Illposedness of C2 Vortex Patches

It is well known that vortex patches are wellposed in C1,α if 0 < α< 1 . In this paper, we prove the illposedness of C2 vortex patches. The setup is to consider the vortex patches in Sobolev spaces W2,p where the curvature of the boundary is Lp integrable. In this setting, we show the persistence of W2,p regularity when 1 < p< ∞ and construct C2 initial patch data for which the curvature of the patch boundary becomes unbounded immediately for t> 0 , though it regains C2 regularity precisely at all integer times without being time periodic. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.