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、细液体线或固体颈的挤压以及液滴的扩散。本研究旨在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性的理解。自相似解可以被认为是缩放坐标系中的稳定平衡，该坐标系“缩放”以匹配现象的规模。这些模型中的大多数都有无限个自相似解。该项目的首要目标是发展一种稳定性理论，该理论将确定模型动态选择的解决方案。在这个项目中发展的思想将为数学家、工程师和物理学家提供一套分析自相似动力学的工具。当时间尺度适当加速或减慢，空间尺度适当调整后，一个现象似乎没有变化，这就是自相似现象。例如，集成电路依赖于在衬底上沉积薄金属痕迹；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，并阐明断裂发生的细节，因为在关键事件附近经常有自相似的行为。在油漆、粘合剂和光学涂层的应用中产生薄流体膜；在这个项目中开发的技术应该描述薄膜何时以及如何破裂，并帮助设计防止这些缺陷的策略，这些缺陷通常以自相似的方式发展。这笔拨款还将资助哈维马德学院本科生的研究，他们将成为本研究的组成部分。

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.