Inertial Manifolds for Nonautonomous Evolution Equations

Koksch, N.; Siegmund, Stefan

doi:10.1142/9789812794253_0062

Cited by 7 publications

(6 citation statements)

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“…Remark 4.3. Both asymptotic completeness and cone invariance are important tools to study the inertial manifold of deterministic infinite dimensional systems [20,21,16]. Here we modified both concepts for random systems.…”

Section: Dynamical Approximationsmentioning

confidence: 99%

A dynamical approximation for stochastic partial differential equations

Wang

Duan

2007

Journal of Mathematical Physics

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Random invariant manifolds provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states ͑invariant measures͒ are considered for a class of stochastic hyperbolic partial differential equations.

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Section: Dynamical Approximationsmentioning

confidence: 99%

A dynamical approximation for stochastic partial differential equations

Wang

Duan

2007

Journal of Mathematical Physics

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“…Proof and further rigorous properties can be found in [1] and [36]. In this article, since we are interested in the computational aspect, we sketch the way that we use to find Ψ, and for more details, we refer readers to our follow-up work [8] and [9].…”

Section: Non-autonomous Inertial Manifold Reductionmentioning

confidence: 99%

“…[24,10,39,38,34,11,30] and [24,10,39,38,34,11,7,30], respectively. Recently, the theory of inertial manifolds has been generalized to non-autonomous dynamical systems [40,4,27,36], and recently, to random dynamical systems [35], and [3] (and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Global error analysis and inertial manifold reduction

Chung

Steyer

Tubbs

et al. 2016

Journal of Computational and Applied Mathematics

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Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to rescaling of time. To determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto-Sivashinsky equation.

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“…Both asymptotic completeness and cone invariance are important tools to study the inertial manifold of deterministic infinite dimensional systems. 24,25,19 Here we modified both concepts for random systems.…”

Section: Invariant Manifold Reductionmentioning

confidence: 99%