项目名称: 耗散型动力系统吸引子的正则性研究
项目编号: No.11201204
项目类型: 青年科学基金项目
立项/批准年度: 2013
项目学科: 数理科学和化学
项目作者: 杨璐
作者单位: 兰州大学
项目金额: 23万元
中文摘要: 本项目主要研究耗散型动力系统吸引子的正则性性质,包括在不同外力项条件下吸引子可能达到的各种最优正则性,耗散性不同的非线性项对正则性的影响,以及各种不同的边界条件对正则性的影响。在具体应用方面,我们将重点考察各种具体的非线性发展方程,特别是各类临界指数问题(如临界波方程、非经典扩散方程、反应扩散方程等),以及带有各种不同的非线性边界条件的耗散方程。作为正则性在动力系统中的应用,将重点考虑吸引子的存在性及其维数估计(指数吸引子的存在性),不同正则空间中吸引性的联系,吸引子的上半连续性等问题。这些问题是无穷维动力系统的主要问题和活跃问题之一,对进一步深入研究吸引子的几何拓扑结构有着重要的理论和实际意义。
中文关键词: 耗散动力系统;吸引子;正则性;非线性边界;非线性发展方程
英文摘要: This project is to study the regularity of attractors for dissipative dynamical systems, we consider the optimal regularity of attractors with the different external forces and the different nonlinear effect on the regularity, meanwhile, we also study the different boundary effect on the regularity. In applications, we will explore different nonlinear evolutionary equations, especially those with critical nonlinearity (such as critical wave equation, nonclassical diffusion equation, reaction-diffusion equation and so on), and dissipative equations with different nonlinear boundary condition. As an application of the regularity, we hope to consider the existence of attractors and estimates of their dimension(the existence of exponential attractors), to establish the relation of attraction of different regular spaces and the upper semicontinuity of attractors. These problems are key and active in dynamical systems, these problems are more helpful for further understanding the structure of attractors.
英文关键词: Dissipative dynamical systems;Attractors;Regularity;Nonlinear boundary;Nonlinear evolutionary equations