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arXiv · 2204.06416

Illposedness of $C^{2}$ vortex patches

Alexander Kiselev·Xiaoyutao Luo
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Abstract

It is well known that vortex patches are wellposed in $C^{1,\alpha}$ if $0<\alpha <1$. In this paper, we prove the illposedness of $C^{2}$ vortex patches. The setup is to consider the vortex patches in Sobolev spaces $W^{2,p}$ where the curvature of the boundary is $L^p$ integrable. In this setting, we show the persistence of $W^{2,p}$ regularity when $1 0$. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.

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Alexander Kiselev, Xiaoyutao Luo. 2022-04-13. Illposedness of $C^{2}$ vortex patches. https://doi.org/10.1007/s00205-023-01892-7

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