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arXiv · 1801.07382

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

Tam Do
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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 $\omega^\theta/r$.

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Tam Do. 2018-01-23. On Vorticity Gradient Growth for the Axisymmetric 3D Euler Equations Without Swirl. https://doi.org/10.1007/s00205-019-01388-3

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