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

Archive for Rational Mechanics and Analysis(2019)

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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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