Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

1. Front Matter

   Pages i-x

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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

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.

• convergence
• differential equation
• integral
• manifold
• partial differential equation
• stability

• 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

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