7R47. Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. - AJ Majda (Courant Inst of Math Sci, New York Univ, New York NY) and AL Bertozzi (Duke Univ, Durham NC). Cambridge UP, Cambridge, UK. 2002. 545 pp. Softcover. ISBN 0-521-63948-4. $40.00. (Hardcover ISBN 0-521-63057-6$100.00).

Reviewed by A Ogawa (Dept of Mech Eng, Col of Eng, Nihon Univ, T 963 Tamura-machi, Kooriyama-city, Japan).

In this book, the theoretical and experimental analyses of the velocity fields with vorticity are applied to explain the physical phenomena of flow patterns of various types of vortex flows and of the flow structures in the boundary layer with high velocity gradient on a solid surface. Further, the vortex flow in nature is not always circular, but also elliptic rotational flows occur under stable and unstable conditions. Therefore, the fundamental mathematical and applied descriptions of vorticity are important factors in fluid mechanics in scientific and engineering applications. From these viewpoints, this textbook is reviewed as follows.

In Chapter 1, Introduction to Vortex Dynamics for Incompressible Fluid Flows, the fundamental descriptions of the Euler and the Navier-Stokes equations, and the important quantities of velocity, vorticity, helicity, impulse, and moment of fluid impulse are defined. Chapter 2, entitled Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations, discusses the vorticity-stream formulations of 2D. Periodic flow, cat’s-eye flow, and also Beltrami flows are discussed with figures.

The existence of a solution to either the Euler or the Navier-Stokes equations on the same time interval, given generally smooth initial velocity fields, is examined in Chapter 3, Energy Method for the Euler and the Navier-Stokes Equations. In Chapter 4, Particle-Trajectory Method for Existence and Uniqueness of Solution to the Euler Equation, the questions of existence, uniqueness, and continuation of solution to the Euler and the Navier-Stokes equations are discussed. Search for Singular Solutions to the 3D Euler Equations, is the title for Chapter 5, where an important unsolved research problem for incompressible flow and the current attempts to make progress on this problem are the main focus. Chapter 6, Computational Vortex Methods, deals with computational methods for simulations of the Euler and the Navier-Stokes equations with high Reynolds number condition for vortex dynamics, and includes a brief historical summary. In order to study the motion of slender tubes of vorticity at high Reynolds numbers, Chapter 7, Simplified Asymptotic Equations for Slender Vortex Filaments, describes formal, but concise, asymptotic expansions to the simplified asymptotic equations; they are seen to emerge with remarkable properties. In Chapter 8, Weak Solutions to the 2D Euler Equations with Initial Vorticity in$L∞,$ a flow field having elliptic vorticity, like Thomson-Rankine combined vortex model, is examined. Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation is the title of Chapter 9, which describes a vortex sheet problem occurring in the vorticity layers. The last four chapters—Weak Solutions and Solution Sequences in Two Dimensions, 2D Euler Equation: Concentrations and Weak Solutions with Vortex Sheet Initial Data (with an example constructed by Greengard and Thomann), Reduced Hausdorf Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions, and Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions (with an analogy between vorticity and electron density)—address the mathematical theory connected with small scale structures and dynamics in high Reynolds number and inviscid flow.

To sum up, this textbook for advanced undergraduate and beginning graduate students, is aimed at mathematicians and physicists to examine mathematical theories and techniques, and to explore their applications. It appears to be a little difficult to grasp the physical phenomena involved and to apply them directly to nature and engineering. Since there is a small gap between the actual fluid flows in nature and engineering and the contents in this textbook with the examples shown, it would be better if the authors had introduced and explained the examples from W Albring, Elementarvorga¨nge Fluider Wirbelbewegungen (Akademische-Verlag, Berlin, 1981), and L Lugt, Introduction to Vortex Theory (Vortex Flow Press, Inc, 1996). However, for graduate students, mathematicians, and physicists who can understand the contents of the books by GK Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1999) and by LD Landau and EM Lifshitz, Fluid Mechanics (Butterworth-Heinemann, Oxford, 1987), this peculiar textbook certainly contributes to understanding the fundamental concepts of fluid flow with vorticity, and to apply them to nature and engineering.