Variational and other methods for nonlinear PDE

这个提议的目的是研究被称为拓扑缺陷的结构，在非常小的尺度（超导体、超流体、微磁性材料）和非常大的尺度（被称为宇宙弦的假设物体的模型，如果它们存在，可能跨越星系）的一系列物理现象中发现。这些拓扑缺陷与人们在日常生活中遇到的普通流体（特别是水和空气）中的涡旋丝具有某些特性。这种漩涡细丝的例子包括烟圈、龙卷风、飞机机翼的拖尾漩涡和独木舟桨产生的漩涡。在量子力学流体中发现的特定缺陷（称为量子化涡旋丝）与这些经典涡旋丝和拓扑缺陷之间的相似性最强。该提案涉及三类问题：1。控制拓扑缺陷行为的有效定律的推导。就像流体中的涡流丝一样，拓扑缺陷只不过是在动态介质中传播的一种特定类型的局部扰动。为了捕捉它的行为，人们从描述环境介质演化的方程开始，而挑战在于对这种特殊类型的扰动在介质中的运动给出严格的数学描述。例如，一个被称为双正态曲率流（BCF）的方程被广泛认为可以控制理想超流体中的量子化涡丝。为这一信念提供严格的证据是一个长期存在的开放性问题，它推动了我们在这一领域的许多研究。2. 将拓扑缺陷研究中发展起来的思想扩展到不同但相关的问题。这类问题中最重要和最有前途的涉及经典流体中的涡旋细丝，如（理想的）空气或水。例如，自19世纪50年代亥姆霍兹（Helmholtz）的基础工作以来，在经典流体中就已经预测到了一种被称为涡旋跃迁的现象。物理学家和应用数学家对此进行了详细的研究，甚至可以在youtube上的视频中看到。但数学上的理解一直难以捉摸。在我和我的合作者D . Smets最近的工作中才首次证明了跃进现象，而且只适用于超流体中的量子化涡流。本研究将探讨我们所开发的技术是否可以应用于研究理想经典流体中的跃迁现象。控制拓扑缺陷行为的有效规律的研究。例如，上面提到的BCF具有显著但鲜为人知的特性，我们将对此进行研究。在解决这些问题时，我们将发展新的数学技术，在某些解析和几何问题之间提供桥梁。

This proposal aims to study structures known as topological defects, found in a range of physical phenomena on both very small scales (superconductors, superfluids, micromagnetic materials) and very large scales (models for hypothetical objects known as cosmic strings which, if they exist, may span galaxies). These topological defects share some properties with vortex filaments in ordinary fluids that one encounters in everyday existence, notably water and air. Examples of such vortex filaments include smoke rings, tornados, trailing vortices from airplane wings, and the vortices shed by a canoe paddle. The analogies between these classical vortex filaments and topological defects are strongest for the particular defects (known as quantized vortex filaments) found in quantum mechanical fluids.The proposal addresses three classes of problems:1. the derivation of effective laws that govern the behaviour of topological defects. Like a vortex filament in a fluid, a topological defect is nothing more than a particular type of localized disturbance that propagates within a dynamic medium. To capture its behaviour, one starts from equations that describe the evolution of the ambient medium, and the challenge is to give a rigorous mathematical description of the motion of this particular type of disturbance within the medium.For example, an equation called the binormal curvature flow (BCF) is widely believed to govern quantized vortex filaments in ideal superfluids. Providing a rigorous proof of this belief is a long-standing open problem that drives much of our research in this area. 2. the extension of ideas developed in the study of topological defects to distinct but related problems. The most important and promising such problems involve vortex filaments in classical fluids such as (idealized) air or water. For example, a phenomenon known as vortex leapfrogging has been predicted in classical fluids since foundational work of Helmholtz in the 1850s. This has been studied in great detail by physicists and applied mathematicians, and it can even be seen in videos on youtube. But a mathematical understanding has been elusive. Leapfrogging was first proved to occur only in recent work of myself and my collaborator D Smets, and only for quantized vortices in superfluids. The proposed research will investigate whether techniques that we developed can be applied to study leapfrogging in ideal classical fluids.3. the study of effective laws that govern the behaviour of topological defects. For example, the above-mentioned BCF has remarkable but poorly-understood propertes that we will investigate.In addressing these problems, we will develop new mathematical techniques to provide a bridge between certain analytic and geometric questions.