流体力学中若干随机偏微分方程模型的研究

Because of combining both features of partial differential equations and random noise, the study of stochastic partial differential equations will be much harder, and many properties of partial differential equations fail in the stochastic case. However, the synchronization property is one of the very few exceptions, it basically says that the random noise can make the long time behavior of some stochastic equations become much simpler than the deterministic case, and this property is the central point of all study in this project. The main subjects of this project are a class of stochastic partial differential equation models which has important applications in fluid mechanics and other fields, and the final target is to study whether the synchronization property holds for this type of models, meanwhile we will intensively investigate the relations between ergodicity and random attractor, which are two main tools to study the long time behavior of stochastic dynamical systems. For this reason, we will first study the well-posedness of this class of stochastic partial differential equations using the generalized variational framework, especially for the linear multiplicative noise case; then we will investigate the large deviation principle, ergodicity and random attractors for those SPDEs; we will also study the influence of random noise on the structure/dimension of random attractors and under which circumstances the noise has the synchronization property.

由于随机偏微分方程结合了偏微分方程和随机噪声两方面的特性，这使得随机偏微分方程问题的研究更加困难，很多确定性方程所具有的性质在随机噪声的影响下往往不再成立。然而噪声的同步性质是少数例外之一，同步性质直观上是指噪声使得某些随机方程的长时间渐近行为比确定性方程变得更为简单，这一性质是本项目主要研究内容围绕的中心点。本项目的研究对象是一类在流体力学等领域具有重要应用的随机偏微分方程模型，目标是研究该类模型中随机噪声是否具有同步性质，并重点探讨遍历性与随机吸引子这两种描述系统长时间渐近行为的工具之间的关系。为此，我们将在推广的变分框架下研究这一类随机偏微分方程解的存在唯一性，特别是线性可乘噪声驱动的情形；并在此框架下研究随机偏微分方程解的大偏差原理，遍历性和随机吸引子的存在性等渐近性质；我们也将探讨随机噪声对随机吸引子结构(维数)的影响，特别是在何种情况下噪声具有同步性质。

本项目的研究对象是一类在流体力学和数学物理等领域具有重要应用的随机偏微分方程模型，主要研究内容是在新的变分框架下研究随机偏微分方程遍历性质与随机吸引子之间的相互关系和(带可乘/退化噪声)随机偏微分方程的遍历性等问题，其中重点探讨随机噪声对动力系统长时间渐近行为(如稳定性等)的影响。经过4年的系统研究，我们取得了如下主要成果：(1) 对于一类带分数阶时间导数的拟线性(随机)偏微分方程，通过引入新的广义狄氏型方法，我们解决了关于解的存在唯一性这一公开问题；(2) 运用弱收敛方法和指数等价性方法证明了一类随机偏微分方程的大偏差原理，刻画了一类具有局部单调系数的随机偏微分方程的解在小噪声情形和小时间情形的渐近性质；(3) 系统研究了若干流体力学随机偏微分方程模型和对应随机动力系统的渐近性质，揭示了不同类型噪声对于随机系统长时间渐近行为的影响。