无穷维随机微分系统的适定性与渐近动力学研究

项目名称： 无穷维随机微分系统的适定性与渐近动力学研究

项目编号： No.11471190

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 陈章

作者单位： 山东大学

项目金额： 60万元

中文摘要： 在现实生活中，时滞和随机现象是普遍存在的，因此研究具有时滞效应和随机扰动的偏微分方程模型的长时间动力学与随机控制等问题是有实际意义的。本项目拟主要研究全局修正的三维非自治随机时滞Navier-Stokes方程的随机指数吸引子的存在性与上半连续性；Lévy噪音驱动的全局修正三维时滞Navier-Stokes方程的弱（强）解的整体适定性、遍历性、弱解与不变测度序列的收敛性；G-布朗运动驱动的全局修正超前倒向随机三维Navier-Stokes方程适应解的存在唯一性、解序列的收敛性、以及与非线性时滞偏微分方程解的随机表示之间的联系。该项目的研究有助于了解原三维随机时滞Navier-Stokes方程的长时间动力学行为和可为流体运动的随机控制、一大类非线性时滞偏微分方程解的随机算法等提供理论基础与依据，因此具有重要的科学意义和应用价值。

中文关键词： 时滞；随机指数吸引子；整体适定性；G-布朗运动；倒向随机微分方程

英文摘要： In the real world, delay and stochastic phenomenon are ubiquitous, so it is significant to investigate problems including long-time dynamics and stochastic control of partial differential equations with delay effects and stochastic perturbations. In this project, globally modified 3D Navier-Stokes equations with delay and stochastic perturbations will mainly be investigated, and the main contents are as follows. Firstly, existence and upper semicontinuity of random exponential attractor will be studied for globally modified non-autonomous 3D Navier-Stokes equations with delay and perturbations of additive noise. Secondly, global well-posedness of weak (strong) solution, ergodicity, and convergence of sequences of weak solutions and invariant measures will be studied for globally modified 3D Navier-Stokes equations with delay and perturbation of Lévy noise. Thirdly, existence and uniqueness of adapted solution and convergence of sequence of solutions will be studied for globally-modified and anticipated backward stochastic 3D Navier-Stokes equations, as well as stochastic representation of solutions for nonlinear delay partial differential equations associated to this anticipated backward stochastic differential systems. The study of this project can contribute to understand long-time dynamical behaviors of original stochastic 3D Navier-Stokes equations with delay, and provide the theoretical foundation for stochastic control of fluid motion and stochastic algorithm of solutions for a large kind of nonlinear delay partial differential equations and other fields, so it has important academic significance and application value.

英文关键词： delay;random exponential attractor;global well-posedness;G-Brownian motion;backward stochastic differential equations