Mathematical Analysis of Fluid Free Boundary Problems

该项目涉及水波运动和其他相关流体自由边界问题的建模。研究的重点是发展数学工具，以了解水波在二维和三维空间的长期行为，有和没有表面张力。研究了自重力不可压缩流体自由边界问题解的长时性。另一个研究课题是研究具有固定刚性边界的水波的相互作用，例如海浪与海岸的相互作用。这项工作将导致更好的建模和更有效的数值模拟现象。该项目的一个组成部分是研究生和博士后的参与，这对下一代劳动力的准备至关重要。该项目将研究水波方程的四次可积性，以及水波在有和没有表面张力的情况下在二维和三维的长期行为。研究了自重力不可压缩流体自由边界问题解的四次可积性、高阶可积性和长时性。目标是揭示方程的隐藏结构，获得关于解的行为的新信息，并解决诸如在一定小条件下解的长期存在性和规律性，以及具有任意大陡度初始界面的解的长期存在性和规律性等开放性问题。该项目还将研究水波与任意固定刚性边界的相互作用。该项目是PI了解水波和其他相关流体运动（包括奇点地层）行为的长期目标的一个组成部分。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。

The project is concerned with modeling the motion of water waves and other related fluid free-boundary problems. The focus of the research is on the development of mathematical tools to understand the long-time behavior of water waves in two and three dimensions, with and without surface tension. The long-time behavior of solutions of the free-boundary problem of a self-gravitating incompressible fluid will also be investigated. Another research topic is devoted to the interaction of water waves with a fixed rigid boundary, such as the ocean's waves with the coast. This work will lead to better modeling and more efficient numerical simulations of the phenomena. An integral part of the project is the involvement of graduate students and postdoctoral fellows, which is crucially important for the preparation of the next generation workforce. The project will study the quartic integrability of the water wave equations and the long-time behaviors of water waves in two and three dimensions, with and without surface tension. It will also investigate the quartic and higher order integrability and long-time behaviors of solutions of the free boundary problem of a self-gravitating incompressible fluid. The goals are to unearth hidden structures of the equations, to yield novel information on the behavior of solutions, and to solve open questions such as the long-time existence and regularity of solutions under certain smallness conditions, and the long-time existence and regularity of solutions with initial interfaces of arbitrary large steepness. The project will also investigate the interaction of water waves with an arbitrary fixed rigid boundary. This project is an integral part of the PI's long-term goal of understanding behaviors of water waves and other related fluid motions, including the singularity formations.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.