随机偏微分方程

项目名称： 随机偏微分方程

项目编号： No.11726627

项目类型： 专项基金项目

立项/批准年度： 2018

项目学科： 数理科学和化学

项目作者： 王凤雨

作者单位： 天津大学

项目金额： 20万元

中文摘要： 在自然界中，任何物体都会受到其他物体的干扰，或因物体自身的振动等原因而产生的自我干扰，我们把这些干扰统称为噪声扰动。有些噪声的影响是微弱的，即不会改变其自身的主要性质。相反，有些噪声的影响则是强烈的，可以发生本质性的改变。假设一个系统在理想的情况下运转，或者假设此系统具有很强的抗干扰能力，则此系统可以用确定型(偏)微分方程来描述。如果一个系统对噪声敏感，则要用随机(偏)微分方程来描述。由此可知，研究随机(偏)微分方程的必要性。本项目拟研究来源于统计物理、经济、生物、神经网络等学科领域中出现的几类随机(偏)微分方程，讨论其解的适定性、正则性，动力学性质和噪声的影响，以及偏微分方程解的概率表示。内容包括:(1) 随机流体方程解的适定性；(2) 随机偏微分方程解的正则性;(3) 随机行波解的存在唯一性和偏微分方程解的概率表示。

中文关键词： 随机偏微分方程；不变测度；随机Navier-Stokes方程；遍历性；适定性

英文摘要： In the real world, any object will be disturbed by others or by itself vibration, and we call these interferences as noise perturbations. Some impacts of noise are weak, that is, the main natures of the object will not be changed. However, some impacts of noise are strong and it can change the main natures. Suppose a system stays in an ideal situation, or suppose the system has strong anti-interference ability, then the system can be described by deterministic (partial) differential equations. But if the system is sensitive for noise perturbations, it will be described by stochastic (partial) differential equations. Therefore, the study of stochastic (partial) differential equations is necessary. This project is planned to study some kinds of stochastic (partial) differential equations which derived from statistical physics, Economics, Biology,.Neural network and so on. We will consider the well-posedness, regularity, dynamic properties and the impact of noise. Moreover, the probabilistic representations of solutions to the partial differential equations will be also considered. The contents contain: (1) the well-posedness of stochastic fluid equations; (2) the regularity of solutions to the stochastic partial differential equations; (3) the existence and uniqueness of stochastic traveling wave solutions and the probabilistic representations of solutions to the partial differential equations.

英文关键词： stochastic partial differential equation;invariant measure;stochastic Navier-Stokes equation;ergodicity;well-posedness