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

这一建议旨在严格研究非线性偏微分方程组中的几个问题，分为两部分。第一部分由研究相变方程所产生的一系列问题组成。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-爱因斯坦凝聚体和液晶。这些模型在奇数渐近区域形成了有趣的缺陷，在这些区域中，方程大大简化。这种简化对于改进数值技术和定性理解是有用的。该项目的第一部分利用变分和调和分析发展起来的技术，研究了这些缺陷在静态和时间相关情况下的行为。建议的第二部分集中于检查流体中具有非平凡涡度的流体方程中自由边界的稳定性。模拟这类流体的方程既有几何成分，也有物理成分。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。用来分析这些问题的技术来自能量方法、光谱理论和调和分析。当氦冷却到接近绝对零度的温度时，它就会变成一种具有不寻常特性的液体。液体失去所有的内耗，如果被搅拌，就会不停地旋转。这种液体被称为超流体。如果搅拌太强，就会形成带有量化动量的小旋涡(或旋涡)，每个旋涡都不再处于超流体状态。了解涡旋的行为和位置是描述超流体所需的基本信息。这一建议的第一部分研究了超流、超导体和其他相关物理问题中这种涡旋的性质，这些问题中的量子力学效应出现了大规模。这样的物理问题在应用中的用途越来越大。该提案的第二部分集中在了解具有自由边界的规则流体的行为，如海洋表面和恒星表面。对这些问题进行建模的方程是复杂的；然而，由于这些问题经常出现，因此它们的分析很重要

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