Defect dynamics in nonlinear Hamiltonian partial differential equations

这个建议将调查有关的流体中的涡丝的行为的数学问题。这是一个令学者们着迷的话题，至少可以追溯到列奥纳多达芬奇的时代，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，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）到极大尺度（假设的宇宙弦）。 在真实的流体中，涡旋结构具有有限的厚度，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非常长而薄的龙卷风的合理数学模型。这种简化的超流体涡旋模型已经被物理学家使用了近50年，但忽略涡旋厚度的近似值的严格数学证明从未被提供过。这项建议的主要目的是提供这样一种理由。另一个目标是发展一个精确的数学理解，当涡丝碰撞时会发生什么。在这方面，物理学家也进行了大量的研究，这表明碰撞的细丝经历了一个称为“重连”的过程，而且这个过程对一系列物理现象有重要的影响。但从数学的角度来看，这个过程是非常不理解的。我们的目标是建立涡旋重联的数学理论基础。 为进行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.