Non-existence and strong ill-posedness in $C^{k,\beta }$ for the generalized Surface Quasi-geostrophic equation

We consider solutions to the generalized Surface Quasi-geostrophic equation ($\gamma$-SQG) when the velocity is more singular than the active scalar function (i.e. $\gamma \in (0,1)$). In this paper we establish strong ill-posedness in $C^{k,\beta }$ ($k\ge 1$, $\beta \in (0,1]$ and $k+\beta >1+\gamma$) and we also construct solutions in ${\mathbb{R}}^2$ that initially are in $C^{k,\beta }\cap L^2$ but are not in $C^{k,\beta }$ for $t>0$. Furthermore these solutions stay in $H^{k+\beta +1-2\delta }$ for some small $\delta$ and an arbitrarily long time.