On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl

TL;DR

This paper investigates the growth of vorticity gradients in the 3D axisymmetric Euler equations without swirl, establishing upper bounds and demonstrating possible double exponential growth at the boundary.

Contribution

It proves that the vorticity gradient can grow at most double exponentially and constructs scenarios for such growth at the boundary of the domain.

Findings

Gradient of vorticity grows at most double exponentially.

Double exponential growth of vorticity gradient is achievable at the boundary.

Well-posedness is confirmed due to the 2D-like geometry.

Abstract

We consider the 3D axisymmetric Euler equations without swirl on some bounded axial symmetric domains. In this setting, well-posedness is well known due to the essentially 2D geometry. The quantity $\omega^\theta/r$ plays the role of vorticity in 2D. First, we prove that the gradient of $\omega^\theta/r$ can grow at most double exponentially with improving a priori bound close to the axis of symmetry. Next, on the unit ball, we show that at the boundary, one can achieve double exponential growth of the gradient of $\omega^\theta/r$.