Long-Time Behavior in Free-Surface Problems in Fluid Dynamics

具有自由表面的流体的运动在工程和科学中的许多问题中出现;然而，这种流动的数学理论只得到了部分发展。 虽然短时间内的平稳流动最近已经得到了相当好的理解，但对于其他情况几乎没有理论。 本计画探讨拓扑转变后的界面Navier-Stokes流模型、涡面小解的整体存在性、非牛顿Hele-Shaw流的数学理论等问题。自由表面流理论在流体动力学的科学与工程上有许多可能的应用，例如石油威尔斯的钻井与固井、了解血液流动、或者理解大气和海洋中流体的混合，以及微流体学中的问题。 总的来说，拟议的研究将提高对波浪破碎，流体混合，湍流和这种流动的稳定性的理解。 调查人员将利用数学分析和科学计算的技术，利用从每种方法中获得的洞察力，更有效地使用另一种方法。

The motion of fluids with free surfaces occurs in many problems in engineering and science; the mathematical theory of such flows is only partially developed, however. While smooth flows for short times have recently become fairly well understood, there is little theory developed for other situations. This project explores models for interfacial Navier-Stokes flows after the onset of topological transitions, global existence of small solutions for vortex sheets, and the mathematical theory of non-Newtonian Hele-Shaw flow, among other problems.There are many possible applications of the theory of free-surface flows in fluid dynamics in science and engineering, such as the drilling and cementing of oil wells, understanding blood flow, or understanding the mixing of fluids in the atmosphere and ocean, as well as problems in microfluidics. In general terms, the proposed research will improve the understanding of wave breaking, fluid mixing, turbulence, and stability of such flows. The investigator will utilize techniques of both mathematical analysis and scientific computing, using insight gained from each approach to more efficiently and more effectively use the other.