CAREER: Quasilinear Dispersive Evolutions in Fluid Dynamics

非线性色散方程模拟了流体动力学（海洋学）、量子力学、等离子体物理、非线性光学以及广义相对论中出现的物理现象。该项目的目的是通过(i)旨在研究这些方程解的长期行为的具体研究项目，以及（ii）旨在向新一代和多样化的数学家介绍这些问题的教育组成部分，提高对这些方程的数学和科学理解。首席研究员的主要重点将是分析二维水波方程，该方程控制着自由流体表面或两种流体之间界面的演变。目的是更好地描述流体流动中的局部动力学（例如低正则解和奇点的形成）和全局动力学。这里，通过自由边界问题中的奇点，首席研究员指的是界面失去平滑的进化，可能形成一个角状奇点，然后是“破波”。这种行为在许多重要的物理现象中都表现出来，比如海浪中的湍流，海啸的形成，只是举几个例子。虽然这种破波现象很容易观察到，在自然界中有很强的表现，但对它的科学认识还很差，基于本构方程的现象的数学证明相当困难，目前是完全开放的。该项目旨在解决与奇点形成和几类色散和双曲方程解的长时间性质相关的关键开放问题，这些问题主要是由流体动力学引起的，它们来自物理或几何环境。该项目包括一个教育组成部分，旨在通过研讨会、研讨会、REU项目等提高年轻一代研究人员对这些基本问题的兴趣。该项目的主要目标之一是了解水波解的长期行为。这包括寿命估计以及全局时间动力学和散射特性，如果可能的话。从这个角度来看，有两个关键性质起作用：一个是“色散衰减”，另一个是“共振分析”。PI和合作者开发的一些关键工具（“拟线性修正能量法”和“波包测试法”）将有助于更好地理解所提出的问题，但尽管如此，为了描述，例如奇点形成机制，改进现有方法和开发新的强大技术仍然是必不可少的。这些方程的拟线性性质在与分析解的长期行为和性质相关的困难中起着至关重要的作用。水波及其相关模型是物理文献中基于启发式考虑导出的有效的海洋动力学方程，对色散偏微分方程的长期行为具有重要意义。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Nonlinear dispersive equations model physical phenomena arising in fluid dynamics (oceanography), quantum mechanics, plasma physics, nonlinear optics, all the way to general relativity. The purpose of this project is to improve the mathematical and scientific understanding of those equations via (i) concrete research projects aimed at studying the long-time behavior of solutions to such equations, and (ii) an educational component aimed at introducing such problems to a new and diverse generation of mathematicians. A primary focus of the principal investigator will be the analysis of the two-dimensional water wave equations, which govern the evolution of a free fluid surface, or of the interface between two fluids. The goal is to provide a better description of both the local dynamics (e.g. low regularity solutions and formation of singularities) and of global dynamics in fluid flows. Here, by singularities in free boundary problems, the principal investigator means evolutions where the interface loses smoothness, possibly forming a corner-like singularity followed by "wave breaking''. Such a behavior is exhibited in many physically important phenomena, like turbulence in ocean waves, tsunami formation, just to mention a few. While this wave breaking is easy to observe and has very strong manifestations in nature, its scientific understanding is rather poor, and the mathematical justification of the phenomena based on the constitutive equations is rather difficult and is fully open at this time. This project aims to tackle a selection of key open problems related to singularity formation and to long-time properties of solutions to several classes of dispersive and hyperbolic equations that arise from a physical or geometric context, largely motivated by fluid dynamics. The project includes an educational component aimed at raising the interest of a younger generation of researchers in those fundamental problems through workshops, seminars, REU projects, etc.One of the main goals of the project is to understand the long-time behavior of solutions of the water waves. This includes lifespan estimates as well as global in time dynamics and scattering properties, where possible. From this perspective there are two key properties that play a role: one is the "dispersive decay'', and the other is the ``resonance analysis''. Some of the key tools developed by the PI and collaborators (the "quasilinear modified energy method'', and the "testing with wave packets method'') will contribute to a better understanding of the proposed problems, but nevertheless, improvements of the existing methods and the need to develop new and robust techniques remain essential in order to describe, for instance, the singularity formation mechanism. The quasilinear nature of these equations plays a crucial role in the difficulty associated with the analysis of the long- time behavior and properties of the solutions. Water waves and related models are effective equations for the ocean dynamics that are derived in the physics literature from heuristic considerations and have very important implications on the long-time behavior of dispersive partial differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.