Two-dimensional stochastic Navier–Stokes equations with fractional Brownian noise
Two-dimensional stochastic Navier–Stokes equations with fractional Brownian noise
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DOI：
10.1515/rose-2013-0008
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发表时间：
2013-06
期刊：
影响因子：
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通讯作者：
L. Fang;P. Sundar;F. Viens
中科院分区：
文献类型：
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作者：
L. Fang;P. Sundar;F. Viens
摘要。研究了二维随机Navier-Stokes方程被hilbert空间值分数布朗噪声扰动的问题。每个希尔伯特分量都是标量分数布朗噪声，具有共同的赫斯特参数H和特定强度。由于噪声是可加性的，简单的维纳型积分足以适当地定义问题。将其分解为确定性非线性偏微分方程和线性随机偏微分方程。在适当的条件下，对所有Hurst参数值的噪声强度建立了温和解的存在唯一性。几乎可以肯定的是，解的路径在时间和空间上是可积的。这种可积性是否扩展到随机参数是一个悬而未决的问题。给出了多重分形模型的推广。
Abstract. We study the perturbation of the two-dimensional stochastic Navier–Stokes equation by a Hilbert-space-valued fractional Brownian noise. Each Hilbert component is a scalar fractional Brownian noise in time, with a common Hurst parameter H and a specific intensity. Because the noise is additive, simple Wiener-type integrals are sufficient for properly defining the problem. It is resolved by separating it into a deterministic nonlinear PDE, and a linear stochastic PDE. Existence and uniqueness of mild solutions are established under suitable conditions on the noise intensities for all Hurst parameter values. Almost surely, the solution's paths are shown to be quartically integrable in time and space. Whether this integrability extends to the random parameter is an open question. An extension to a multifractal model is given.