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

Archive for Rational Mechanics and Analysis（2019）

Abstract

We consider solutions of the 3D axisymmetric Euler equations without swirl. In this setting, well-posedness is well-known due to the essentially 2D geometry. The quantity ω ^θ /r plays an analogous role as vorticity in 2D. For our first result, we prove that the gradient of ω ^θ /r can grow with at most double exponential rate with an 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 ω ^θ /r .

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Key words

vorticity gradient growth,axisymmetric 3d euler equations,swirl

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