Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory
Modulation equation for SPDEs in unbounded domains with space–time white noise — Linear theory
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DOI：
10.1016/j.spa.2016.04.024
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发表时间：
2015-07
影响因子：
1.4
通讯作者：
L. Bianchi;D. Blomker
中科院分区：
文献类型：
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作者：
L. Bianchi;D. Blomker
我们通过调制或振幅方程研究了稳定性变化附近整条真实的线上的SPDE的近似，它可以替代扩展域上缺乏随机不变流形的问题。由于基础域的无界性，无穷多个本征函数的整个带改变稳定性。因此，我们期望不仅是一个慢动作的时间，但也是一个缓慢的空间调制的主导模式，这是所描述的调制equation.As的第一步，对一个完整的理论调制方程的非线性SPDEs上无界域，我们专注于，在这里提出的结果，在线性理论的一个特定的例子，Swift-Hohenberg方程。这些线性结果是将确定性近似结果推广到具有附加强迫的随机情况的关键技术工具之一。建立误差估计的一个技术问题来自于无界域上时空白色噪声的空间平移不变性质，这意味着在任何时候，我们都可以预期误差在空间的某个地方总是非常大的。
We study the approximation of SPDEs on the whole real line near a change of stability via modulation or amplitude equations, which acts as a replacement for the lack of random invariant manifolds on extended domains. Due to the unboundedness of the underlying domain a whole band of infinitely many eigenfunctions changes stability. Thus we expect not only a slow motion in time, but also a slow spatial modulation of the dominant modes, which is described by the modulation equation.As a first step towards a full theory of modulation equations for nonlinear SPDEs on unbounded domains, we focus, in the results presented here, on the linear theory for one particular example, the Swift–Hohenberg equation. These linear results are one of the key technical tools to carry over the deterministic approximation results to the stochastic case with additive forcing. One technical problem for establishing error estimates rises from the spatially translation invariant nature of space–time white noise on unbounded domains, which implies that at any time we can expect the error to be always very large somewhere in space.