Non-existence and Strong lll-posedness in \(C^{k,\beta }\) for the Generalized Surface Quasi-geostrophic Equation

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