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

本提案旨在严格研究非线性偏微分方程中出现的几个问题，并分为两部分。第一部分是关于相变方程研究的一系列问题。这些方程模拟了几种类型的现象，包括超导体、超流体、玻色-爱因斯坦凝聚和液晶。这些模型在奇异渐近状态下形成了有趣的缺陷，在这种状态下方程得到了极大的简化。这种简化有助于改进数值技术和定性理解。项目的第一部分研究了这些缺陷在静态和时间依赖情况下的行为，采用了从变分法和谐波分析中发展出来的技术。建议的第二部分集中在检查流体中具有非平凡涡量的流体方程中自由边界的稳定性。模拟这种流体的方程既有几何成分，也有物理成分。该项目的这一部分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、频谱理论和谐波分析。当氦冷却到接近绝对零度的温度时，就会变成一种具有不同寻常特征的液体。液体失去了所有的内摩擦，如果搅拌，就会不停地旋转。这种液体被称为超流体。如果搅拌太强烈，就会形成带有量子化动量的小漩涡（或漩涡），每个漩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。本提案的第一部分研究了超流体、超导体和其他相关物理问题中量子力学效应在大尺度上出现的性质。这类物理问题在实际应用中有越来越多的用途。该提议的第二部分集中在理解具有自由边界的规则流体的行为，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，对它们的分析是很重要的

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