Modulation Equation for Stochastic Swift--Hohenberg Equation

Mohammed

2013

Z. Angew. Math. Phys.

Self Cite

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.

Section: 2mentioning

confidence: 65%

Amplitude equation for the generalized Swift–Hohenberg equation with noise

Klepel

Mohammed

2013

Z. Angew. Math. Phys.

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“…The first results for modulation equations for Swift-Hohenberg on the whole real line were presented by Klepel, Mohammed, and Blömker [27,20]. Here the authors used spatially constant noise of a strength of order ε, which is stronger than the one treated here.…”

Section: Modulation Equations For Spdesmentioning

confidence: 96%

Modulation Equation and SPDEs on Unbounded Domains

Bianchi

Schneider

2019

Commun. Math. Phys.

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We consider the approximation via modulation equations for nonlinear SPDEs on unbounded domains with additive space time white noise. Close to a bifurcation an infinite band of eigenvalues changes stability, and we study the impact of small space-time white noise on this bifurcation.As a first example we study the stochastic Swift-Hohenberg equation on the whole real line. Here due to the weak regularity of solutions the standard methods for modulation equations fail, and we need to develop new tools to treat the approximation.As an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic Swift-Hohenberg and the complex Ginzburg Landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions, which is natural for translation invariant noise like space-time white noise. Moreover we use energy estimates to show that solutions of the Ginzburg-Landau equation are Hölder continuous and have moments in those functions spaces. This gives just enough regularity to proceed with the error estimates of the approximation result.

“…With these bounds and some less optimal bounds for t 1 from [31], we immediately obtain the following Lemma:…”

Section: Semigroups and Green's Functionmentioning

confidence: 97%

Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory

Bianchi

Stochastic Processes and their Applications

2016

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We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation.As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift-Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space-time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.Date: November 5, 2018. 2010 Mathematics Subject Classification. 60H15, 60H05, 60G15.