Geometric aspects of hydrodynamic blowup

点击翻译按钮获取中文摘要

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.