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

Abstract

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.

Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

“…Moreover, it would be interesting to extend the present considerations to SPDEs on unbounded domains that are intensely studied over the past few years, cf. [4,11]. Such studies are currently in progress and will be reported in future publications.…”

“…Models of modulation or amplitude equations [4,12,21,38] have proven to be rather universal and efficient in describing the dynamics associated with a qualitative change of stability (bifurcation). Such structures emerge in fields ranging from spatially and temporarily oscillating wave packets [23] to long waves in dispersive media [36] and spatio-temporal pattern in dissipative systems [31].…”

Modulation Equation and SPDEs on Unbounded Domains

“…The key result towards a full result for amplitude equations on the whole real line with space-time white noise is by Blömker and Bianchi [2]. Here the full approximation result for linear SPDEs, namely the Swift-Hohenberg and Ginzburg-Landau equations without cubic terms, is established.…”

“…This is a monotone increasing sequence of spaces of continuous functions with growth condition at ±∞ for ̺ > 0. See also Bianchi, Blömker [2].…”

“…We refrain from giving all the lengthy details of this proof here. More details on the estimates used can for example be found in Lemma 2.4 and Lemma 3.1 in [2], where all tools necessary to prove this lemma are presented.…”