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Infinite-Dimensional Dynamical Systems

Infinite-Dimensional Dynamical Systems
An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors


Part of Cambridge Texts in Applied Mathematics

  • Date Published: April 2001
  • availability: Available
  • format: Paperback
  • isbn: 9780521635646


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About the Authors
  • 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 which generate the infinite-dimensional dynamical systemss 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.

    • Develops theory of PDEs as dynamical systems, theory of global attractors, and some consequences of that theory
    • Only a low level of previous knowledge of functional analysis is assumed, so accessible to the widest possible mathematical audience
    • Numerous exercises, with full solutions available on the web
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    Reviews & endorsements

    "The book is written clearly and concisely. It is well structured, and the material is presented in a rigorous, coherent fashion...[it] constitutes an excellent resource for researchers and advanced graduate students in applied mathematics, dynamical systems, nonlinear dynamics, and computational mechanics. Its acquisition by libraries is strongly recommended." Applied Mechanics Reviews

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    Product details

    • Date Published: April 2001
    • format: Paperback
    • isbn: 9780521635646
    • length: 480 pages
    • dimensions: 229 x 152 x 27 mm
    • weight: 0.7kg
    • contains: 14 b/w illus.
    • availability: Available
  • 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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    These resources are provided free of charge by Cambridge University Press with permission of the author of the corresponding work, but are subject to copyright. You are permitted to view, print and download these resources for your own personal use only, provided any copyright lines on the resources are not removed or altered in any way. Any other use, including but not limited to distribution of the resources in modified form, or via electronic or other media, is strictly prohibited unless you have permission from the author of the corresponding work and provided you give appropriate acknowledgement of the source.

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  • Author

    James C. Robinson, University of Warwick

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