Mathematical Analysis of Fluid Free Boundary Problems

该项目关注的是模拟水波运动和其他相关的流体自由边界问题。研究的重点是数学工具的发展，以了解在二维和三维水波的长期行为，有和没有表面张力。我们也将研究自引力不可压缩流体的自由边界问题的解的长时间行为。 另一个研究主题是致力于水波与固定刚性边界的相互作用，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的现象。该项目的一个组成部分是研究生和博士后研究员的参与，这对培养下一代劳动力至关重要。 该项目将研究水波方程的四次可积性和二维和三维水波的长时间行为，有和没有表面张力。它还将研究自引力不可压缩流体自由边界问题解的四次和更高阶可积性以及长时间行为。我们的目标是挖掘隐藏的结构的方程，以产生新的信息的行为的解决方案，并解决开放的问题，如长期存在和正则性的解决方案在一定的小条件下，长期存在和正则性的解决方案与初始界面的任意大的陡度。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长期目标的一个组成部分，该目标是了解水波和其他相关流体运动的行为，包括奇点形成。该奖项反映了NSF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.