Variational Theories for Defects and Patterns

本提案中研究的项目的主要物理动力是获得对模式形成的分析性理解。具体地说，PI试图描述当一个缺陷远远高于出现图案的参数阈值时出现的缺陷类别。这些问题是通过一个非凸变分问题被称为正则化的Cross-Newell(RCN)模型来研究的。与最近研究的类似模型不同，RCN包含了非平凡扭曲的特征，这使得存在更丰富的分类？缺陷(特别是凹面和凸面的倾斜)。这些特征确实可以在实验和基础微观物理方程的数值模拟中看到。问题在于确定变分模型是否能够捕捉到这些缺陷。在最近的一项工作中，PI证明了有扭曲的变分模型的极小化必然不同于没有扭曲的变分模型。该建议的主要项目集中在建立具有某些几何对称性的RCN模型版本中微结构形成和总体拓扑诱导的能量应力之间的竞争的精确解析表征。图案形成系统的研究是科学研究的一个基本领域，其中现代数学分析工具可以用于物理系统的建模，特别是在系统行为的关键转变附近。对于本提案中研究的项目，人们主要对在临界阈值下将连续的平移对称降低到离散的周期对称时产生的图案感兴趣，从而产生通常所说的“条纹”图案。例如，这种模式在瑞利-贝纳德对流(RBC)中是通用的，这是形成天气模式的主要理论模型。在RBC中，条纹图案对应于在临界温度下形成具有均匀特征宽度的周期性“对流卷”。这项研究的一个主要目标是不仅研究在临界阈值下形成的模式，而且描述当一个人远离阈值时在这些模式中出现的缺陷的类型。这项工作将对可以在实验室测试的缺陷形成做出明确的预测。这将对液晶、动物皮毛图案(包括指纹)和植物图案的进化中的缺陷结构进行建模。

The principal physical impetus for the projects studied in this proposal is to gain an analytical understanding of pattern formation. In particular the PI seeks to characterize the classes of defects that appear when one is far above the parameter threshold at which patterns emerge. These issues are studied through a non-convex variational problem known as the regularized Cross-Newell (RCN) model. Unlike similar models that have been studied recently, RCN incorporates features of non-trivial twist which enable the existence of a richer ?taxonomy? of defects (in particular, concave and convex disclinations). Such features are indeed seen in experiments and numerical simulations of the underlying microscopic physical equations. The issue has been to determine whether or not the variational model can capture these defects. In a recent work the PI has demonstrated that the minimizers of a variational model with twist must necessarily differ from those without twist. The main projects of this proposal are centered on establishing a precise analytical characterization of the competition between microstructure formation and overall topologically induced energetic stress in versions of the RCN model having some geometric symmetry. The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis can be brought to bear on the modeling of physical systems especially near a critical transition in the behavior of the system. For the projects studied in this proposal one is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a principal theoretical model for the formation of weather patterns. In RBC the striped pattern corresponds to the formation, at a critical temperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. This work will make definite predictions on defect formation that can be tested in the laboratory. This will have relevance for modeling defect structure in liquid crystals, in animal coat patterns (including fingerprints) and in the evolution of plant patterns.