Dynamics of Evolutionary Equations

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.

The theory and applications of infinite dimensional dynamical systems have attracted the attention of scientists for quite some time. This book serves as an entree for scholars beginning their journey into the world of dynamical systems, especially infinite dimensional spaces. The main approach involves the theory of evolutionary equations.

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.