Modulation Equation and SPDEs on Unbounded Domains

Abstract

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 the dynamics close to 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. 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.