Infinite-Dimensional Dynamical Systems

An Introduction to Dissipative Parabolic Pdes and the Theory of Global Attractors
Erscheinungsjahr 2009
ISBN: 978-0-521-63204-1
Verlag: Cambridge University Press

Buch, Englisch, Band 28, 480 Seiten, Format (B × H): 157 mm x 235 mm, Gewicht: 924 g

Reihe: Cambridge Texts in Applied Mathematics

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.

Autoren/Hrsg.

Robinson, James C.

Crighton, D. G.

Ablowitz, M. J.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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.