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Klappentext This book develops the theory of global attractors for a class of parabolic PDEs that 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 systemss 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. Zusammenfassung 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 a quick but directed introduction to some fundamental concepts! and by the end proceeds to current research problems. Inhaltsverzeichnis Part I. Functional Analysis: 1. Banach and Hilbert spaces; 2. Ordinary differential equations; 3. Linear operators; 4. Dual spaces; 5. Sobolev spaces; Part II. Existence and Uniqueness Theory: 6. The Laplacian; 7. Weak solutions of linear parabolic equations; 8. Nonlinear reaction-diffusion equations; 9. The Navier-Stokes equations existence and uniqueness; Part II. Finite-Dimensional Global Attractors: 10. The global attractor existence and general properties; 11. The global attractor for reaction-diffusion equations; 12. The global attractor for the Navier-Stokes equations; 13. Finite-dimensional attractors: theory and examples; Part III. Finite-Dimensional Dynamics: 14. Finite-dimensional dynamics I, the squeezing property: determining modes; 15. Finite-dimensional dynamics II, The stong squeezing property: inertial manifolds; 16. Finite-dimensional dynamics III, a direct approach; 17. The Kuramoto-Sivashinsky equation; Appendix A. Sobolev spaces of periodic functions; Appendix B. Bounding the fractal dimension using the decay of volume elements....
