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: $ 2.04万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=682964
• 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（假设物体的模型被称为宇宙串的模型）中发现了一系列物理现象。它们存在，可能涵盖星系）。这些拓扑缺陷与普通流体中的涡流丝共享某些特性，这些涡流在日常存在中遇到，尤其是水和空气。这种涡旋细丝的例子包括烟环，龙卷风，飞机翼的尾随涡流以及独木舟桨脱落的涡流。这些经典涡流细丝和拓扑缺陷之间的类比对于在量子机械流体中发现的特定缺陷（称为量化涡流丝）中最强。 。控制拓扑缺陷行为的有效法律的推导。就像流体中的涡旋细丝一样，拓扑缺陷无非就是在动态介质中传播的特定类型的局部干扰。为了捕获其行为，一个从描述环境介质演变的方程式开始，而挑战是对介质中这种特定类型的干扰运动的严格数学描述。例如，******，例如人们普遍认为，称为双向曲率流（BCF）的方程式可以控制理想超流体中量化的涡旋细丝。提供这种信念的严格证明是一个长期的开放问题，它推动了我们在这一领域的大部分研究。 ****** 2。在研究拓扑缺陷研究到不同但相关的问题的研究中发展的思想的扩展。此类问题最重要，最有希望的问题涉及经典液体中的涡流丝，例如（理想化的）空气或水。例如，自Helmholtz在1850年代的基础工作以来，已经在经典流体中预测了一种称为Vortex Leaprogging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。但是数学理解是难以捉摸的。首先证明LeapFrogging仅在我自己和我的合作者D SMET的最新工作中才发生，并且仅用于量化超级流体的涡流。拟议的研究将研究我们开发的技术是否可以应用于理想的经典流体中的跨越。****** 3。控制拓扑缺陷行为的有效法律的研究。例如，上述BCF具有我们将研究的显着但不理想的属性。