Infinite-dimensional dynamical systems : an introduction to dissipative parabolic PDEs and the theory of global attractors
Author(s)
Bibliographic Information
Infinite-dimensional dynamical systems : an introduction to dissipative parabolic PDEs and the theory of global attractors
（Cambridge texts in applied mathematics）
Cambridge University Press, c2001
- : pbk
- Other Title
-
Infinite-dimensional dynamical systems : from basic concepts to actual calculations
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
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.
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.
by "Nielsen BookData"
