Geometric aspects of hydrodynamic blowup

这个项目涉及使用一个新的表征爆破的三维欧拉方程的理想流体，在黎曼几何的一组体积保持同构（最初由阿诺德开创）。标准是这一组中的测地线，它代表流体的拉格朗日运动，不能最小化长度越来越短的时间间隔达到爆破时间。这个条件可以理解为截面曲率的正爆破，也可以理解为保体积映射空间的弱几何。我们建议使用三种更简单的近似几何来研究几何，以试图从几何上排除爆破。这个项目涉及研究欧拉方程，它描述了三维流体的运动。证明这些方程可以永远用来描述流体的问题，即使运动变得非常湍流，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，它涉及到观看流体几何，作为一个最短的路径在一个无限维的弯曲空间（在很大程度上相同的方式，飞机的痕迹出一个长度最小化的路径上的二维曲面的地球）。虽然这种流体的几何图像自20世纪60年代以来就已为人所知，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，该项目将使用这种方法来帮助确定方程是否总是有效，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.