非高斯噪声驱动的无穷维随机动力系统的动力学研究

The proposal is devoted to the study on dynamics of infinite-dimensional random dynamical systems driven by non-Gaussian noise on time-varying domain or lower-regularity space respectively. The proposal is firstly projected to study on the random dynamical systems driven by both Pure Levy noise and fractional Brownian motion, we focus on the following three parts: 1) Dynamics of stochastic fluid equation driven by both Levy noise and fractional Browmian motion on time-varyng domain, 2) Well-posedness and random dynamics of stochastic Ostrovsky equation driven by fractional Brownian motion or Levy noise in lower regularity space, and 3) basic theory of random dynamical systems generated by strochastic fractional Boussiesq equation driven by both Levy noise and fractional Brownian motion, which include the local well-posedness and global well-posedness, time and space regularity, the existence and uniqueness of the invariant measure, ergodicity, compactness of random dynamical systems, the existence of random attractor and random invaiant maniflod and so on. Based on the systematic research projected in this proposal, the effects on the solution of stochastic evolution equation can be provoded, include the regularity of initial values, time-varying domain, the type and property of non-Gaussian noise and the order of the fractional operators. Finaly, some new phenomenon and new technique are found or improved to deal with the difficulity occurred in the proposal research.

本项目主要研究时变区域上和低正则空间中非高斯噪声驱动无穷维随机动力系统的动力学。首次提出研究由纯跳Levy过程和分数布朗运动共同驱动的无穷维随机动力系统，研究包括三个方面：1）时变区域上Levy过程和分数布朗运动共同驱动的随机流体方程的动力学；2）低正则空间中Levy过程和分数布朗运动驱动的随机Strovsky方程的适定性和随机动力学研究；3）Levy过程和分数布朗运动共同驱动的空间分数次随机Boussinesq方程生成随机动力系统的基本理论。包括解的局部和整体适定性、时间和空间正则性、不变测度的存在唯一性、遍历性、随机动力系统的随机吸引子和随机不变流形的存在性和紧性等。通过对几类具有物理和流体背景的随机系统研究，分析随机系统的初值正则性、时变区域、非高斯噪声类型和分数阶的指数对随机发展方程生成的随机动力系统动力学的实质影响，改进和探索新的方法和技巧，并在研究中探索出现新现象。

该项目主要研究了高斯噪声、Levy噪声和alpha平稳噪声、退化噪声驱动的分数阶Boussinesq方程、MHD方程、随机分数阶耦合Ginzburg-Landau方程以及一类流体发展方程的适定性和不变测度的存在唯一性;在低正则空间中研究了分数布朗运动驱动的随机Ostrovsky方程的适定性. 研究了时变区域上非自治和随机部分耗散系统的适定性和随机吸引子的存在性,也研究了时间Caputo分数阶,空间分数阶随机Ginzburg-Landau方程和Navier-Stokes方程以及Boussinesq方程在高斯噪声、分数布朗运动和alpha平稳噪声驱动下mild解的存在性和正则性.最后研究了随机快慢系统的随机稳定流形、不变叶层以及随机不变流形的近似性质等，以及在Hilbert空间中加性高斯白噪声驱动的随机发展方程的解的Wong-Zakai逼近及随机稳定不变流形的逼近问题。采用随机平均方法研究了基于数据的非高斯alpha平稳噪声驱动的随机多尺度系统的低阶降维方法和参数估计. 相关研究结果发表在J.Differential Equations, Journal of Function Analysis, J.Math.Anal.Appl.,Nonlinear Analysis, 中国科学和Scientific Reports等杂志上.出版学术专著1部,发表标注基金资助的SCI论文39篇.培养毕业博士研究生3名,硕士研究生2名,出站博士后4名. 顺利完成了项目资助书的研究内容.