无穷维随机微分系统的适定性与渐近动力学研究

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.

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。

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。该项目主要研究了全局修正的倒向随机三维Navier-Stokes的适定性，建立了适应解的存在唯一性定理；研究了与二维时滞Navier-Stokes方程的随机控制有关的超前倒向随机二维Navier-Stokes方程的适应强解的存在唯一性；研究了非线性项满足局部Lipschitz条件的统一混沌系统在不连续随机干扰下的混沌控制；研究了带或不带随机扰动的反应扩散型时滞神经网络耦合系统的同步动力学；通过带Poisson跳的复Itô公式研究了带跳的倒向随机Schrödinger方程适应解的存在唯一性；引入了非线性期望下随机变量指数独立的概念，研究了非线性期望下的强大数定律等，研究结果对发展和丰富倒向随机偏微分方程与随机控制、复杂网络系统的动力学分析、非线性随机分析等理论有重要的科学意义和应用价值。