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Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities
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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 ISSN 0951-7715.
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Official URL: http://dx.doi.org/10.1088/0951-7715/20/7/009
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 |
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| 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: |
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| 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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