Variational and other methods for nonlinear PDE

* 这项提议旨在研究被称为拓扑缺陷的结构，这些结构存在于一系列非常小尺度（超导体、超流体、微磁材料）和非常大尺度（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）的物理现象中。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）中的涡丝具有某些特性。这种涡丝的例子包括烟圈、龙卷风、飞机机翼的尾涡和独木舟桨产生的涡。这些经典涡丝和拓扑缺陷之间的类比对于量子力学流体中发现的特定缺陷（称为量子化涡丝）来说是最强的。该提案解决了三类问题：*1。导出支配拓扑缺陷行为的有效定律。就像流体中的涡丝一样，拓扑缺陷只不过是在动态介质中传播的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，挑战在于对介质中这种特殊类型的扰动的运动给出严格的数学描述。例如，一个被称为副法向曲率流（BCF）的方程被广泛认为是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明是一个长期存在的开放问题，推动了我们在这一领域的大部分研究。2.在拓扑缺陷的研究中发展出来的思想的延伸，使其延伸到不同但相关的问题。最重要和最有前途的问题涉及涡丝在经典流体，如（理想化）空气或水。例如，自19世纪50年代亥姆霍兹的基础工作以来，在经典流体中已经预测了一种称为涡蛙跳的现象。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是难以捉摸的。蛙跳现象首次被证明是在我和我的合作者D Smets最近的工作中发生的，而且只适用于超流体中的量子化涡旋。拟议的研究将调查我们开发的技术是否可以应用于研究理想经典流体中的蛙跳。3.对控制拓扑缺陷行为的有效定律的研究。例如，上面提到的BCF具有我们将要研究的显著但知之甚少的性质。在解决这些问题时，我们将发展新的数学技术，以提供某些分析和几何问题之间的桥梁。

***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.**