Validity of the stochastic Ginzburg-Landau approximation in higher space dimensions -A Wiener algebra approach-

TL;DR

This paper rigorously justifies the use of a stochastic Ginzburg-Landau equation as an approximation for pattern formation near the first instability in higher-dimensional Swift-Hohenberg models with additive noise, using a Wiener algebra approach.

Contribution

It provides a mathematical proof of the approximation's validity in higher dimensions and extends existing results to allow larger stable noise components.

Findings

Validates the stochastic Ginzburg-Landau approximation in higher dimensions.

Extends approximation validity to larger stable noise sets.

Uses Wiener algebra techniques for rigorous proof.

Abstract

We consider an anisotropic $d$-dimensional Swift-Hohenberg model $ \mathcal{O}(\varepsilon^2) $-close to the first instability, where $ 0 < \varepsilon \ll 1 $ is a small perturbation parameter. This model for pattern formation is perturbed with additive noise in time and space. By a multiple scaling ansatz we derive a stochastic $ d $-dimensional Ginzburg-Landau equation for the approximate description of the bifurcating solutions. We prove the validity of the approximation by this amplitude equation on its natural time scale in case of $ d $-dimensional periodic domains of length $ \mathcal{O}(1/\varepsilon) $ for the Swift-Hohenberg model under suitable conditions on the additive noise. In detail, we prove the validity of this approximation for noise whose set of Fourier coefficients with respect to $ x $ is in $ \ell^1 $ for fixed $ t \geq 0 $. Moreover, we improve existing approximation results in the sense that the stable part of the noise can be larger.