Defect dynamics in nonlinear Hamiltonian partial differential equations

本提案将研究与流体中涡旋细丝行为有关的数学问题。这个话题让学者们着迷，至少可以追溯到达芬奇时代，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，而且存在于其他广泛的物理现象中，范围从非常小的尺度（量子力学流体，如玻色-爱因斯坦凝聚物、超导体、微磁性材料）到非常大的尺度（假设的宇宙弦）。在真实的流体中，涡旋结构的厚度是有限的，但它们有时可以用更简单的一维模型来描述。例如，在某些情况下，人们可以通过忽略其厚度并将其视为随时间演变的一维曲线来构建一个很长很薄的龙卷风的合理数学模型。物理学家们使用这种简化的超流体涡旋模型已有近50年的历史，但是对于忽略涡旋厚度所涉及的近似，从来没有提供过严格的数学证明。本建议的一个主要目标就是提供这样一个理由。另一个目标是发展一个精确的数学理解，当涡旋细丝碰撞时会发生什么。在这里，物理学家也进行了大量的研究，这些研究表明，碰撞的细丝经历了一个被称为“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意义。但从数学的角度来看，这个过程还很不为人所知。我们的目的是建立涡旋重联的数学理论基础。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这里讨论的具体问题。

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.