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

Bibliographic Information

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

James C. Robinson

(Cambridge texts in applied mathematics)

Cambridge University Press, c2001

  • : pbk

Other Title

Infinite-dimensional dynamical systems : from basic concepts to actual calculations

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Note

Includes bibliographical references and index

Description and Table of Contents

Description

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.

Table of Contents

  • 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.

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