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.
I studied the equations of "ideal" fluid motion (without viscosity) from a geometric perspective, along with several other equations with a related geometric structure. These equations are a limiting case of the Navier-Stokes equations that form one of the Millennium Problems in mathematics, and the big question in this field is whether solutions of the Euler or Navier-Stokes equations in three space dimensions exist for all time. If they did not (due for example to very drastic turbulence), the equations would need to be replaced with a totally new theory. The practical significance of these equations is that they are widely used to predict the behavior of fluids (both gas and liquid) in situations such as weather prediction, mixing in engineering, geology, and any other situation where a fluid moves in a nontrivial way. Our approach (pioneered by Vladimir Arnold in 1966) is to view the Euler equations of fluid motion as a shortest-path problem in an infinite-dimensional curved space, in much the same way that an airplane flies on a shortest path on the two-dimensional surface of the earth. This geometric approach is well-known, but has been used seriously by few mathematicians since it is considered too complicated; however we have been able to make significant progress. Since the equations are too complicated to have a complete analysis, we sometimes study simpler differential equations with related properties in hopes that we can generalize their features. The present project involved two undergraduates, four graduate students, and a postdoctoral associate, as well as myself and other faculty collaborators at other universities. We studied geometric aspects of fluids and simpler models, particularly related to the "blow-up" problem in axisymmetric fluids (that is, when the flow of the fluid in a cylinder is the same regardless of how the cylinder is rotated). In addition we derived results for models of fluids such as the inextensible string, and other partial differential equations which can also be viewed as shortest-path problems on other infinite-dimensional curved spaces. The project led to seven published papers and funded travel to two conferences the PI organized, along with funding travel to present the work of the PI and pay salary and equipment expenses for undergraduates.