The $α$-SQG patch problem is illposed in $C^{2,β}$ and $W^{2,p}$

Abstract: We consider the patch problem for the $\alpha$-SQG system with the values $\alpha=0$ and $\alpha= \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint $C^{k,\beta}$ H\"older spaces, as well as in $W^{2,p},$ $1<p<\infty$ spaces. In stark contrast to the Euler case, we prove that for $0<\alpha< \frac{1}{2}$, the $\alpha$-SQG patch problem is strongly illposed in \emph{every} $C^{2,\beta} $ H\"older space with $\beta<1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for \emph{every} $W^{2,p}$ Sobolev space unless $p=2$.