Long Time Behaviour for Dispersive PDEs with Large Initial Data

本课题主要研究某些色散偏微分方程的长时间行为。该项目中考虑的所有方程都有一个物理起源：薛定谔映射方程在铁磁中被称为海森堡模型，自旋模型也有类似的起源，而扎哈罗夫系统来自等离子体物理学。从数学的角度来看，首席研究员打算解决的大多数问题都处于偏微分方程研究的前沿。各种大数据方程的动力学问题是该领域的一个重要问题。在过去几年中取得了一些重大突破，所涉及的分析非常重要。理解目标流形的几何形状对方程演化的影响（在薛定谔映射和自旋模型的情况下）是非常有趣的。研究缺乏尺度的系统（如Zakharov系统）是极具挑战性的，特别是在缺乏可用的守恒定律的情况下。虽然严格来说，本项目所探索的问题属于偏微分方程领域，但研究需要使用其他数学领域的优秀工具，特别是谐波分析和黎曼几何。数学对广泛的科学界有用，进而对整个社会有用的主要原因之一是，它为构建解释我们周围世界和预测未来事件的理论提供了最严格的框架之一。偏微分方程领域在很大程度上是对由物理学产生的模型的研究。每个人都知道光、热、流体流动、磁力等的存在。这些都是自然现象，一旦它们被很好地理解，可以导致重大发现，对人类生活的影响是巨大的。研究自然现象的科学方法遵循一个标准模式。人们研究现象的复杂性，确定其基本特征，并写下描述所研究对象随时间演变的微分方程。接下来，研究数学模型的长期行为，并用定性术语描述其演变。（如果这个过程确定了潜在的奇点，这是主要研究者的主要兴趣，那么这个现象就与当前的项目直接相关。）最后，如果数学分析与现象的经验观察一致，那么数学模型就得到了验证，这往往打开了广泛的应用可能。另一方面，如果数学和经验观察之间出现了差异，那么人们就会寻求改进数学模型，通常是通过考虑更大的复杂性。新模型经过类似的数学分析，直到找到一个与物理现实相匹配的好的数学模型。

This project focuses mainly on analyzing the long-time behavior of certain dispersive partial differential equations. All the equations considered in the project have a physical origins: the Schrodinger maps equation is known as the Heisenberg model in ferro-magnetism, the spin-models have a similar origin, while the Zakharov system comes from plasma physics. From a mathematical point of view, most of the problems the principal investigator intends to address lie at the cutting-edge of research in partial differential equations. The dynamics of various equations with large data is a very important problem in the field. Some major breakthroughs have been achieved during the past few years, and the analysis involved is highly nontrivial. Understating the impact of the geometry of the target manifold on the evolution of the equation (in the case of Schrodinger maps and spin-models) is of great interest. Research into systems that lack scaling (like the Zakharov system) is extremely challenging, especially in the absence of usable conservation laws. While, strictly speaking, the problems to be explored in this project belong to the field of partial differential equations, the research requires the use of fine tools from other areas of mathematics, notably harmonic analysis and Riemannian geometry.One of the main reasons that mathematics is useful to the broad scientific community, and in turn to society as a whole, is that it provides one of the most rigorous frameworks for constructing theories that explain the world around us and predict future events. The field of partial differential equations is, to a great extent, the study of models arising from physics. Everyone is aware of the existence of light, heat, fluid flow, magnetism, etc. These are all natural phenomena that, once they are well understood, can lead to major discoveries whose impact on human lives is tremendous. A scientific approach to the study of a natural phenomenon follows a standard pattern. One investigates the complexity of the phenomenon, determines its essential features, and writes down a differential equation that describes the evolution in time of the object under study. Next, one studies the long-time behavior of the mathematical model and describes its evolution in qualitative terms. (In the event that this process identifies potential singularities, a prime interest of the principal investigator, then the phenomenon becomes directly relevant to the current project.) Finally, if the mathematical analysis agrees with the empirical observation of the phenomenon, then the mathematical model is validated, which often opens a wide range of possible applications. On the other hand, if discrepancies arise between the mathematics and the empirical observations, then one seeks to refine the mathematical model, usually by allowing for greater complexity. The new model undergoes a similar mathematical analysis and so on, until a good mathematical model matching the physical reality is found.