Variational Theories for Defects and Patterns

本提案中研究的项目的主要物理推动力是获得对模式形成的分析理解。特别是PI的目的是表征出现的缺陷，当一个是远远高于出现模式的参数阈值的类别。这些问题的研究通过一个非凸变分问题称为正则化交叉纽厄尔（RCN）模型。与最近研究的类似模型不同，RCN采用了非平凡扭曲的功能，使更丰富的？分类学？缺陷（特别是凹向错和凸向错）。这些特征在实验和微观物理方程的数值模拟中确实可以看到。问题是要确定是否变分模型可以捕捉到这些缺陷。在最近的一项工作中，PI证明了具有扭曲的变分模型的最小值必须与没有扭曲的最小值不同。该提案的主要项目集中在建立一个精确的分析表征之间的竞争微观结构的形成和整体拓扑引起的能量应力的版本的RCN模型具有一定的几何对称性。模式形成系统的研究是科学研究的一个基本领域，其中现代数学分析的工具可以用于对物理系统的建模，特别是在系统行为的临界转变附近。对于本提案中研究的项目，人们主要感兴趣的是当连续平移对称性在临界阈值处减少到离散周期对称性时产生的图案，从而导致通常被称为“条纹”图案。这种模式例如在瑞利-贝纳德对流（RBC）中是通用的，其是用于天气模式形成的主要理论模型。在RBC中，条纹图案对应于在临界温度下形成的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要研究在临界阈值形成的模式，而且要描述当一个人远离阈值时，这些模式中出现的缺陷类型。这项工作将作出明确的预测缺陷的形成，可以在实验室进行测试。这将与液晶中的缺陷结构、动物皮毛图案（包括指纹）和植物图案的进化中的建模相关。

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.