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

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

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