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 $C^k$ ($k\geq 2$) but that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$. Moreover, we prove strong ill-posedness in the critical space $H^{2}$.

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