Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities
Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities
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DOI：
10.1088/0951-7715/20/7/009
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发表时间：
2007-07-01
期刊：
影响因子：
1.7
通讯作者：
Pavliotis, G. A.
中科院分区：
文献类型：
--
作者：
Bloemker, D.;Hairer, M.;Pavliotis, G. A.
在本文中，我们推导出严格的振幅方程的随机偏微分方程的二次非线性，假设噪声只作用于稳定的模式和适当的缩放之间的距离从分歧和噪声的强度。我们发现，由于存在两个不同的时间尺度在我们的系统中，噪声（它只作用于快模式）被传输到慢模式，因此，振幅方程包含加性和乘性noise.As应用我们研究的情况下，一维Burgers方程的正交子空间中的加性噪声迫使其主导模式。因此，在本文件中开发的理论，使我们能够从理论上解释最近的一些数值观测稳定加性噪声。
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.