PDE Dynamics: An Introduction

This book is intended to illustrate how dynamical systems theory can be applied to partial differential equations (PDE's) and, conversely, how to take advantage of PDE methods in dynamical systems. It is primarily designed for beginning graduate students and researchers who might benefit from the exchange of ideas between the two fields. The author feels that the two areas have drifted apart and tend to follow their own traditional paths. While PDE courses tend to focus on existence and regularity questions, treatments of dynamical systems tend to emphasize ordinary differential equations or iterated mappings in discrete time. Furthermore, PDE courses tend more to the functional analytic side, while dynamical systems theory is inclined to be more geometric. There is a lot of merit in trying to relate the two.

 

The book has thirty-six chapters, none of them longer than seven pages. A background in PDE's at the level of a first course is desirable as is comparable experience with ordinary differential equations. The author says his plan is to present key concepts and to promote analytic strategies that are at the interface between the two fields. He provides a very broad overview that includes a large selection of topics. A consequence of this broad reach is that the pace is fast and no topics are treated very deeply.

 

Much of the work implicitly revolves around the interplay between infinite and finite-dimensional solution spaces and understanding the behavior of PDE's by reduction to finite-dimensional subspaces. The idea of an inertial manifold is a good example. In the context of dissipative dynamical systems, an inertial manifold is a finite-dimensional, smooth invariant manifold that contains a global attractor. Inertial manifolds are finite-dimensional even if the original system (a reaction-diffusion equation, for example) is infinite-dimensional. Another example is Fenichel theory, a geometric singular perturbation theory developed originally for finite-dimensional dynamical systems but important for studying the infinite-dimensional systems of traveling waves.

 

The author uses concrete PDE examples to illustrate the results he presents. In so doing he develops a collection of benchmark PDEss, those equations that have motivated and guided the general theory. Among these are the Fokker-Planck, Korteweg-deVries, Navier-Stokes and Ginzburg-Landau equations.

 

The author provides proofs when the ideas behind them are important and new strategies are contained in the proof. He gives specific references for proofs that are omitted. Each chapter comes with a good set of references to enable further investigation. The exercises in the book are minimal – only three per chapter. These are designed to reinforce the concepts and techniques presented in the chapter. Should an instructor choose to use this book for a course, many more exercises would be needed.

 

This would be a difficult book to use as a text for a course. It has too much material, presented too fast, with too little depth. It might serve as a useful source of ideas for future research, or as a self-study guide for those with the necessary background. Another book that addresses many of the same topics is Nonlinear PDEs: A Dynamical Systems Approach by Schneider and Uecker. It is better suited for use as a course text, and the book under review might nicely complement it.

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Sujeewa Hapuarachchi is a visiting assistant professor at the University of Toledo in Ohio. His primary interests include partial differential equations and functional analysis.