A potential two-scale traveling wave singularity for 3D incompressible Euler equations

Abstract

In this paper, we investigate a potential two-scale traveling wave singularity of the 3D incompressible axisymmetric Euler equations with smooth initial data of finite energy. The two-scale feature is characterized by the property that the center of the traveling wave approaches to the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields. Without any viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. In order to stabilize the rapidly decreasing thickness of the sharp front, we apply a vanishing first order numerical viscosity to the Euler equations. We present numerical evidence that the 3D Euler equations with this first order numerical viscosity develop a locally self-similar blowup at the origin.