紹介
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.
目次
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.
