Approximation of stochastic invariant manifolds : stochastic manifolds for nonlinear SPDEs I / by Mickaël D. Chekroun, Honghu Liu, Shouhong Wang.

Format:

Book

Author/Creator:

Chekroun, Mickaël D., Author.

Liu, Honghu, Author.

Wang, Shouhong, Author.

Series:

SpringerBriefs in Mathematics, 2191-8198

Language:

English

Subjects (All):

Dynamics.

Ergodic theory.

Differential equations, Partial.

Probabilities.

Differential equations.

Dynamical Systems and Ergodic Theory.

Partial Differential Equations.

Probability Theory and Stochastic Processes.

Ordinary Differential Equations.

Local Subjects:

Dynamical Systems and Ergodic Theory.

Partial Differential Equations.

Probability Theory and Stochastic Processes.

Ordinary Differential Equations.

Physical Description:

1 online resource (136 p.)

Edition:

1st ed. 2015.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2015.

Language Note:

English

Summary:

This first volume is concerned with the analytic derivation of explicit formulas for the leading-order Taylor approximations of (local) stochastic invariant manifolds associated with a broad class of nonlinear stochastic partial differential equations. These approximations take the form of Lyapunov-Perron integrals, which are further characterized in Volume II as pullback limits associated with some partially coupled backward-forward systems. This pullback characterization provides a useful interpretation of the corresponding approximating manifolds and leads to a simple framework that unifies some other approximation approaches in the literature. A self-contained survey is also included on the existence and attraction of one-parameter families of stochastic invariant manifolds, from the point of view of the theory of random dynamical systems.

Contents:

General Introduction

Stochastic Invariant Manifolds: Background and Main Contributions

Preliminaries

Stochastic Evolution Equations

Random Dynamical Systems

Cohomologous Cocycles and Random Evolution Equations

Linearized Stochastic Flow and Related Estimates

Existence and Attraction Properties of Global Stochastic Invariant Manifolds

Existence and Smoothness of Global Stochastic Invariant Manifolds

Asymptotic Completeness of Stochastic Invariant Manifolds

Local Stochastic Invariant Manifolds: Preparation to Critical Manifolds

Local Stochastic Critical Manifolds: Existence and Approximation Formulas

Standing Hypotheses

Existence of Local Stochastic Critical Manifolds

Approximation of Local Stochastic Critical Manifolds

Proofs of Theorem 6.1 and Corollary 6.1

Approximation of Stochastic Hyperbolic Invariant Manifolds

A Classical and Mild Solutions of the Transformed RPDE

B Proof of Theorem 4.1

References.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

3-319-12496-X

OCLC:

898892853