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Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

  • Book
  • © 1989

Overview

Authors:
  1. P. Constantin

    1. Department of Mathematics, University of Chicago, Chicago, USA

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  2. C. Foias

    1. Department of Mathematics, Indiana University, Bloomington, USA

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  3. B. Nicolaenko

    1. Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, USA

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  4. R. Teman

    1. Department de Mathematiques, Université de Paris-Sud, Orsay, France

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Part of the book series: Applied Mathematical Sciences (AMS, volume 70)

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 1-3

  3. Presentation of the Approach and of the Main Results

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 4-14

  4. The Transport of Finite-Dimensional Contact Elements

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 15-20

  5. Spectral Blocking Property

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 21-24

  6. Strong Squeezing Property

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 25-28

  7. Cone Invariance Properties

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 29-32

  8. Consequences Regarding the Global Attractor

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 33-35

  9. Local Exponential Decay Toward Blocked Integral Surfaces

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 36-37

  10. Exponential Decay of Volume Elements and the Dimension of the Global Attractor

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 38-41

  11. Choice of the Initial Manifold

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 42-46

  12. Construction of the Inertial Manifold

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 47-51

  13. Lower Bound for the Exponential Rate of Convergence to the Attractor

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 52-54

  14. Asymptotic Completeness: Preparation

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 55-60

  15. Asymptotic Completeness: Proof of Theorem 12.1

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 61-67

  16. Stability with Respect to Perturbations

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 68-71

  17. Application: The Kuramoto—Sivashinsky Equation

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 72-81

  18. Application: A Nonlocal Burgers Equation

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 82-90

  19. Application: The Cahn—Hilliard Equation

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 91-104

  20. Application: A Parabolic Equation in Two Space Variables

    • P. Constantin, C. Foias, B. Nicolaenko, R. Teman
    Pages 105-110
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Keywords

About this book

This work was initiated in the summer of 1985 while all of the authors were at the Center of Nonlinear Studies of the Los Alamos National Laboratory; it was then continued and polished while the authors were at Indiana Univer­ sity, at the University of Paris-Sud (Orsay), and again at Los Alamos in 1986 and 1987. Our aim was to present a direct geometric approach in the theory of inertial manifolds (global analogs of the unstable-center manifolds) for dissipative partial differential equations. This approach, based on Cauchy integral mani­ folds for which the solutions of the partial differential equations are the generating characteristic curves, has the advantage that it provides a sound basis for numerical Galerkin schemes obtained by approximating the inertial manifold. The work is self-contained and the prerequisites are at the level of a graduate student. The theoretical part of the work is developed in Chapters 2-14, while in Chapters 15-19 we apply the theory to several remarkable partial differ­ ential equations.

Authors and Affiliations

  • Department of Mathematics, University of Chicago, Chicago, USA

    P. Constantin

  • Department of Mathematics, Indiana University, Bloomington, USA

    C. Foias

  • Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, USA

    B. Nicolaenko

  • Department de Mathematiques, Université de Paris-Sud, Orsay, France

    R. Teman

Bibliographic Information

  • Book Title: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

  • Authors: P. Constantin, C. Foias, B. Nicolaenko, R. Teman

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-1-4612-3506-4

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1989

  • Hardcover ISBN: 978-0-387-96729-5Due: 25 October 1988

  • Softcover ISBN: 978-1-4612-8131-3Published: 22 September 2011

  • eBook ISBN: 978-1-4612-3506-4Published: 06 December 2012

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 1

  • Number of Pages: X, 123

  • Topics: Manifolds and Cell Complexes (incl. Diff.Topology), Analysis

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