Defect dynamics in nonlinear Hamiltonian partial differential equations

非线性哈密顿偏微分方程中的缺陷动力学

• 批准号: 261955-2013
• 负责人: Jerrard, Robert
• 金额: $ 2.48万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571185
• 关键词: Defect; dynamics; nonlinear; Hamiltonian; partial

This proposal will investigate mathematical issues related to the behaviour of vortex filaments in fluids. This is a topic that has fascinated scholars dating back at least to the days of Leonardo da Vinci, and over the past 60 years, physicists have realized that vortex-like objects are present not only in everyday fluids such as water and air, but also in a wide range of other physical phenomena, ranging from very small scales (quantum mechanical fluids such Bose-Einstein condensates, superconductors, micromagnetic materials) to extremely large scales (hypothetical cosmic strings). In real fluids, vortex structures have finite thickness, but they can sometimes be described by simpler, 1-dimensional models. For example, under some circumstances, one might be able to construct a reasonable mathematical model of a very long, thin tornado by neglecting its thickness and treating it as a 1-dimensional curve that evolves in time. Such reduced models for vortices in superfluids have been used by physicists for close to 50 years, but a rigorous mathematical justification of the approximation involved in neglecting the vortex thickness has never been supplied. A main goal of this proposal is to supply such a justification. Another goal is to develop a precise mathematical understanding of what happens when vortex filaments collide. Here, too, physicists have carried out a huge amount of research, which suggests that colliding filaments undergo a process called "reconnection", and moreover that this process has important implications for a range of physical phenomena. But from a mathematical point of view, this process is very poorly understood. We aim to build the foundations of the mathematical theory of vortex reconnection. The mathematical techniques that will be developed to carry out this research will have important implications for a range of problems that extends far beyond the specific questions that we address here.

该建议将研究与流体中涡旋细丝的行为有关的数学问题。 This is a topic that has fascinated scholars dating back at least to the days of Leonardo da Vinci, and over the past 60 years, physicists have realized that vortex-like objects are present not only in everyday fluids such as water and air, but also in a wide range of other physical phenomena, ranging from very small scales (quantum mechanical fluids such Bose-Einstein condensates, superconductors, micromagnetic materials)到极大的尺度（假设的宇宙字符串）。 在实际流体中，涡流结构具有有限的厚度，但有时可以通过更简单的一维模型来描述它们。例如，在某些情况下，人们可能能够通过忽略其厚度并将其视为及时演变的一维曲线来构建一个非常长而细的龙卷风的合理数学模型。物理学家已经使用了近50年的超流体涡流模型，但是从未提供过忽略涡旋厚度所涉及的近似值的严格数学理由。该提案的主要目标是提供这样的理由。另一个目标是对涡旋细丝发生碰撞时发生的事情进行精确的数学理解。在这里，物理学家也进行了大量研究，这表明碰撞细丝经历了一个称为“重新连接”的过程，此外，此过程对一系列物理现象具有重要意义。但是从数学的角度来看，这个过程非常了解。我们旨在建立涡旋重新连接数学理论的基础。 将开发为进行这项研究的数学技术将对一系列问题具有重要意义，这些问题远远超出了我们在这里解决的特定问题。