Amplitude Equations for Stochastic Partial Differential Equations

Blömker, Dirk

doi:10.1142/6440

Cited by 49 publications

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“…There have been rapid progresses in this area. 12,22,9,26,2,14 More recently, SPDEs have been investigated in the context of random dynamical systems ͑RDSs͒; 1 see, for example Refs. 3,5,4,8,7,23,10, and 11, among others.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

A dynamical approximation for stochastic partial differential equations

Wang

Duan

2007

Journal of Mathematical Physics

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“…The second main results studies the higher order correction for the solution of equation (1). As indicated for the fast OU-process in Section 2.1, we obtain additional Martingale terms that lead to additive noise in an equation for the higher order correction of the amplitude, but the strength of the noise depends on the first order approximation.…”

Section: Higher Order Correctionsmentioning

confidence: 68%

“…When the order of the noise strength is comparable to the order of the distance from the change of stability, the impact of noise can be seen. See for example [1] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Other norms like the supremum-norm or the L p -norm would lead to similar results. Our aim of this paper is to establish rigorously an amplitude equation and their higher order corrections for this quite general class of SPDEs with cubic nonlinearities given by (1). In the examples we will show that additive degenerate noise leads to stabilization of the solutions.…”

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confidence: 99%

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Amplitude equations for SPDEs with cubic nonlinearities

Blömker

Mohammed

2011

Stochastics

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For a quite general class of SPDEs with cubic nonlinearities we derive rigorously amplitude equations describing the essential dynamics using the natural separation of time-scales near a change of stability. Typical examples are the Swift-Hohenberg equation, the Ginzburg-Landau (or Allen-Cahn) equation and some model from surface growth.We discuss the impact of degenerate noise on the dominant behavior, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.

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“…If the additive noise acts on the dominant modes, then we need to change scaling and consider smaller noise. See for example [BH04] or [Blö07] Thus we consider the following stochastic generalized Swift Hohenberg equation:…”

Section: Introductionmentioning

confidence: 99%

Amplitude equation for the generalized Swift–Hohenberg equation with noise

Klepel

Blömker

Mohammed

2013

Z. Angew. Math. Phys.

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Abstract. We derive an amplitude equation for a stochastic partial differential equation (SPDE) of Swift-Hohenberg type with a nonlinearity that is composed of a stable cubic and an unstable quadratic term, under the assumption that the noise acts only on the constant mode. Due to the natural separation of timescales, solutions are approximated well by the slow modes. Nevertheless, via the nonlinearity, the noise gets transmitted to those modes too, such that multiplicative noise appears in the amplitude equation.

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Amplitude Equations for Stochastic Partial Differential Equations

Cited by 49 publications

References 0 publications

A dynamical approximation for stochastic partial differential equations

A dynamical approximation for stochastic partial differential equations

Amplitude equations for SPDEs with cubic nonlinearities

Amplitude equation for the generalized Swift–Hohenberg equation with noise

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