Amplitude equations for SPDEs with cubic nonlinearities

Blömker, Dirk; Mohammed, Wael W.

doi:10.1080/17442508.2011.624628

Cited by 40 publications

(50 citation statements)

References 14 publications

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“…In a previous paper [6] we considered a similar setting. We studied the stochastic Swift-Hohenberg equation (SH) near its change of stability but on bounded domains.…”

Section: Introductionmentioning

confidence: 99%

Modulation Equation for Stochastic Swift--Hohenberg Equation

Mohammed¹,

Blömker²,

Klepel³

2013

SIAM J. Math. Anal.

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Abstract. The purpose of this paper is to study the influence of large or unbounded domains on a stochastic PDE near a change of stability, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider the stochastic Swift-Hohenberg equation and derive rigorously the Ginzburg-Landau equation as a modulation equation for the amplitudes of the dominating modes. We verify that small global noise has the potential to stabilize the modulation equation, and thus to destroy the dominant pattern.

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“…In a previous paper [6] we considered a similar setting. We studied the stochastic Swift-Hohenberg equation (SH) near its change of stability but on bounded domains.…”

Section: Introductionmentioning

confidence: 99%

Modulation Equation for Stochastic Swift--Hohenberg Equation

Mohammed¹,

Blömker²,

Klepel³

2013

SIAM J. Math. Anal.

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“…Using the higher order corrections (see, Lemma 18 in [8]) in the averaging result for the fast OU-process, one obtains an additional semi-martingale term in the amplitude equation.…”

Section: Higher Order Effectsmentioning

confidence: 99%

“…But then only weak convergence of the approximation is established, and no path-wise error bounds are available (cf. [8]). …”

Section: Higher Order Effectsmentioning

confidence: 99%

“…In [8] higher order corrections are studied for the solution of the stochastic Swift-Hohenberg Equation (13) with degenerate noise. Using the higher order corrections (see, Lemma 18 in [8]) in the averaging result for the fast OU-process, one obtains an additional semi-martingale term in the amplitude equation.…”

Section: Higher Order Effectsmentioning

confidence: 99%

“…Surprisingly, for global noise the amplitude a ∈ R 2 solves a deterministic ODE (cf. Blömker & Mohammed [8])…”

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confidence: 99%

See 2 more Smart Citations

Numerical study of amplitude equations for SPDEs with degenerate forcing

Blömker¹,

Mohammed

Nolde

et al. 2012

International Journal of Computer Mathematics

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Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

Mohammed

2015

Math Methods in App Sciences

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In this paper, we consider a quite general class of reaction-diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg-Landau equation) is used to illustrate our result.In the case of nonhomogeneous boundary conditions (i.e., ¤ 0). Sowers [2] has studied parabolic reaction diffusion equation with Neumann boundary conditions by a detailed analysis involving fundamental solutions. Da Prato and Zabczyk [3,4] explained the difference between the problems with Dirichlet and Neumann boundary noises. Moreover, several authors, see for example Chueshov,

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Amplitude equations for SPDEs with cubic nonlinearities

Cited by 40 publications

References 14 publications

Modulation Equation for Stochastic Swift--Hohenberg Equation

Modulation Equation for Stochastic Swift--Hohenberg Equation

Numerical study of amplitude equations for SPDEs with degenerate forcing

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

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