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.

该提案将研究与流体中涡丝行为相关的数学问题。这是一个至少可以追溯到达芬奇时代的学者们一直着迷的话题，在过去的 60 年里，物理学家们已经意识到，漩涡状物体不仅存在于水和空气等日常流体中，而且还存在于水和空气中。在广泛的其他物理现象中，从非常小的尺度（量子力学流体，如玻色-爱因斯坦凝聚体、超导体、微磁材料）到极大的尺度（假设的宇宙弦）。 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。例如，在某些情况下，通过忽略其厚度并将其视为随时间演变的一维曲线，人们可能能够构建一个非常长且薄的龙卷风的合理数学模型。物理学家使用这种超流体涡流简化模型已近 50 年，但从未对忽略涡流厚度的近似值提供严格的数学论证。该提案的主要目标是提供这样的理由。另一个目标是对涡流细丝碰撞时发生的情况进行精确的数学理解。在这里，物理学家也进行了大量的研究，表明碰撞的细丝经历了一个称为“重新连接”的过程，而且这个过程对一系列物理现象具有重要的影响。但从数学的角度来看，人们对这个过程知之甚少。我们的目标是建立涡重联数学理论的基础。 为开展这项研究而开发的数学技术将对一系列问题产生重要影响，这些问题远远超出了我们在此解决的具体问题。