Vanishing viscosity in the presence of a boundary

这个项目的目的是为了更好地理解边界附近几乎无粘性流体的行为。工程师和物理学家对这种行为感兴趣至少有一百年了，而在其中至少一半的时间里，研究流体的数学家一直迫切关注这种行为。关于边界行为有很多实用的知识，但很少有精确的数学理解。该建议是研究流体的边界行为在有限设置的二维有界域，集中在一个磁盘，其中的重要方面的分析是简化的，但在那里的行为仍然是物理相关。目标是加强现有的关于流体边界行为的部分结果，并试图将工程师和物理学家的部分启发式方法整合到数学设置中。许多流体具有相当低的粘度，而且几乎所有流体都与边界接触。血管壁缺陷对血流有何影响？机翼的升力是如何由它的形状决定的？当潜艇以一定速度运行时，螺旋桨是如何引起湍流的？在所有这些问题以及更多的问题中，流体的流动可能受到流体与边界的相互作用的限制，而这些限制目前只被部分地理解。在理解这种相互作用方面取得的几乎任何进展最终都将有利于解决所有这些问题，无论是通过更好地控制数值计算中的误差界限，还是通过对相互作用进行定性理解，甚至通过证明边界行为比目前认为的还要复杂。

The purpose of this project is to better understand the behavior of nearly inviscid fluids near a boundary. Such behavior has been of interest to engineers and physicists for at least a hundred years and has been a pressing concern of mathematicians working in fluids for at least half of that time. There is much practical knowledge about the boundary behavior, but little precise mathematical understanding. The proposal is to investigate the boundary behavior of the fluid in the limited setting of a two dimensional bounded domain, focusing on a disk, where important aspects of the analysis are simplified, but where the behavior is still physically relevant. The goal is to strengthen existing partial results concerning the boundary behavior of the fluid and to attempt to integrate the partially heuristic methods of engineers and physicists into a mathematical setting.Many fluids have reasonably low viscosity and nearly all fluids are in contact with a boundary. How is the flow of blood affected by imperfections in vessel walls? How is the lift of a wing determined by its shape? How do a submarine's propellers induce turbulence when running at certain speeds? In all of these problems and many more the flow of the fluid can be constrained by the interaction of the fluid with the boundary in ways that are currently only partially understood. Almost any progress that can be made in the understanding of this interaction would ultimately be beneficial in addressing all of these problems, whether by giving better control on the error bounds in numerical computations, by giving a qualitative understanding of the interaction, or even by demonstrating that the boundary behavior is even more complex than is currently believed.