Dispersive PDE and Interfacial Fluid Dynamics

首席研究员和他的合作者将讨论界面流体动力学和色散偏微分方程中的几个问题。以下是将要进行的具体活动：(A)将从计算和分析两方面研究时间周期界面流体流动，例如具有表面张力的水波和漩涡片。(B)将研究界面流动的光滑解的分解，特别是解决摩尔奇点是否会出现在具有表面张力的流动中的问题。(C)旋转界面Hele-Shaw流的适定性将得到证明。(D)将对具有退化色散的方程，如Rosenau-Hyman方程的适定性或病态性进行严格的研究。在科学和工程中，有许多实际情况涉及运动流体之间的界面，例如海洋中流体的混合，或地下水中污染物的运动。在这个项目中进行的研究将深入了解这些运动的可预测性，以及波浪破碎和流体混合等现象。研究者将在适当的情况下使用数学分析和科学计算的方法来进行研究。本科生和研究生都将参与研究。

The Principal Investigator and his collaborators will address several problems in interfacial fluid dynamics and dispersive partial differential equations.The following are the specific activities to be undertaken: (A) Time-periodic interfacial fluid flows, such as water waves and vortex sheets with surface tension, will be investigated, both computationally and analytically. (B) Breakdown of smooth solutions for interfacial flows will be investigated, specifically addressing the question of whether a Moore singularity can occur for flows with surface tension. (C) Well-posedness of rotational interfacial Hele-Shaw flows will be proved. (D) Rigorous studies of the well-posedness or ill-posedness of equations with degenerate dispersion, such as the Rosenau-Hyman equations, will be performed.There are many practical situations in science and engineering which involve interfaces between moving fluids, such as the mixing of fluids in the ocean, or the motion of contaminants in groundwater. The research to be undertaken in this project will provide insight into the predictability of such motions, as well as phenomena such as wave breaking and fluid mixing. The investigator will use methods of both mathematical analysis and scientific computing as appropriate in carrying out the research. Both undergraduate and graduate students will participate in the research.