Developing Robust Techniques for the Analysis of Multiple-Scale Behaviors

该研究项目的长期目标是开发对多尺度现象进行稳健和统计描述所需的数学工具。该项目综合了最初发展于动力系统的思想和变分问题（伽玛收敛）的方法，研究(1)薄片的弹性，(2)瑞利-贝纳德对流中的条纹模式，(3)二维静电以及与Hele-Shaw流和扩散限制聚集的联系，以及(4)扩展周期性强迫系统中的模式形成。从研究这些特殊问题中收集到的见解将用于解决激励本研究的一般问题。在非线性偏微分方程和扩展系统的物理学中，理解和分析多时空尺度上的现象是一个重要的问题。这个项目是一个长期研究项目的一部分，旨在理解多尺度行为，特别强调非凸变分问题（能量驱动的模式形成）。一个简单的例子是一张皱巴巴的纸的行为。所有皱巴巴的纸“看起来”都是一样的，但要把两张纸揉成一模一样的形状是不可能的。这提出了许多自然的问题，包括“描述系统全局行为的统计数据是什么？”和“这些统计数据有理论吗？”这类重要问题在物理、地球物理和生物系统的研究中反复出现。该项目旨在开发回答这些问题所需的工具。

The long term goal of this research project is to develop the mathematical tools needed for a robust and statistical description of multiple scale phenomena. Synthesizing ideas first developed for dynamical systems and methods for variational problems (Gamma-convergence), this project studies (1) elasticity of thin sheets, (2) stripe patterns in Rayleigh-Benard convection, (3) two-dimensional electrostatics and connections to Hele-Shaw flow and diffusion-limited aggregation, and (4) pattern formation in extended, periodically-forced systems. The insights gleaned from studying these particular problems will be used to address the general questions motivating this research.A significant problem in nonlinear partial differential equations and in the physics of extended systems is to understand and analyze phenomena on multiple spatial and temporal scales. This project is part of a long term research program to understand multiple scale behaviors with particular emphasis on non-convex variational problems (energy driven pattern formation). A simple example is the behavior of a crumpled sheet. All crumpled sheets "look" the same "statistically", but it is impossible to crumple two sheets of paper into identical configurations. This suggests many natural questions, including "What are the statistics that describe the global behavior of the system?" and "Is there a theory for these statistics?" Important questions of this type arise repeatedly in the study of physical, geophysical, and biological systems. This project aims to develop tools needed to answer such questions.