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产生的几个问题，并分为两部分。第一部分包括来自相过渡方程的研究的一系列问题。这样的方程模拟了几种类型的现象，包括超导体，超流体，玻色菌冷凝水和液晶。这些模型在奇异渐近方案中形成了有趣的缺陷，其中方程可简化。这种简化对于改进的数值技术和定性理解很有用。该项目的第一部分研究这些缺陷在静态和时间依赖性情况下的行为，并从变异和谐波分析的计算中开发出来的技术。提案的第二部分集中于检查流体非平凡涡度的流体方程中游离边界的稳定性。模拟此类流体的方程式具有几何和物理成分。该项目的主要目的是研究这种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。当冷却至接近绝对零的温度时，将成为具有异常特征的液体。液体失去所有内部摩擦，如果搅拌，将旋转而不会结束。这样的液体称为超流体。如果搅拌太强，则具有量化动量的很小的涡流（或涡旋）形成，每个涡流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了这些涡旋在超流体，超导体和其他相关物理问题中的特性，在这些物理问题中，量子机械效应以大规模出现。这些物理问题是在应用中使用的越来越多。该提案的第二部分集中在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。建模这些问题的方程式很复杂；但是，由于这些问题的频率很高，因此他们的分析很重要