CAREER: Mathematics of Vorticity in Ginzburg-Landau Theory and Fluids

职业：金兹堡-朗道理论和流体中的涡度数学

• 批准号: 0955687
• 负责人: Daniel Spirn
• 金额: $ 47万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0955687&HistoricalAwards=false
• 关键词: CAREER; Mathematics; Vorticity; Ginzburg; Landau

This project aims to establish rigorous qualitative behavior of problems arising in nonlinear partial differential equations and comprises three parts. The first part consists of a family of problems arising in the study of phase transition equations, including those that model superconductivity and superfluidity. When a certain parameter becomes asymptotically large, these materials form vortices, which are regions of high energy and spin. This part of the project studies the dynamical behavior of asymptotically large numbers of vortices, each of which contain an asymptotically large amount of energy. The second part of the project analyzes the nucleation and dynamical behavior of vortices in models for thin micromagnetic materials. Such problems arise in magnetic information storage devices. The analytical methods used in the first and second part of the project come from geometric measure theory and the calculus of variations. The third part of the research project entails an analysis of water waves over variable topographies. The analytical methods employed will include bifurcation theory, harmonic analysis, and spectral theory, along with numerical experiments. Tied to the three research areas are several educational projects that aim to train undergraduate and graduate students in analysis, along with the goal of raising interest in applied mathematics.In fluid dynamics and materials science, vorticity roughly measures the local rotation of either a fluid or a phase function. In many important physical contexts, vorticity is a crucial component, and in many cases understanding the vorticity leads to understanding of the entire physical problem. The first two parts of the project study the qualitative behavior of vorticity in two physics problems of importance to materials science, superconductors and thin micromagnetic films. These materials are of increasing use in applications, such as in powerful magnets and information storage devices. The third part of the proposal centers on understanding the behavior of ideal fluids with free boundaries, which arise in the study of waves on the surface of the ocean or fast-moving streams. The equations that model these problems are complicated; however, since these problems arise with great frequency, their analysis is important. Educational and training objectives are clearly and closely tied to the research projects.

本课题旨在建立非线性偏微分方程问题的严格定性行为，包括三个部分。第一部分包括相变方程研究中出现的一系列问题，包括那些模拟超导和超流体的问题。当某一参数渐近变大时，这些材料形成涡流，这是高能量和自旋的区域。这部分项目研究了渐近大量涡旋的动力学行为，其中每个涡旋都包含渐近大量的能量。项目第二部分分析了薄微磁性材料模型中涡旋的成核和动力学行为。这类问题在磁性信息存储装置中出现。项目的第一部分和第二部分使用的分析方法来自几何测量理论和变分法。研究项目的第三部分需要对不同地形上的水波进行分析。所采用的分析方法将包括分岔理论、谐波分析和谱理论，以及数值实验。与这三个研究领域相关的还有几个教育项目，旨在培养本科生和研究生的分析能力，同时提高他们对应用数学的兴趣。在流体动力学和材料科学中，涡度大致测量流体或相函数的局部旋转。在许多重要的物理环境中，涡度是一个关键的组成部分，在许多情况下，理解涡度会导致对整个物理问题的理解。项目的前两部分研究涡旋在材料科学的两个重要物理问题——超导体和微磁薄膜中的定性行为。这些材料在诸如强力磁铁和信息存储设备等方面的应用越来越广泛。提案的第三部分集中在理解具有自由边界的理想流体的行为，这种情况出现在对海洋表面的波浪或快速流动的溪流的研究中。模拟这些问题的方程很复杂；然而，由于这些问题经常出现，对它们的分析很重要。教育和培训目标与研究项目密切相关。