Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors: 28 (Cambridge Texts in Applied Mathematics, Series Number 28)
Cambridge University Press
Edition: Illustrated, 4/23/2001
EAN 9780521632041, ISBN10: 0521632048
Hardcover, 480 pages, 22.9 x 15.2 x 3 cm
Language: English
Originally published in English
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.
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.
'... 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