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.
- Develops theory of PDEs as dynamical systems, theory of global attractors, and some consequences of that theory
- Only a low level of previous knowledge of functional analysis is assumed, so accessible to the widest possible mathematical audience
- Numerous exercises, with full solutions available on the web
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Reviews & endorsements
'… will certainly benefit young researchers entering the described field.' Jan Cholewa, Zentralblatt MATH
'This impressive book offers an excellent, self-contained introduction to many important aspects of infinite-dimensional systems … At the outset, the author states that his aim was to produce a didactic text suitable or first-year graduate students. Unquestionably he has achieved his goal. This book should prove invaluable to mathematicians wishing to gain some knowledge of the dynamical-systems approach to dissipative partial differential equations that has been developed during the past 20 years, and should be essential reading for any graduate student starting out on a PhD in this area.' W. Lamb, Proceedings of the Edinburgh Mathematical Society
'The book is written clearly and concisely. It is well structured, and the material is presented in a rigorous, coherent fashion. A number of example problems are treated, and each chapter is followed by a series of problems whose solutions are available on the internet. … constitutes an excellent resource for researchers and advanced graduate students in applied mathematics, dynamical systems, nonlinear dynamics, and computational mechanics. Its acquisition by libraries is strongly recommended.' Applied Mechanics Reviews
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Product details
- Date Published: June 2001
- format: Paperback
- isbn: 9780521635646
- length: 480 pages
- dimensions: 229 x 152 x 27 mm
- weight: 0.7kg
- contains: 14 b/w illus.
- availability: Available
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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