ISBN-10:
0521635640
ISBN-13:
9780521635646
Pub. Date:
06/30/2010
Publisher:
Cambridge University Press
Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors / Edition 1

Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors / Edition 1

by James C RobinsonJames C Robinson

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Overview

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.

Product Details

ISBN-13: 9780521635646
Publisher: Cambridge University Press
Publication date: 06/30/2010
Series: Cambridge Texts in Applied Mathematics Series , #28
Edition description: New Edition
Pages: 480
Product dimensions: 6.00(w) x 8.90(h) x 1.10(d)

Table of Contents

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.

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