Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

1989. Auflage 1988
ISBN: 978-0-387-96729-5
Verlag: Springer

Buch, Englisch, Band 70, 123 Seiten, Gewicht: 360 g

Reihe: Applied Mathematical Sciences

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.

Zielgruppe

Research

Autoren/Hrsg.

Constantin, P.

Foias, C.

Nicolaenko, B.

Temam, R.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Topologie Analytische Topologie

Inhaltsverzeichnis

Preface.- Acknowledgments.- 1 Presentation of the Approach and of the Main Results.- 2 The Transport of Finite-Dimensional Contact Elements.- 3 Spectral Blocking Property.- 4 Strong Squeezing Property.- 5 Cone Invariance Properties.- 6 Consequences Regarding the Global Attractor.- 7 Local Exponential Decay Toward Blocked Integral Surfaces.- 8 Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- 9 Choice of the Initial Manifold.- 10 Construction of the Inertial Manifold.- 11 Lower Bound for the Exponential Rate of Convergence to the Attractor.- 12 Asymptotic Completeness: Preparation.- 13 Asymptotic Completeness: Proof of Theorem 12.1.- 14 Stability with Respect to Perturbations.- 15 Application: The Kuramoto—Sivashinsky Equation.- 16 Application: A Nonlocal Burgers Equation.- 17 Application: The Cahn—Hilliard Equation.- 18 Application: A Parabolic Equation in Two Space Variables.- 19 Application: The Chaffee—Infante Reaction—Diffusion Equation.- References.