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 $C^{2}$ patches. The setup to consider Sobolev spaces $W^{2,p}$ where curvature boundary $L^p$ integrable. setting, show persistence regularity when $10$. key ingredient evolution equation curvature, dominant term turns out be linear dispersive.