RUI: Stability and Dynamics of Self-Similarity in Evolution Equations

自相似行为在许多物理系统中都很明显，包括薄膜的破裂、薄液丝或固体颈部的挤压以及液滴的扩散。这项研究试图通过在动力系统理论的背景下观察这些现象的数学模型来增加对自相似现象的理解。自相似解可以被认为是在缩放的坐标系中的稳定平衡，该坐标系正在放大以匹配现象的尺度。这些模型中的大多数都有无数个自相似解。这个项目的首要目标是发展一种稳定性理论，它将确定模型动态选择的解。在这个项目中提出的想法将为数学家、工程师和物理学家提供一套分析自相似动力学的工具。一种现象是自相似的，如果一旦时间尺度被适当地加速或减慢，并且空间尺度也被调整，它看起来是不变的。例如，集成电路依赖于在衬底上沉积薄的金属痕迹；这项研究将有助于确定这些痕迹在制造过程中何时会断裂，并阐明断裂是如何发生的细节，因为在临界事件附近往往存在自相似行为。在油漆、粘合剂和光学涂层的应用中会产生薄膜；本项目中开发的技术应该描述薄膜何时以及如何破裂，并帮助制定防止这些缺陷的策略，这些缺陷通常以自相似的方式发展。这笔赠款还将资助哈维穆德学院本科生的研究，他们将成为这项调查的组成部分。

Self-similar behavior is evident in many physical systems includingthe rupture of thin films, the pinching of thin liquid threads orsolid necks, and the spreading of liquid drops. This research seeksto increase the understanding of self-similarity by viewing the mathematical models for these phenomena in the context of dynamical systems theory.Self-similar solutions can be thought of as steady equilibria in arescaled coordinate system that is "zooming" to match the scale ofthe phenomena. Most of these models have an infinite number ofself-similar solutions. The foremost goal of this project is thedevelopment of a stability theory that will identify which solution isdynamically selected by the model.The ideas developed in this project will provide mathematicians,engineers and physicists with a set of tools for analyzingself-similar dynamics. A phenomenon is self-similar if it appears to be unchanging once the time scale is suitably accelerated or slowed down and the spatial scale is also adjusted. For example, integrated circuits rely on thedeposition of thin metallic traces on a substrate; this researchwill help determine when these traces will break during fabricationand elucidate the details of how the break occurs, since there is often self-similarbehavior near the critical event. Thin fluid films are created in the application of paint, adhesives and optical coatings; the techniques developed in this project should describe when and how films rupture and help devise strategies for preventing these defects, which often develop in a self-similar way. This grant will also underwrite the research of Harvey Mudd College undergraduates who will become an integral part of this investigation.