Illposedness of $C^2$ vortex patches

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 $1<p<\mathrm{\infty }$ and construct $C^2$ initial patch data for which the curvature of the patch boundary becomes unbounded immediately for $t>0$. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.