CAREER: Analysis of Surface Water Waves

职业：地表水波分析

• 批准号: 1352597
• 负责人: Vera Mikyoung Hur
• 金额: $ 41.98万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-05-15 至 2021-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352597&HistoricalAwards=false
• 关键词: CAREER; Analysis; Surface; Water; Waves

The PI will develop new technical tools in partial differential equations and other branches of mathematics, and she will extend and combine existing tools, in order to tackle several long-standing open problems in theoretical aspects of water waves. They include (1) the global regularity versus finite-time blowup for the initial value problem, (2) the existence of traveling waves and their classification, (3) the stability and instability of traveling waves. Emphasis is placed upon the large scale dynamics and genuinely nonlinear behaviors, an acute understanding of which ultimately hinges upon analytical proofs. Emphasis is placed upon the use of the Euler equations in hydrodynamics rather than simple approximate models such as the Korteweg-de Vries equation. The PI proposes to foster applied mathematics at her host institution. She will continue organizing seminars and conferences, and she will disseminate her research through conference presentations, seminars and colloquia. The PI proposes to enhance the undergraduate ODE curriculum and develop new graduate courses. She plans to involve undergraduate and graduate students in her research and mentor graduate students and postdoctoral researchers. The PI will encourage women and minorities to pursue careers in mathematics, science and engineering, and improve the pipeline for women research mathematicians.The problem of water waves concerns the wave motion at the interface separating in two or three dimensions an incompressible inviscid fluid below a body of air, acted upon by gravity and possibly surface tension. Describing in an idealized fashion what may be observed in an ocean or a lake, water waves are a perfect specimen of applied mathematics. They host a wealth of wave phenomena, ranging in length scale from ripples driven by surface tension to tsunamis and to rogue waves. They provide source and inspiration to several branches of mathematics. Furthermore they impact outside of mathematics, from hydraulics to weather prediction. The water wave problem, notwithstanding, presents profound and subtle difficulties for rigorous analysis, modeling and numerical simulations. For one thing, the interface between the water and the air is a priori unknown and to be determined as part of the solution, namely a free boundary. Incidentally, free boundaries are mathematically challenging in their own right and they occur in numerous situations such as the melting of ice and the stretching of a flexible membrane over an obstacle. To make things worse, boundary conditions at the free surface are severely nonlinear. This project will develop new tools to advance understanding of these challenging problems.

PI将在偏微分方程和其他数学分支中开发新的技术工具，她将扩展和结合现有的工具，以解决水波理论方面的几个长期开放问题。它们包括：(1)初值问题的全局正则性与有限时间爆破；(2)行波的存在性及其分类；(3)行波的稳定性与不稳定性。重点放在大规模的动力学和真正的非线性行为，一个敏锐的理解，最终取决于分析证明。重点放在欧拉方程在流体力学中的应用，而不是简单的近似模型，如Korteweg-de Vries方程。PI建议在她的主办机构培养应用数学。她将继续组织研讨会和会议，并将通过会议演讲、研讨会和座谈会传播她的研究成果。《纲领》建议加强本科生的ODE课程，并发展新的研究生课程。她计划让本科生和研究生参与她的研究，并指导研究生和博士后研究人员。该计划将鼓励女性和少数族裔从事数学、科学和工程方面的职业，并改善女性研究数学家的培养渠道。水波问题涉及的是在重力和可能的表面张力作用下，将一种不可压缩的无粘流体在两维或三维空间中分离出来的界面波的运动。水波以一种理想化的方式描述了在海洋或湖泊中可以观察到的东西，是应用数学的完美样本。它们承载着丰富的波浪现象，从表面张力驱动的涟漪到海啸再到巨浪，其长度范围不等。它们为数学的几个分支提供了来源和灵感。此外，它们还影响着数学以外的领域，从水力学到天气预报。然而，水波问题对严格的分析、建模和数值模拟提出了深刻而微妙的困难。首先，水和空气之间的界面是先验未知的，需要作为解的一部分来确定，即自由边界。顺便说一句，自由边界本身在数学上是具有挑战性的，它们发生在许多情况下，例如冰的融化和柔性膜在障碍物上的拉伸。更糟糕的是，自由表面的边界条件是严重非线性的。该项目将开发新的工具来促进对这些具有挑战性问题的理解。