Infinite-Dimensional Dynamical Systems

This book develops the theory of global attractors for a class of parabolic PDEs which includes reaction-diffusion equations and the Navier-Stokes equations two examples that are treated in detail. A lengthy chapter on Sobolev spaces provides the framework that allows a rigorous treatment of existence and uniqueness of solutions for both linear time-independent problems (Poisson''s equation) and the nonlinear evolution equations which generate the infinite-dimensional dynamical systems of the title. Attention then switches to the global attractor a finite-dimensional subset of the infinite-dimensional phase space which determines the asymptotic dynamics. In particular the concluding chapters investigate in what sense the dynamics restricted to the attractor are themselves ''finite-dimensional''. The book is intended as a didactic text for first year graduates and assumes only a basic knowledge of Banach and Hilbert spaces and a working understanding of the Lebesgue integral.