Long Time Behaviour for Dispersive PDEs with Large Initial Data

这个项目主要集中在分析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物理起源：薛定谔映射方程被称为铁磁性中的海森堡模型，自旋模型具有类似的起源，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位于偏微分方程研究的前沿。大数据条件下各类方程的动力学问题是该领域的一个重要问题。在过去的几年里，已经取得了一些重大突破，所涉及的分析是非常重要的。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。对缺乏标度的系统（如扎哈罗夫系统）的研究极具挑战性，特别是在缺乏可用的守恒定律的情况下。虽然严格来说，这个项目中要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精细工具，特别是调和分析和黎曼几何。数学对广泛的科学界有用的主要原因之一，反过来对整个社会，它提供了一个最严格的框架来构建解释我们周围世界和预测未来事件的理论。偏微分方程领域在很大程度上是研究物理学中产生的模型。每个人都知道光、热、流体流动、磁等的存在，这些都是自然现象，一旦被很好地理解，就可以导致对人类生活产生巨大影响的重大发现。研究自然现象的科学方法遵循标准模式。一个人研究现象的复杂性，确定其基本特征，并写下一个微分方程，描述所研究对象的时间演化。接下来，研究数学模型的长期行为，并以定性的方式描述其演变。(In如果这个过程识别出潜在的奇点，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，如果数学分析与对现象的经验观察一致，那么数学模型就得到了验证，这往往会带来广泛的可能应用。另一方面，如果数学和经验观察之间出现差异，那么人们就会寻求改进数学模型，通常是通过允许更大的复杂性。对新模型进行类似的数学分析等，直到找到符合物理实际的良好数学模型。

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.