9R8. Dynamics of Evolutionary Equations. Applied Math Sciences, Vol 143. - GR Sell (Sch of Math, Univ of Minnesota, Minneapolis MN 55455) and Yuncheng You (Dept of Math, Univ of S Florida, Tampa FL 33620-5700). Springer-Verlag, New York. 2002. 670 pp. ISBN 0-387-98347-3. $79.00.

Reviewed by K Anderson (Dept of Mech Eng, Aeronaut Eng, and Mech (JEC4006), RPI, Troy NY 12180-3590).

This book treats the modeling and analysis of linear and nonlinear dynamics systems in a mathematically rigorous manner. Considerable emphasis is given to explaining key concepts such as semiflow and flow, attractors and global attractors. The authors present how the time-varying solutions of the partial differential equations associated with a generally nonlinear dynamics system can be viewed as a trajectory in the phase space of the problem. The equation of motion of this trajectory in this Banach space is the “evolutionary equation” associated with this system. Loosely speaking, this equation is being an ODE on this space.

The book begins with some brief historical notes and then proceeds with a discussion of fundamentals associate with Dynamic Systems Theory in Chapter 2. Here, the concepts of semiflow, semigroups, Limit sets, Compact, κ-compact, and singular semiflows, attractors and the like are presented. The discussion then proceeds to a very detailed presentation of Linear Semigroups in Chapter 3.

In Chapter 4, the basic theory of Evolutionary Equations is presented. Here particular emphasis is placed on the difference in treatment which arises when one considers a dynamics system in an infinite dimension as opposed to a finite dimensional setting. Chapter 4 is particularly nice in that it collects and presents many of the important aspects of evolutionary equations. These include solutions concepts, linear theory, nonlinear theory, well-posedness, regularity and compactness, and the linearized equation.

The basic application of semiflow theory to nonlinear partial differential equations is presented in Chapter 5. Here a good variety of applications is discussed which include, but are not limited to, reaction diffusion equations, nonlinear wave equations, convection equations, Kuramoto-Sivahinsky Equation, etc. Chapter 6 continues to the discussion of nonlinear partial differential equations, but is dedicated to the use of this material in the treatment of the Navier-Stokes equation.

Several aspects of linear and nonlinear dynamics systems theory and how these relate to evolution equations are presented in Chapters 7 and 8. These aspects include perturbation theory near a saddlepoint, the reduction theory and center manifold, periodic orbits and invariant manifolds, and inertial manifolds. Applications are also provided here involving Couette-Taylor flow and Bobnov-Galkerkin approximations. Other topics/theory of dynamic systems such as bifurcation theory, ergodic theory, dimension theory, singular perturbations, Hamiltonian systems, etc, are only mentioned or are very briefly presented.

The book is intended to introduce the subject to “scholars,” who wish to learn about and analyze dynamic systems at a sophisticated level. The text is well written and is very much of the style and form of the other members of Springer’s Applied Mathematical Sciences series, of which this book is the 143rd volume. As such, the book is written for the applied mathematician, as opposed to the engineer, even though their experience and expertise may well be in nonlinear dynamics systems. This is, in some degree, evidenced by the terminology “Evolutionary Equations,” which use is largely restricted to the mathematicians working the field as opposed to the engineers (refer to works by Holmes, Wiggins, Nayfeh, and others). As such, this book assumes that the reader is familiar with the rudiments of nonlinear and linear systems analysis, from a mathematics, not engineering, background. For readers not so familiar with Banach and Fre´chet spaces or the basic of functional analysis, the book’s appendices offer some relief, for those whose interest and mathematics experience are otherwise up to the presentation.

As a text, this reviewer finds the general layout of the book good. The index is thorough, with the diagrams which appear in the text being extremely sparse and simple, but clear. This reviewer feels that Dynamics of Evolutionary Equations would serve well as a reference to any mathematician or well-seasoned mathematically inclined engineer actively performing research in the field.