Variational and other methods for nonlinear PDE

非线性偏微分方程的变分法和其他方法

• 批准号: RGPIN-2018-05691
• 负责人: Jerrard, Robert
• 金额: $ 4.08万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758678
• 关键词: Variational; other; methods; nonlinear; PDE

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.

该提案旨在研究被称为拓扑缺陷的结构，这些结构在非常小尺度（超导体、超流体、微磁性材料）和非常大尺度（称为宇宙弦的假设物体的模型，如果存在的话）的一系列物理现象中发现。可能跨越星系）。这些拓扑缺陷与人们在日常生活中遇到的普通流体（尤其是水和空气）中的涡旋丝具有一些相同的特性。这种涡丝的例子包括烟圈、龙卷风、飞机机翼的尾涡以及独木舟桨产生的涡流。对于量子力学流体中发现的特定缺陷（称为量子化涡丝），这些经典涡丝和拓扑缺陷之间的相似性最强。该提案解决了三类问题：1。导出控制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一种特定类型的局部扰动。为了捕捉它的行为，我们从描述环境介质演化的方程开始，挑战是对介质内这种特定类型的扰动的运动给出严格的数学描述。例如，一个称为副法线曲率的方程人们普遍认为流动（BCF）控制着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悬而未决的问题，它推动了我们在这一领域的大部分研究。 2. 将拓扑缺陷研究中发展的思想扩展到不同但相关的问题。最重要和最有前途的此类问题涉及经典流体（例如（理想的）空气或水）中的涡丝。例如，自 1850 年代亥姆霍兹的基础工作以来，人们就在经典流体中预测了一种称为涡越级的现象。物理学家和应用数学家对此进行了非常详细的研究，甚至可以在 YouTube 上的视频中看到。但数学理解一直难以捉摸。蛙跳现象首次被证明仅在我和我的合作者 D Smets 最近的工作中出现，并且仅适用于超流体中的量子化涡流。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．研究控制拓扑缺陷行为的有效规律。例如，上述 BCF 具有我们将研究的显着但鲜为人知的属性。在解决这些问题时，我们将开发新的数学技术，以在某些分析问题和几何问题之间架起一座桥梁。