非线性偏微分方程稳态统计性质的研究

It is well known that the nonlinear partial differential equations have attracted more and more attention owing to their widespread existence and potential applications in "Nature and Human life", and the corresponding stationary statistical properties serve also as important issues in modern mathematics. Although in many cases such properties may not be known explicitly, there are enough firm results available assuring that many of the widely accepted experimental results are meaningful. This project aims on solving the open problem proposed by Prof. C. Foias, including how to construct the invariant measure for the multi-solution systems and how to get the high-order integrability and regularity of the attractors of the special systems. Firstly, we will begin with the construction of invariant measure for the nonlinear weakly dissipative wave equation and three-dimensional Navier-Stokes equation. Then, based on these special and important systems, we are supposed to establish the framework for studying the stationary statistical properties for multi-solution systems. Secondly, we will study the high-order integrability and regularity of the attractors of the special systems. Much of our effort is directed in proving the existence and uniqueness of weak global solutions as well as establishing the continuity and pullback attractions in high-order integrable and regular spaces for the nonlinear semilinear degenerate parabolic equation and fractional reaction-diffusion equation on time-varying domains. In a word, all of above work will not only partly answer the open problem proposed by C. Foias, but also play a positive role for further investigation on the stationary statistical properties of complex systems.

非线性偏微分方程因在“人与自然”中的广泛存在而备受关注，其稳态统计性质已经成为当今数学研究的重要课题之一。然而由于某些偏微分方程的复杂结构，给系统解的长时间行为研究带来了许多困难。本项目拟针对数学家C.Foias等提出的开问题，致力于复杂非线性偏微分方程稳态统计性质的研究，包括：(1)多解系统不变测度的构造性问题研究。针对两类经典多解模型——弱耗散波方程和三维Navier-Stokes方程，探究系统不变测度的构造方式，建立多解系统稳态统计性质研究的理论框架；(2)特性系统吸引子的高阶可积性和正则性问题研究。针对两类具体变区域上的退化抛物方程和分数阶反应扩散方程，讨论系统解的适定性、连续依赖性以及吸引子在高阶可积空间和正则空间中的存在性等问题。这些工作的展开不仅部分回答了C.Foias等人提出开问题，还将为复杂系统稳态统计性质的深入探究奠定重要的理论基础。

非线性偏微分方程因在“人与自然”中的广泛存在而备受关注，其稳态统计性质已经成为当今数学研究的重要课题之一。从动力学角度讲，系统稳态统计性质的研究侧重于对动力学吸引子存在性问题的讨论。作为刻画系统渐近性态最重要的概念之一，吸引子是相空间中具有“紧性、吸引性和不变性”的集合。它不仅包含了系统所有可能的极限状态，还可以在不丢失任何信息的前提下对原方程实现简化和降维。与此同时，系统稳态统计性质还可以从随机变化的角度描述，并以不变Borel概率测度的形式表现出来。这种测度不仅可以刻画方程的物理性质，还可以为系统轨道的运动趋势提供重要信息。本项目主要以“吸引子内部结构”和“不变测度”两方面内容为出发点，探究经典复杂特性系统的渐近性态，主要包括解的连续依赖性、不变测度的构造性以及动力学吸引子在高阶可积空间和正则空间中的存在性等问题。一方面，提出了“连续依赖性”和“高阶吸引性和正则性”研究框架。针对全空间、有界柱型区域以及两类变区域（同胚区域和单调区域）上的分数阶反应扩散方程和退化抛物方程，建立了新型Nash-Moser-Alikakos估计，通过利用稠密性分析等方法，给出了系统“解的连续依赖性”和“吸引子的高阶吸引性和正则性”结果；另一方面，基于Krylov-Bogoliubov定理和广义“极限解”的基本思想，通过引入Birkhoof遍历性假设和收敛性讨论等方法，针对有界区域上非局部抛物方程的稳态统计性质进行了系统地思考：给出了具有不变结构时间平均测度的构造方式和系统稳态统计解的合理定义，证明了该不变测度与系统稳态统计解的等价关系。本课题共撰写、完成学术论文7篇，其中4篇已在SCI期刊发表，3篇已完成、在投稿过程中。