Infinite Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors: 28

This book treats the theory of global attractors, a recent development in the theory of partial differential equations, in a way that also includes many traditional elements of the subject. It gives an introduction to some fundamental concepts, and by the end proceeds to current research problems. 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.