Analytical and Computational Approaches to Free-Surface Problems in Fluid Dynamics

理解具有自由表面的流体的运动是流体动力学中的一个基本问题。目前的部分挑战在于将二维结果扩展到三维环境。这个项目探索了几个自由表面问题，包括单个流体的三维问题和两个剪切流体之间的涡片界面的适定性问题，三维表面张力问题的高效数值方法的发展，以及小解和弱解的整体存在性的研究。流体界面的运动预测是许多科学和工程分支的核心。这个项目的目的是通过分析基本的数学结构来促进流体力学研究的进步。通过研究偏微分方程组的适定性，这项工作有助于确定潜在的流体动力学模型是否能够准确地描述物理现实。通过开发有效的数值方法，这项工作有助于深入了解奇点形成的长期行为和机制。所获得的见解将对未来的分析和实验研究产生重大影响。

Understanding the motion of fluids with free surfaces is a fundamental problem in fluid dynamics. Part of the current challenge lies in extending two-dimensional results to the three-dimensional setting. This project explores several free-surface problems, including well-posedness of three-dimensional problems for single fluids and for vortex-sheet interfaces between two shearing fluids, development of efficient numerical methods to deal with surface tension in three dimensions, and investigation of global existence of small solutions and of weak solutions.Predicting the motion of fluid interfaces is central to many branches of sciences and engineering. The aim of this project is to facilitate advances in fluid dynamics research through analysis of underlying fundamental mathematical structures. By investigating well-posedness of systems of partial differential equations, this work helps determine whether the underlying fluid-dynamical models can accurately describe physical reality. By developing efficient numerical methods, this work facilitates insight into long-time behavior and mechanisms through which singularities may form. The insight gained will significantly influence future analysis and experimental studies.