CAREER: Quasilinear Dispersive Evolutions in Fluid Dynamics

职业：流体动力学中的拟线性色散演化

基本信息

批准号：
1845037
负责人：
Mihaela Ifrim
金额：
$ 40万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1845037&HistoricalAwards=false
关键词：
CAREER Quasilinear Dispersive Evolutions Fluid

项目摘要

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.

非线性分散方程模型在流体动力学（海洋学），量子力学，血浆物理学，非线性光学元件中产生的物理现象，一直到一般相对论。该项目的目的是通过（i）旨在研究解决方案的长期行为的具体研究项目来改善对这些方程式的数学和科学理解，以及（ii）旨在将这些问题引入新一代数学家的教育成分。主要研究者的主要重点是分析二维水波方程，该方程控制了自由流体表面的演变或两个流体之间的界面。目的是更好地描述局部动力学（例如低规律性解决方案和奇异性的形成）和流体流动中全球动力学。在这里，通过自由边界问题的奇点，主要研究者意味着界面失去光滑度，可能形成类似角落的奇点，然后是“波浪破裂”。在许多物理上重要的现象中表现出这种行为，例如在海浪中的湍流，像海浪中的湍流一样，可以轻松地观察到少数人的自然风险。基于本构方程的现象是相当困难的，目前是完全开放的。通过研讨会，研讨会，REU项目等的根本问题。项目的主要目标之一是了解水浪解决方案的长期行为。这包括寿命估计以及时间动态和散射属性的全局。 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强大的技术仍然必须描述这些方程式的奇异性形成机制。奖项反映了NSF的法定任务，并通过使用基金会的智力优点和更广泛的影响审查标准评估值得支持。