Geometric aspects of hydrodynamic blowup

这个项目涉及到使用一种新的特性来描述理想流体的三维欧拉方程的放大，根据黎曼几何的保体积微分同态群（最初是由Arnold首创的）。这组测地线代表流体的拉格朗日运动，当达到爆破时间时，测地线不能在更短的间隔内使长度最小化。这个条件可以用截面曲率的正膨胀来理解，也可以用保体映射空间的弱几何来理解。我们建议用三种更简单的近似几何来研究几何，以便尝试在几何上排除爆炸。这个项目包括研究欧拉方程，它描述了流体在三维空间中的运动。证明这些方程可以永远用来描述流体的问题，即使运动变得非常动荡，已经研究了数百年，但仍然没有解决。我们提出了一种新的方法，它包括从几何上观察流体，作为无限维弯曲空间中的最短路径（与飞机在地球的二维曲面上追踪长度最小路径的方式非常相似）。尽管这种流体的几何图形早在20世纪60年代就被人们所知，但直到最近才有可能将湍流运动与弯曲空间中的路径长度联系起来，该项目是利用这种方法来帮助确定这些方程是否总是有效，或者当流体运动变得过于复杂时，它们是否必须“爆炸”。

This project involves using a new characterization of blowup for the three-dimensional Euler equations for an ideal fluid, in terms of the Riemannian geometry of the group of volume-preserving diffeomorphisms (as originally pioneered by Arnold). The criterion is that geodesics in this group, which represent Lagrangian motions of fluids, fail to minimize length on ever-shorter intervals as the blowup time is reached. This condition can be understood in terms of positive blowup of the sectional curvature, as well as in terms of the weak geometry of the space of volume-preserving maps. We propose to investigate the geometry using three simpler approximate geometries in order to try to rule out blowup geometrically.This project involves studying the Euler equations, which describe the motion of a fluid in three dimensions. The problem of showing that these equations can be used to describe the fluid forever, even if the motion becomes very turbulent, has been studied for hundreds of years but remains unsolved. We propose a new approach which involves viewing the fluid geometrically, as a shortest path in an infinite-dimensional curved space (in much the same way that an airplane traces out a length-minimizing path on the two-dimensional curved surface of the earth). Although this geometric picture of a fluid has been known since the 1960s, only recently has it been possible to relate turbulent motion to path length in this curved space, and the project is to use this approach to help decide whether the equations are always valid or whether they have to "blow up" when the fluid motion gets too complicated.