Multiscale Analysis for SPDEs with Quadratic Nonlinearities

TL;DR

This paper rigorously derives amplitude equations for stochastic PDEs with quadratic nonlinearities, revealing how noise on fast modes influences slow modes through multiscale analysis, with applications to the stochastic Burgers equation.

Contribution

It introduces a novel multiscale framework for deriving amplitude equations in SPDEs with quadratic nonlinearities, accounting for noise transmission between modes.

Findings

Amplitude equations include both additive and multiplicative noise.

Noise on fast modes affects slow modes via multiscale interactions.

Application to stochastic Burgers equation explains recent numerical results.

Abstract

In this article we derive rigorously amplitude equations for stochastic PDEs with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one dimensional Burgers equation forced by additive noise in the orthogonal subspace to its dominant modes. The theory developed in the present article thus allows to explain theoretically some recent numerical observations from [Rob03].