence-free vector fields, the authors present several original results as refinements or generalizations of known dynamical systems theorems, or by introducing new ideas like the block structure and block stability for incompressible flows on two-dimensional manifolds with non-zero genus.

The last three chapters deal with classification, stability and evolution of topological structures of specified vector fields considered as solutions of Euler and Navier-Stokes equations. Not all sections and sub-sections of these chapters could be further commented here in more detail. However, a particular mention deserves section 5.4 on the boundary layer separation (a structural bifurcation of index zero) of planar incompressible flows governed by full Navier-Stokes equations and initial and Dirichlet boundary conditions. A separation equation is presented and mathematically examined without resorting to Prandtl’s boundary layer equations. The proof links the point and time of separation with Reynolds number, external forcing, the curvature of the boundary, and the initial velocity field. The authors’ opinion that this theory will indeed be useful in future theoretical and engineering studies of boundary layer separation. However, it cannot be applied to determine the separation point a priori, as was wrongly stated in a paper by the authors (see reference [64] in the book). The results of two numerical simulations (chapter 6) produced by CFD-techniques confirm the validity of the global qualitative theory presented in this book.

The book is self-contained in the sense that almost every theorem or lemma is provided with a proof. However, various topics are presented in a rather brief manner only for experts in the field; for those, like this reviewer, whose main activity is not in dynamical systems theory, somewhat more prerequisite material would have been welcome for better understanding. The book is not free from typographical errors, and it would be easier readable when bold-face vector notation would have been used. Though this book is not for easy reading, it is a good contribution that presents material at the edge of the current research in the field.

Reviewer: Tomislav Zlatanovski (Skopje)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76Dxx | Incompressible viscous fluids |

76Bxx | Incompressible inviscid fluids |

76Exx | Hydrodynamic stability |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |