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
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Infinite-dimensional dynamical systems : from basic concepts to actual calculations
Available at 46 libraries
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"