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

在这个项目中，PI将研究一系列与连续介质力学相关的重要现象的数学模型。 这包括研究在平面或曲面（如球体）上的某些能量守恒流动，这些流动包含表面上的旋涡点，流动围绕这些旋涡点旋转。 第二个项目涉及到一个液晶模型。这些都是棒状分子，其行为部分像液体，而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相模型适应表面上液晶的这种相对未探索的设置。 最后一个项目涉及一个几何问题，作为周期性图案形成的范例，这是对于物理系统而言的，在这些物理系统中，人们期望物质的两种不同状态被界面分开，一方面，界面趋于使界面表面积最小化（就像肥皂泡一样）另一方面，倾向于发展在大样本中以小规模复制自己的模式。非线性偏微分方程和连续介质力学及相关领域的变分问题。这些包括Gross-Pitaevskii系统，Landau-deGennes模型的液晶和一个非局部的变种的经典等周问题有关的模型为二嵌段共聚物。在这些调查的目标是描述这些系统的解决方案的行为在低维对象-涡，缺陷或相边界在适当的渐近制度的模型中出现的参数。一个主要的主题是分析如何解决这些模型的行为时，提出了曲面。这些物体--涡流、缺陷或相边界--在很大程度上表征了整个系统的状态。对于Gross-Pitaevskii项目，另一个目的是在这个重要的非线性薛定谔方程和研究得很好的点涡问题之间建立更深层次的联系。在流体力学的背景下，点涡问题更常见于不可压缩欧拉流，但在这里，专注于周期性的解决方案，我们打算加强桥梁的量子力学的设置。对于表面上的向列相问题，我们希望为材料科学家研究沉积在曲面上的液晶的工作提供严格的数学支持。对于非局部等周问题，我们的目标是部分照亮如何细尺度周期性结构出现的强度的非局部性的增长。 在这个程序中采用的方法包括约束最小化技术，伽玛收敛和几何测量理论的组合。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.