7R7. Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors. Cambridge Texts in Applied Mathematics. - JC Robinson (Math Inst, Univ of Warwick, UK). Cambridge UP, Cambridge, UK. 2001. 461 pp. (Softcover). ISBN 0-521-63564-0. $110.00.
Reviewed by C Pierre (Dept of Mech Eng and Appl Mech, Univ of Michigan, 2250 GG Brown Bldg, Ann Arbor MI 48109-0001).
This volume in applied mathematics is concerned with the study of the longterm dynamics of partial differential equations (PDEs), with the aim of reducing dynamic behavior complexity. The approach is based on ideas from the theory of dynamical systems, which has proven successful for the study of finite-dimensional systems and for the past two decades or so has been developed for infinite-dimensional systems. The focus of this book is on dissipative parabolic PDEs, and particularly on the investigation of their asymptotic behavior by means of global attractors whose dynamics can be reproduced by a finite-dimensional system. As described in the Preface, the author aims to provide a systematic and rigorous treatment of the theory of global attractors. To do so he assumes minimum analytical background from the reader and begins with the basic foundation of functional analysis. As a result, the book provides both an in-depth, educated coverage of the tools necessary for the study of global attractors of PDEs and an authoritative account of the state-of-the-art in this evolving field.
The volume is organized into four parts of four or five sequential chapters each, and two appendices. In the first part, the reader is given a rigorous exposition to the topic of functional analysis. Since this is required to understand the rest of the book, coverage is comprehensive and this part constitutes about one third of the volume. It includes chapters on Banach and Hilbert spaces, existence and uniqueness of solutions for ordinary differential equations, linear operators, and dual spaces, and concludes with a thorough treatment of Sobolev spaces. The second part of the book is concerned with the existence and uniqueness of solutions of time-dependent PDEs. Galerkin’s method is first introduced on an example linear parabolic equation as a means of proving these properties. Then this approach is applied to investigate existence and uniqueness for scalar nonlinear reaction-diffusion equations and for two-dimensional Navier-Stokes equations with periodic boundary conditions. In the third part of the book, finite-dimensional global attractors are introduced: a general result to prove their existence is given, fractal and Hausdorff measures of their dimension are defined, and a method to estimate their dimension is proposed. These ideas are then applied to the examples of the reaction-diffusion and the Navier-Stokes equations. The fourth, and last, part investigates how the finite dimensionality of the global attractor can be exploited to reduce the complexity of the asymptotic dynamics of PDEs, namely, how the dynamics on the attractor are finite-dimensional. One chapter covers the squeezing property and its implications for finite-numbered “determining modes,” approximate inertial manifolds, and exponential attractors. Another chapter introduces the “strong squeezing property” for the analysis of inertial manifolds. A third chapter gives a proof that the attractor dynamics can be reproduced by a finite-dimensional system, and a fourth, and final, chapter applies the various methods developed in the book to the analysis of the Kuramoto-Sivashinsky equation, which is used to model instabilities such as flame fronts. In addition, two useful appendices on Sobolev spaces of periodic functions and the bounding of fractal dimension support the material presented in the 17 chapters.
The book is written clearly and concisely. It is well structured, and the material is presented in a rigorous, coherent fashion. A number of example problems are treated, and each chapter is followed by a series of problems whose solutions are available on the internet. Although this is not an easy book to read, the informal style in which it is written and the pedagogical presentation of the material make it accessible to the reader. In summary, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors constitutes an excellent resource for researchers and advanced graduate students in applied mathematics, dynamical systems, nonlinear dynamics, and computational mechanics. Its acquisition by libraries is strongly recommended.