面向流体力学的随机偏微分方程的分析和渐近性质研究

Turbulence is considered as classic problems in physics, but also raises many challenging mathematical questions. The models are typically described by Navier-Stokes equations. The objectives of the statistical study of turbulence is to understand the mechanism of turbulence from the basic physical conservation laws --- N-S equation and the continuity equation. From the physical point of view, the limiting behavior of the dynamical system generated by stochastic partial differential equations correspond to different states in the real world.The ergodicity of stochastic dynamical systems is to describe the long time behavior of the corresponding systems. The supporting side of the traverse area is the area where turbulence occurred. One can get a good understanding of the turbulence through analyzing the properties of the ergodic measures. It is therefore fundamental to study the ergodicity (long time bahaviour)of the stochastic partial differential equations in fluid mechanics.The objectives of this project want to build and develop necessary mathematical methods to study the analysis and progressive property of stochastic partial differential equations, particularly for equations related to hydrodynamics.The main contents to study are the ergodicty and convergence rate, large deviation,functional inequalities, concentration of measure ets.

湍流是重大基础科学问题。湍流的控制方程是Navier-Stokes方程。湍流的统计理论的目标是从最基本的物理守恒定律---N-S方程出发和连续性方程出发，探讨湍流的机理。从物理上看，随机偏微分方程对应的动力系统的极限状态对应现实世界不同状态。遍历性是描述随机动力系统的极限状态的一种数学方法。支撑遍历测度的区域是湍流发生的区域。湍流发生的规律可以从遍历测度的性质加以分析。所以研究随机扰动下的流体力学方程的遍历性是认识湍流的一种重要手段。本项目以流体力学方程为背景，研究Levy过程驱动的随机偏微分方程的解的各种性质，包括遍历性、在不同拓扑下的指数收敛性、小扰动大偏差、占位时大偏差、随机流、 半群的梯度估计与Harnack不等式、Dirichlet型的泛函不等式、不变测度的集中不等式等。本项目的研究不仅局限于随机扰动下的流体力学方程，并力争对一般的随机偏微分方程发展出有效的研究方法。

本项目研究了两类相关问题： 随机流体力学方程；奇异的以及退化、路径和分布依赖的随机微分方程的分析性质...研究的问题之一: 随机流体力学方程.在随机扰动下，基于流体力学方程解的分析性质和长时间行为，项目在随机Navier-Stokes方程、随机洋流体方程、随机非牛顿流体方程、无粘性守恒率方程以及一般随机偏微分方程的大偏差、遍历性等问题做了深入的研究。获得了系列创新性成果。..研究的问题之二: 奇异的以及退化、路径和分布依赖的随机微分方程的分析性质..关于随机偏微分方程各种性质的研究通常需要系数具有一定的正则性，而大量的现实模型是奇异的，并不满足这些条件。因此刻画奇异随机系统的有关性质非常重要，近年来在有限维情形已取得许多进展。我们在奇异的随机偏微分方程的正则性和遍历性、退化、路径和分布依赖的随机偏微分方程的研究方面取得了一系列新进展。