非线性偏微分方程解的渐近性态研究

项目名称： 非线性偏微分方程解的渐近性态研究

项目编号： No.11471148

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： 孙春友

作者单位： 兰州大学

项目金额： 72万元

中文摘要： 在不求解偏微分方程的情况下能直接由方程来研究其解的性质，无疑对了解非线性系统的整体特征有重要的指导意义。本项目就是运用非线性分析和无穷维动力系统的思想方法对来自控制论、反常扩散流体等领域的耗散偏微分方程解的渐近性态进行深入研究。对定义在非柱形区域上的偏微分方程，将研究所对应的无穷维动力系统的吸引子相关问题，建立研究该类方程吸引子问题的针对性理论框架，探索处理这类方程的先验估计方法，并通过构造具体方程和区域来研究吸引子的结构变化，丰富无穷维动力系统的理论和方法。对非局部方程，先重点研究临界2D拟地转方程，将围绕解决黏性系数趋于 0 时解的长时间平均极限这一公开问题展开，并研究固定耗散系数时解的长时间平均、吸引子的存在性和正则性问题，再建立能反映这类方程特性的吸引子相关问题的研究框架和应用。这些研究工作的开展，无论是对应用问题的深入理解，还是对无穷维动力系统理论和应用的发展，都将有积极的推动。

中文关键词： 非线性分析；无穷维动力系统；非线性偏微分方程；吸引子

英文摘要： It is undoubtedly meaningful to understand the characters of nonlinear systems on the condition that we study the properties of the partial differential equations by exploring the equations themselves directly without any explicit solutions. In this project, we investigate the asymptotic behaviors of solutions to the partial differential equations that originate mainly from the control theory, the anomalous diffusion fluid and so on by applying the ideas and methodology developed in nonlinear analysis and infinite-dimensional dynamical systems. Concerning the partial differential equations defined on non-cylindrical domains, we study its asymptotic behaviors by characterizing the corresponding attractors and the relative problems, establish the particular dynamical theory, explore a priori estimates for this kind partial differential equations, investigate the variation of the structure of attractors by constructing some special equations and domains, and push the development of infinite-dimensional dynamical systems. With respect to the asymptotic behaviors of nonlocal partial differential equations, we focus firstly on the critical 2D surface quasi-geostrophic equation by studying the open problem about the limitation of anomalous dissipation of kinetic energy as the kinematic viscosity goes to 0, the long time average of solutions and the existence and its regularity of the attractors for the case that the kinematic viscosity is fixed; then we establish some new frameworks and methodology which can and will manifest the characters of nonlocal. The results of this project will have significant impacts not only on the deep understanding of the original application problems, but also on the development of theory as well as applications for nonlinear analysis and infinite-dimensional dynamical systems.

英文关键词： Nonlinear Analysis;Infinite-Dimensional Dynamical Systems;Nonlinear Partial Differential Equations;Attractor