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.
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Editorial Reviews
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"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
Book Description
This book treats the theory of global attractors, a recent development in the theory of partial differential equations, in a way that also includes much of the traditional elements of the subject. As such it gives a quick but directed introduction to some fundamental concepts, and by the end proceeds to current research problems. Since the subject is relatively new, this is the first book to attempt to treat these various topics in a unified and didactic way. It is intended to be suitable for first year graduate students.
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Key Phrases - Statistically Improbable Phrases (SIPs):
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strong squeezing property, compact absorbing set, spectral gap condition, compact symmetric operator, interior regularity result, exponential attractors, inertial manifolds, semidynamical system, global attractor, approximate inertial manifold, uniform differentiability, injectivity property, asymptotic dynamics, integral bound, absorbing sets, compactness theorem, attractor dimension, higher regularity, interpolation inequality, compact invariant set, bounded linear map, contraction mapping theorem, embedding theorem, category theorem, uniform boundedness principle Key Phrases - Capitalized Phrases (CAPs): (learn more)
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strong squeezing property, compact absorbing set, spectral gap condition, compact symmetric operator, interior regularity result, exponential attractors, inertial manifolds, semidynamical system, global attractor, approximate inertial manifold, uniform differentiability, injectivity property, asymptotic dynamics, integral bound, absorbing sets, compactness theorem, attractor dimension, higher regularity, interpolation inequality, compact invariant set, bounded linear map, contraction mapping theorem, embedding theorem, category theorem, uniform boundedness principle Key Phrases - Capitalized Phrases (CAPs): (learn more)
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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This book cites 32
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- Applied Analysis of the Navier-Stokes Equations (Cambridge Texts in Applied Mathematics) by Charles R. Doering in Front Matter, and Back Matter
- Sobolev spaces, Volume 65 (Pure and Applied Mathematics) by Author Unknown on page 109, and Back Matter
- A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge Texts in Applied Mathematics) by R. S. Johnson in Front Matter
- Hausdorff Measures (Cambridge Mathematical Library) by C. A. Rogers in Back Matter
- Mathematical Models in the Applied Sciences (Cambridge Texts in Applied Mathematics) by A. C. Fowler in Front Matter
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- Elements of Applied Bifurcation Theory (Applied Mathematical Sciences) by Yuri Kuznetsov in Back Matter
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