FRG: Collaborative Research: Singularity Formation for the Three-Dimensional Euler Equations and Related Problems

The question of singularity formation for the three-dimensional Euler equations of incompressible inviscid fluid flow is a celebrated open problem in mathematics and physics. The existence of Euler singularities is likely to have substantial implications for physical fluid dynamics, in particular a role in the onset and structure of turbulence. This research will utilize a combination of numerical and analytical methods to study Euler singularity formation, as well as examine the significance of these singularities for fluid dynamic turbulence. A centerpiece of the project is a new method for constructing singular Euler solutions, starting from a semi-analytic approach using complex space-time and singularities in the complex plane. The results should be amenable to rigorous analysis and direct numerical validation. A full numerical and analytic validation of the singularity construction forms a main component of this proposal. Several related projects involving singularity formation in interfacial internal waves, and in incompressible nondissipative magnetohydrodynamics will also be undertaken.

The incompressible Euler equations are a system of partial differential equations that describe the flow of inviscid fluids. Although these equations have been known for nearly 250 years, basic mathematical questions concerning the nature of solutions are still open. In particular, it is still not known whether solutions of the three dimensional Euler equations can form a singularity, that is, an infinite value in a flow quantity such as the velocity or vorticity (which measures circulation) in finite time. Due to its implications in turbulence theory, the question of Euler singularities has received intense attention over the last 20 years. Successful construction of Euler singularities would solve a major problem of mathematics and would establish a new method for addressing singularity formation. A fluid dynamic understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open problems of classical physics. This in turn could lead to important new methods for understanding and simulating turbulent flows, essential in a wide range of engineering applications, including the design of aircraft and watercraft.