Long Range Behavior in Dynamical Systems and Partial Differential Equations

动力系统和偏微分方程中的长程行为

• 批准号: 0099399
• 负责人: Rafael de la Llave
• 金额: $ 11.05万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0099399&HistoricalAwards=false
• 关键词: Long; Range; Behavior; Dynamical; Systems

We want to investigate several projects in Dynamical Systems and in Partial Differential equations. We will also consider multi-particle systems that share properties with both/One unifying thread is that we would like to understand the relation between variational approaches and more geometric ones. In particular, we would like to produce proofs of diffusion that use geometric and variational methods and to extend results in dynamics obtained by variational methods to partial differential equations. An important tool for geometric methods is the theory of normally hyperbolic manifolds and we would like to extend it to Partial differentialequations and infinite particle systems.Sometimes, small short range causes may have large long term effects. For example, small periodic forces may build up large changes in energy. Some small changes in the local properties of a material may lead to the emergence of patterns that cover large distances. In other situations, however, local effects just average out.We would like to device a broad based array of methods (including numerical studies and geometric techniques)that can be used to decide whether build up or averaging occurs. We would also like to pay special attention tosome concrete models appearing in technological applications.

我们想研究动力系统和偏微分方程的几个项目。我们还将考虑多粒子系统，共享属性/一个统一的线程是，我们想了解变分方法和更多的几何之间的关系。特别是，我们要生产的扩散证明，使用几何和变分方法，并延长结果在动力学变分方法偏微分方程。几何方法的一个重要工具是正规双曲流形理论，我们想把它推广到偏微分方程和无限粒子系统。有时，小的短期原因可能会产生大的长期影响。例如，小的周期性力可以建立能量的大变化。材料局部属性的一些微小变化可能导致出现覆盖大距离的图案。然而，在其他情况下，局部效应只是平均化。我们想设计一个基础广泛的方法阵列（包括数值研究和几何技术），可以用来决定是否建立或平均化发生。我们还想特别关注一些技术应用中出现的具体模型。