Vortices, phase boundaries and defects arising in nonlinear PDE and variational models

在这个项目中，PI将研究与连续介质力学相关的重要现象的数学模型的集合。这包括研究某些能量守恒的流动，在平面内或曲面上，如球体上，含有涡流--流动绕其旋转的表面上的点。第二个项目涉及向列相液晶的模型。这些是棒状分子，部分像液体，在其他方面像固体晶体。最近，材料科学界开展了许多活动，旨在设计液晶，使其在沉积在曲面上时具有理想的材料特性，PI将探索如何使已知的向列学模型适应这种相对未被探索的表面液晶设置。最后一个项目涉及一个几何问题，它是周期性图案形成的范例，即对于物理系统，人们希望两种不同状态的物质被界面分开，这些界面一方面倾向于最小化界面表面积(如肥皂泡所做的那样)，另一方面倾向于在大样本中形成小范围复制的图案。这项建议涉及从连续介质力学和相关领域引出的非线性偏微分方程组和变分问题的研究。其中包括Gross-Pitaevskii系统，向列型液晶的Landau-deGennes模型，以及与两嵌段共聚物模型相关的经典等周问题的非局部变体。这些研究的目的是根据低维对象描述这些系统的解的行为--对于模型中出现的参数，在适当的渐近范围内的涡旋、缺陷或相边界。一个主要的主题是分析其中一些模型的解决方案在曲面上设置姿势时的行为。这些物体--漩涡、缺陷或相界--在很大程度上表征了整个系统的状态。对于Gross-Pitaevskii上的项目，另一个目的是在这个重要的非线性薛定谔方程和非常研究的点涡旋问题之间建立更深层次的联系。在流体力学的背景下，点涡旋问题更常与不可压缩的欧拉流联系在一起，但在这里，我们将重点放在周期解上，我们打算加强到量子力学背景的桥梁。对于曲面上的向列学问题，我们希望为材料科学家研究沉积在曲面上的液晶的越来越多的工作提供严格的数学支持。对于非局域等周问题，我们的目标部分是为了阐明细微尺度周期结构是如何随着非局域强度的增长而出现的。该程序采用的方法包括约束极小化技术、伽玛收敛和几何测度论的组合。教育部分将包括博士生参与许多项目。

In this project the PI will investigate a collection of mathematical models for important phenomena related to continuum mechanics. These include the study of certain energy-conserving flows, in the plane or on a curved surface such as a sphere, that contain vortices-points on the surface around which the flow rotates. A second project involves a model for nematic liquid crystals. These are rod-like molecules that behave in part like a liquid and in other ways like solid crystals. Recently there has been much activity in the materials science community aimed at designing liquid crystals to take on desirable material properties when deposited on a curved surface and the PI will explore how to adapt the known models for nematics to this relatively unexplored setting of liquid crystals on surfaces. A final project concerns a geometric problem that stands as a paradigm for periodic pattern formation, that is for physical systems where one expects two different states of matter to be separated by interfaces that on the one hand tend to minimize interfacial surface area (like a soap bubble does) and on the other hand tend to develop patterns that replicate themselves on a small scale throughout a large sample.This proposal concerns the study of nonlinear partial differential equations and variational problems drawn from continuum mechanics and related fields. These include the Gross-Pitaevskii system, the Landau-deGennes model for nematic liquid crystals and a nonlocal variant of the classic isoperimetric problem related to models for diblock co-polymers. The goal in these investigations is to describe the behavior of solutions to these systems in terms of lower-dimensional objects--vortices, defects or phase boundaries within appropriate asymptotic regimes for the parameters arising in the models. A major theme is to analyze how solutions to some of these models behave when posed on curved surfaces. These objects--vortices, defects or phase boundaries--largely characterize the state of the overall system. For the project on Gross-Pitaevskii, an additional purpose is to draw a deeper connection between this important nonlinear Schrodinger equation and the very well-studied point-vortex problem. The point-vortex problem is more commonly associated with incompressible Euler flow in the context of fluid mechanics but here, focusing on periodic solutions, we intend to strengthen a bridge to a setting in quantum mechanics. For the problem of nematics on surfaces, we hope to give rigorous mathematical support to the growing body of work by materials scientists studying liquid crystals deposited on curved surfaces. For the nonlocal isoperimetric problem our goal is in part to illuminate how fine scale periodic structures emerge as the strength of the nonlocality grows. The methods to be employed in this program include a combination of constrained minimization techniques, Gamma-convergence and geometric measure theory. The educational component will include the involvement of doctoral students on many of the projects.