%(For Math review)


Vorticity and Incompressible Flow, by Andrew Majda and Andrea Bertozzi,
Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002.
pp. 545+ xii, $38(paperback).




The book is on incompressible flows that are inviscid or have
high Reynolds numbers. The equations involved are called the incompressible 
Euler's equations for the inviscid cases (i.e., idealized materials) and the 
incompressible Navier-Stokes' equations for viscous cases. 
These equations are physically important and mathematically challenging and
interesting. They have stood the test of time and remain the most
important in the field and for years to come. Physically their solutions are 
predictions of physical phenomena. Mathematically, their solutions are not 
yet proven to exist in the function class in which they are derived,
notwithstanding numerous efforts by brilliant minds. One outstanding 
difficulty is associated with vorticity, i.e., the rotation
(curl) of the velocity. Mathematically it seems that vorticity can become 
out of control everywhere, which corresponds to the physical phenomena of 
turbulence. The control of vorticity lies in the heart of the regularity
problem for these equations, and the regularity
problem for the Navier-Stokes equations has recently been listed 
as one of the seven Millennium Prize Problems in the Clay Mathematics 
Institute (http://www.claymath.org/prize_problems/index.htm).


This long-awaited book is a ``comprehensive introduction to the mathematical
theory of vorticity and incompressible flow ranging from elementary 
introductory material to current research topics. Although the contents
center on mathematical theory, many parts of the book showcase the 
interactions among rigorous mathematical theory, numerical, asymptotic,
and qualitative simplified modeling, and physical phenomena.'' (quoted from 
the Preface of the book under review). 

The first half of the book can be used for an introductory graduate course 
on vorticity and incompressible flow. It contains many excellent exact
and explicit solutions. It sets up the equations in various modern
forms for further research. In particular, the work in Chapter 5 on 
search for singular solutions is a novel approach.  The work in Chapter
7 on interacting asymptotic filaments is also very novel in many respects.

The second half can be used for a graduate topics course on the 
theory for weak solutions for incompressible flow with an emphasis on 
modern applied mathematics. There are six chapters covering
the following topics. The first is the elegant vortex patch
theory in which Chemin's theorem (Theorems 8.7-8) is presented with 
a proof given by Bertozzi and Constantin. The second is the subtle 
theoretical and computational issues of vortex sheets, in particular 
Delort's theorem. The third is the concentration-cancellation phenomena 
in sequences of weak solutions. The fourth is the concept of
 measure-valued solutions. This chapter (Chapter 12)
has many open problems. The last one is a chapter on one-dimensional 
Vlasov-Poisson equations which serves as a simplified model for the 
two dimensional Euler equations.

A key feature of the book is the natural and seemless incorporation of
rigorous mathematical theory, numerical, asymptotic, and qualitative 
simplified modeling, and physical phenomena. This symbiotic interaction 
is apparent in 7 chapters out of 13. 

This book is an outgrowth of several lecture courses by Majda, enriched by
the authors' own research in the field and Majda's many students', 
collaborators' and other researchers. Various student's lecture notes were 
in circulation and popular. The book form is thus long-awaited and 
represents the best form.


The chapters of the book are often supplemented with an Appendix, as well
as Notes and references. The Appendices are especially helpful for students.
Interspersed reviews such as singular integral operators are selected just 
right in presenting a complete and understandable exposition of the analysis.
The book goes into deep mathematics without sacrificing much in 
the relative ease of understanding. 


The book has a useful but short index. Maybe in the next edition, the index
can be expanded so that the terms Hausdorff, measure-valued solution, 
Kirchhoff, concentration-cancellation, etc., etc, can be included, and indexed 
in each occurrence.  There are inevitable typoes, but it is 
a graduate level book and the reader should be able to discern the typoes.
The book has no exercises, but there are plenty of open problems. 
Topics such as incompressible limits, Hamiltonian/variational type structures,
and boundary layers are not included.

There are about 11 books available currently on the market covering
incompressible flows. Majda and Bertozzi's book is unique in covering 
both the classical and weak solutions for the incompressible and inviscid 
flows and it is excellently done.

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