Infinite-dimensional dynamical systems : an introduction to dissipative parabolic PDEs and the theory of global attractors

✍ Scribed by James C Robinson

• Publisher: Cambridge University Press
• Year: 2001
• Tongue: English
• Leaves: 473
• Series: Cambridge texts in applied mathematics
• Category: Library

✦ Synopsis

"This book develops the theory of global attractors for a class of parabolic PDEs that 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 that generate the infinite-dimensional dynamical systems of the title. Attention then turns to the global attractor, a finite-dimensional subset of the infinite-dimensional phase space that 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 graduate students and assumes only a basic knowledge of elementary functional analysis."--BOOK JACKET. Read more... 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