Dynamics of Evolutionary Equations: (143 Applied Mathematical Sciences)

The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. Dynamical issues arise in equations that attempt to model phenomena that change with time. The infi­ nite dimensional aspects occur when forces that describe the motion depend on spatial variables, or on the history of the motion. In the case of spatially depen­ dent problems, the model equations are generally partial differential equations, and problems that depend on the past give rise to differential-delay equations. Because the nonlinearities occurring in thse equations need not be small, one needs good dynamical theories to understand the longtime behavior of solutions. Our basic objective in writing this book is to prepare an entree for scholars who are beginning their journey into the world of dynamical systems, especially in infinite dimensional spaces. In order to accomplish this, we start with the key concepts of a semiflow and a flow. As is well known, the basic elements of dynamical systems, such as the theory of attractors and other invariant sets, have their origins here.

Table of Contents:
1. The Evolution of Evolutionary Equations.- 2. Dynamical Systems: Basic Theory.- 3. Linear Semigroups.- 4. Basic Theory of Evolutionary Equations.- 5. Nonlinear Partial Differential Equations.- 6. Navier-Stokes Dynamics.- 7. Major Features of Dynamical Systems.- 8. Inertial Manifolds: The Reduction Principle.- Appendices: Basics of Functional Analysis.- A Banach Spaces and Fréchet Spaces.- B Function Spaces and Sobolev Imbedding Theorems.- C Calculus of Vector-Valued Functions.- D Basic Inequalities.- E Commentary.- Notation Index.

Review :
From the reviews: "The book under review is a fundamental introduction into the field of infinite-dimensional dynamical systems, especially those generated from evolutionary partial differential equations. It is addressed to all those who embark on their journey into this fascinating topic … . the book is nicely written, there are given useful commentaries on all steps and to original and surrounding works. It can be highly recommended to all those who wish to enjoy the world of dynamics of evolutionary equations." (J. Synnatzschke, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (3), 2002) "The dynamics of evolutionary equations are a nice example of the interaction between the theory of ordinary and of partial differential equations. The present volume is a significant contribution to this theory, and should become soon a reference book. … The book can undoubtedly be recommended and used as a basis for graduate courses in partial differential equations, as well as a reference volume for researchers in the important area of evolutionary equations." (Jean Mawhin, Bulletin of the Belgian Mathematical Society, Vol. 11 (4), 2004) "Two well-known experts are the authors of this book on the theory and applications of infinite dimensional dynamical systems. … Some basic results from functional analysis are included in the Appendices. This makes the book self-contained. … The whole text is very carefully written. This excellent book will be certainly a basic reference on infinite dimensional dynamical systems. I can recommend it to researchers and graduate students in mathematics, physical sciences, and engineering." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 69, 2003) "The present book gives an introduction to infinite-dimensional dynamical systems suitable for graduate students with a solid background from functional analysis. … It is well-written and contains many results which are otherwise hard to findin the text book literature." (G. Teschl, Monatshefte für Mathematik, Vol. 138 (3), 2003)