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Beschreibung
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.
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.
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.
Details
Genre: Importe
Medium: Taschenbuch
ISBN-13: 9780521635646
ISBN-10: 0521635640
Sprache: Englisch
Einband: Kartoniert / Broschiert
Autor: Robinson, James C.
Hersteller: Cambridge University Press
Verantwortliche Person für die EU: Libri GmbH, Europaallee 1, D-36244 Bad Hersfeld, gpsr@libri.de
Maße: 229 x 152 x 28 mm
Von/Mit: James C. Robinson
Erscheinungsdatum: 30.06.2010
Gewicht: 0,773 kg
Artikel-ID: 102551603

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