Non existence and strong ill-posedness in $C^k$ and Sobolev spaces for SQG

We construct solutions in ${\mathbb{R}}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\ge 2$) but that are not in $C^k$ for $t>0$. We prove a similar result also for $H^s$ in the range $s\in (\frac{3}{2},2)$. Moreover, we prove strong ill-posedness in the critical space $H^2$.