CAREER: Analysis of Surface Water Waves

PI将开发偏微分方程和其他数学分支的新技术工具，她将扩展和结合现有工具，以解决水波理论方面的几个长期悬而未决的问题。它们包括(1)初值问题的整体正则性与有限时间爆破的关系，(2)行波的存在及其分类，(3)行波的稳定性和不稳定性。重点放在大尺度动力学和真正的非线性行为上，对这些行为的敏锐理解最终取决于分析证据。重点是在流体力学中使用欧拉方程，而不是简单的近似模型，如Korteweg-de Vries方程。国际学生联合会建议在她所在的机构培养应用数学。她将继续组织研讨会和会议，并将通过会议演讲、研讨会和座谈来传播她的研究成果。国际教育协会建议加强本科远程教育课程和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、科学和工程领域的工作，并改善女性研究数学家的渠道。水波问题涉及到界面上的波动，在重力和可能的表面张力的作用下，空气下的不可压缩无粘流体在界面上以二维或三维形式分离。水波以理想化的方式描述了在海洋或湖泊中可能观察到的东西，是应用数学的完美样本。它们拥有丰富的海浪现象，从表面张力引发的涟漪到海啸再到无赖海浪，在长度上不一而足。它们为数学的几个分支提供了源泉和灵感。此外，它们还影响了数学以外的领域，从水力学到天气预报。尽管如此，水波问题仍然给严格的分析、建模和数值模拟带来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，需要作为解决方案的一部分来确定，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，它们在许多情况下都会发生，例如冰的融化和柔性薄膜在障碍上的拉伸。更糟糕的是，自由表面的边界条件是严重的非线性。该项目将开发新的工具，以促进对这些具有挑战性的问题的理解。

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.