Analysis of singular structures in elliptic and parabolic PDE with curvature effects

具有曲率效应的椭圆和抛物线偏微分方程中的奇异结构分析

• 批准号: 1101290
• 负责人: Peter Sternberg
• 金额: $ 24.5万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101290&HistoricalAwards=false
• 关键词: Analysis; singular; structures; elliptic; parabolic

This project focuses on the investigation of singular structures characterizing the solutions to various partial differential equations and variational problems that arise in significant physical models. There are two types of singular structures to be studied here: vortices (i.e., zeros of complex-valued order parameters) and phase boundaries (i.e., nodal sets of real-valued order parameters). Much of the work concerns the setting where the physical model--Ginzburg-Landau, Gross-Pitaevskii, Allen-Cahn, or Kawasaki-Ohta--is considered on a curved surface. The general goal is to further our understanding of what role curvature plays in the stabilization or destabilization of vortices and phase boundaries. The principal investigator plans to attack these problems through a combination of techniques including gamma-convergence, renormalized energy methods, second-variation arguments, constrained minimization, and parabolic nonlinear partial differential equations methods.The investigations to be pursued in this project arise from mathematical models for superconductors and superfluids, grain boundaries in alloys, and di-block copolymers. The main thrust of the research is to gain a better understanding of how curvature may play a role in affecting the behavior of these physical systems. For example, in recent years physicists have succeeded in producing superconductors in the shape of spherical shells, rather than just being flat, but at this point there is little theoretical work devoted to understanding how this shape may affect the response of the superconductor to an applied magnetic field. There may be hidden benefits or surprising behavior induced by the curvature of the superconducting surface that can be revealed through a careful mathematical analysis of the underlying models. The principal investigator will carry out much of this research in collaboration with Ph.D. students who will gain valuable experience studying these applications of mathematics as part of their dissertation work.

该项目的重点是研究奇异结构，这些奇异结构表征了各种偏微分方程和变分问题的解，这些问题出现在重要的物理模型中。这里有两种类型的奇异结构要研究：涡（即，复值序参数的零）和相位边界（即，实值序参数的节点集）。大部分的工作涉及的设置中的物理模型-金斯堡-朗道，大Pitaevskii，艾伦-卡恩，或川崎-太田-被认为是在一个曲面上。总的目标是进一步了解曲率在涡和相边界的稳定或不稳定中所起的作用。主要研究者计划通过伽马收敛，重整化能量方法，二次变分参数，约束最小化和抛物型非线性偏微分方程方法的组合来解决这些问题。在这个项目中要追求的研究来自超导体和超流体的数学模型，合金中的晶界，和二嵌段共聚物。这项研究的主要目的是更好地了解曲率如何影响这些物理系统的行为。例如，近年来，物理学家已经成功地制造出了球壳形状的超导体，而不仅仅是扁平的，但在这一点上，几乎没有理论工作致力于理解这种形状如何影响超导体对外加磁场的响应。超导表面的曲率可能会带来隐藏的好处或令人惊讶的行为，通过对底层模型的仔细数学分析可以揭示这些好处或行为。 首席研究员将与博士合作进行大部分研究。学生谁将获得宝贵的经验，研究这些应用数学作为他们的论文工作的一部分。