Blömker, D., Hairer, Martin and Pavliotis, G. A. (2007) Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities. Nonlinearity, Vol.20 (No.7). pp. 1721-1744. doi:10.1088/0951-7715/20/7/009

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Abstract

In this paper we derive rigorously amplitude equations for stochastic partial differential equations 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 paper thus allows us to explain theoretically some recent numerical observations on stabilization with additive noise.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Nonlinearity
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0951-7715
Official Date:	July 2007
Dates:	Date Event July 2007 Published
Volume:	Vol.20
Number:	No.7
Number of Pages:	24
Page Range:	pp. 1721-1744
DOI:	10.1088/0951-7715/20/7/009
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	EPSRC Fellowship
Grant number:	EP/DO71593/10

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