Mathematical Study of Ginzburg-Landau Asymptotics and the Stability of Fluids

Ginzburg-Landau渐近性和流体稳定性的数学研究

• 批准号: 0707714
• 负责人: Daniel Spirn
• 金额: $ 12.28万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707714&HistoricalAwards=false
• 关键词: Mathematical; Study; Ginzburg; Landau; Asymptotics

This proposal aims to rigorously study several problems arising in nonlinear partial differential equations and is split into two parts. The first part consists of a family of problems coming from the study of phase transition equations. Such equations model several types of phenomena, including superconductors, superfluids, Bose-Einstein condensates, and liquid crystals. These models form interesting defects in singular asymptotic regimes, in which the equations simplify substantially. This simplification is useful for improved numerical techniques and qualitative understanding. The first part of the project studies the behavior of these defects in both static and time dependent situations with techniques developed from the calculus of variations and harmonic analysis. The second part of the proposals centers on examining the stability of free boundaries in fluid equations with nontrivial vorticity in the fluid. The equations that model such fluids have both geometrical and physical components. The primary aim of this part of the project is to study the stability of such fluids under perturbations. The techniques used to analyze these problems come from energy methods, spectral theory, and harmonic analysis.Helium, when cooled to temperatures close to absolute zero, becomes a liquid with unusual characteristics. The liquid loses all internal friction, and if stirred, will rotate without end. Such liquids are called superfluids. If the stirring is too strong, little eddies (or vortices) form with quantized momentum, each of which are no longer in the superfluid state. Knowledge of the behavior and location of the vortices is the fundamental piece of information needed to describe the superfluid. The first part of this proposal studies properties of such vortices in superfluids, superconductors, and other related physics problems where quantum mechanical effects appear in large scales. Such physics problems are of increasing use in applications. The second part of the proposal centers on understanding the behavior of regular fluids with free boundaries, such as the surface of the ocean and the surface of stars. The equations that model these problems are complicated; however, since these problems arise with great frequency that their analysis is important

该提案旨在严格研究非线性偏微分方程中出现的几个问题，分为两部分。第一部分由相变方程研究中的一系列问题组成。这些方程模拟了多种类型的现象，包括超导体、超流体、玻色-爱因斯坦凝聚体和液晶。这些模型在奇异渐进机制中形成了有趣的缺陷，其中方程大大简化。这种简化对于改进数值技术和定性理解很有用。该项目的第一部分利用变分演算和谐波分析开发的技术研究这些缺陷在静态和时间相关情况下的行为。该提案的第二部分集中于检查具有非平凡涡度的流体方程中自由边界的稳定性。模拟此类流体的方程同时具有几何和物理分量。该项目这一部分的主要目的是研究此类流体在扰动下的稳定性。用于分析这些问题的技术来自能量方法、光谱理论和谐波分析。氦当冷却到接近绝对零的温度时，会变成具有不寻常特性的液体。液体失去所有内摩擦力，如果搅拌，就会不停地旋转。这种液体称为超流体。如果搅拌太强，就会形成具有量子化动量的小涡流（或漩涡），每个涡流都不再处于超流体状态。了解涡流的行为和位置是描述超流体所需的基本信息。该提案的第一部分研究超流体、超导体和其他大规模出现量子力学效应的相关物理问题中的涡流特性。此类物理问题在应用中的使用越来越广泛。该提案的第二部分集中于理解具有自由边界的常规流体的行为，例如海洋表面和恒星表面。模拟这些问题的方程很复杂；然而，由于这些问题出现的频率很高，因此对其进行分析很重要