Beyond the Singularity: A Study on the Large-time Behavior of Potentially Singular Solutions to the 3D Axisymmetric Euler Equations

Whether an initially smooth, incompressible ideal flow field can develop singular behaviors (such as a perfect storm) in finite time is one of the most fundamental questions in fluid dynamics. It is closely related to the Clay Millennium Prize Problem on Navier-Stokes equations, and has remained open for more than two and a half centuries. In a recent numerical study, a class of rotationally symmetric, incompressible ideal flows that experiences tremendous growth in local spin rate (vorticity) is proposed and carefully examined. With strong supporting evidence, the study suggests the existence of a finite-time singularity, but the nature of the flow field after the singularity time remains unclear. The purpose of this study is to investigate the large-time behavior of the potentially singular flow field by extending the previously reported computation beyond the singularity time. If successfully worked out, this study would yield important insights in not only the nature of the singularity, but also the potential connection between the singularity and the onset of turbulence.