薄区域上无穷维随机动力系统的动力学行为

Infinite dimensional random dynamical systems play a very important role in nonlinear science. Thin domains problems are encountered in solid mechanics and fluid dynamics. In this project, we plan to study the dynamical behavior of infinite dimensional random dynamical systems on thin domains, including (1) Random attractor of stochastic dissipative non-autonomous partial differential equations on thin domains; (2) Invariant manifolds for stochastic partial differential equations on thin domains. Since the stochastic equations are defined on thin domains, we will study that the dynamical behavior of infinite dimensional random dynamical systems generated by the stochastic equations depends continuously on the domain when thin domains collapsing onto a lower dimensional subspace. Since the domain is singularly perturbed, it’s difficult to study such problem. We need to combine methods and techniques in stochastic analysis, functional analysis, and partial differential equations to study it. Because this project arises from many practical problems, it has a very important theoretical significance and application value. By studying this project, we will perfect and deepen the study of infinite dimensional random dynamical systems, so it is a very meaningful work.

无穷维随机动力系统在非线性科学中占有极为重要的地位。在固体力学以及流体动力学中，有许多薄区域问题。本项目计划研究薄区域上无穷维随机动力系统的动力学行为，主要包括：（1）薄区域上耗散型随机非自治偏微分方程的随机吸引子；（2）薄区域上随机偏微分方程的不变流形。 由于随机方程在薄区域上，我们将研究当区域从高维坍塌至低维时，由随机方程所产生的无穷维随机动力系统动力学行为对区域的依赖性。此类问题研究的主要困难是区域奇异摄动，故我们需要通过结合随机分析，泛函分析和偏微分方程等领域的思想方法和工具对它们进行研究。本项目来源于实际问题，具有重要的理论意义与应用价值。本课题的完成将完善和深化对无穷维随机动力系统的研究， 是十分有意义的工作。

无穷维随机动力系统在非线性科学中占有极为重要的地位。在固体力学以及流体动力学中，有许多薄区域问题。我们已经研究薄区域上无穷维随机动力系统的动力学行为，主要包括：（1）薄区域上耗散型随机非自治偏微分方程的随机吸引子；（2）薄区域上随机偏微分方程的不变流形。 由于随机方程在薄区域上，我们研究当区域从高维坍塌至低维时，由随机方程所产生的无穷维随机动力系统动力学行为对区域的依赖性。此类问题研究的主要困难是区域奇异摄动，故我们需要通过结合随机分析，泛函分析和偏微分方程等领域的思想方法和工具对它们进行研究。本项目来源于实际问题，具有重要的理论意义与应用价值。本课题的完成完善和深化对无穷维随机动力系统的研究， 是十分有意义的工作。