Ebook: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

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.

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This work, the main results of which were announced in (CFNT), focuses on a new geometric explicit construction of inertial manifolds from integral manifolds generated by some initial dimensional surface. The method covers a large class of dissipative PDEs. The existence of a smooth integral manifold the closure of which in an inertial manifold M (i.E. containing X and uniformly exponentially attracting) requires a more detailed analysis of the geometric properties of the infinite dimensional flow. The method is explicity constructive, integrating forward in time and avoiding any fixed point theorems. The key geometric property upon which we base the construction of our integral inertial manifold M is a Spectral Blocking Property of the flow, which controls the evolution of the position of surface elements relative to the fixed reference frame associated to the linear principal part of the PDE.

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This work, the main results of which were announced in (CFNT), focuses on a new geometric explicit construction of inertial manifolds from integral manifolds generated by some initial dimensional surface. The method covers a large class of dissipative PDEs. The existence of a smooth integral manifold the closure of which in an inertial manifold M (i.E. containing X and uniformly exponentially attracting) requires a more detailed analysis of the geometric properties of the infinite dimensional flow. The method is explicity constructive, integrating forward in time and avoiding any fixed point theorems. The key geometric property upon which we base the construction of our integral inertial manifold M is a Spectral Blocking Property of the flow, which controls the evolution of the position of surface elements relative to the fixed reference frame associated to the linear principal part of the PDE.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-3
Presentation of the Approach and of the Main Results....Pages 4-14
The Transport of Finite-Dimensional Contact Elements....Pages 15-20
Spectral Blocking Property....Pages 21-24
Strong Squeezing Property....Pages 25-28
Cone Invariance Properties....Pages 29-32
Consequences Regarding the Global Attractor....Pages 33-35
Local Exponential Decay Toward Blocked Integral Surfaces....Pages 36-37
Exponential Decay of Volume Elements and the Dimension of the Global Attractor....Pages 38-41
Choice of the Initial Manifold....Pages 42-46
Construction of the Inertial Manifold....Pages 47-51
Lower Bound for the Exponential Rate of Convergence to the Attractor....Pages 52-54
Asymptotic Completeness: Preparation....Pages 55-60
Asymptotic Completeness: Proof of Theorem 12.1....Pages 61-67
Stability with Respect to Perturbations....Pages 68-71
Application: The Kuramoto—Sivashinsky Equation....Pages 72-81
Application: A Nonlocal Burgers Equation....Pages 82-90
Application: The Cahn—Hilliard Equation....Pages 91-104
Application: A Parabolic Equation in Two Space Variables....Pages 105-110
Application: The Chaffee—Infante Reaction—Diffusion Equation....Pages 111-118
Back Matter....Pages 119-126

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This work, the main results of which were announced in (CFNT), focuses on a new geometric explicit construction of inertial manifolds from integral manifolds generated by some initial dimensional surface. The method covers a large class of dissipative PDEs. The existence of a smooth integral manifold the closure of which in an inertial manifold M (i.E. containing X and uniformly exponentially attracting) requires a more detailed analysis of the geometric properties of the infinite dimensional flow. The method is explicity constructive, integrating forward in time and avoiding any fixed point theorems. The key geometric property upon which we base the construction of our integral inertial manifold M is a Spectral Blocking Property of the flow, which controls the evolution of the position of surface elements relative to the fixed reference frame associated to the linear principal part of the PDE.
Content:
Front Matter....Pages i-x
Introduction....Pages 1-3
Presentation of the Approach and of the Main Results....Pages 4-14
The Transport of Finite-Dimensional Contact Elements....Pages 15-20
Spectral Blocking Property....Pages 21-24
Strong Squeezing Property....Pages 25-28
Cone Invariance Properties....Pages 29-32
Consequences Regarding the Global Attractor....Pages 33-35
Local Exponential Decay Toward Blocked Integral Surfaces....Pages 36-37
Exponential Decay of Volume Elements and the Dimension of the Global Attractor....Pages 38-41
Choice of the Initial Manifold....Pages 42-46
Construction of the Inertial Manifold....Pages 47-51
Lower Bound for the Exponential Rate of Convergence to the Attractor....Pages 52-54
Asymptotic Completeness: Preparation....Pages 55-60
Asymptotic Completeness: Proof of Theorem 12.1....Pages 61-67
Stability with Respect to Perturbations....Pages 68-71
Application: The Kuramoto—Sivashinsky Equation....Pages 72-81
Application: A Nonlocal Burgers Equation....Pages 82-90
Application: The Cahn—Hilliard Equation....Pages 91-104
Application: A Parabolic Equation in Two Space Variables....Pages 105-110
Application: The Chaffee—Infante Reaction—Diffusion Equation....Pages 111-118
Back Matter....Pages 119-126
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