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

Abstract

We consider the patch problem for the $\alpha$-(surface quasi-geostrophic) 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 $C^{k,\beta }$ Hölder spaces, as well as in $W^{2,p}$, spaces. In stark contrast to the Euler case, we prove that for , the $\alpha$-SQG patch problem is strongly illposed in every $C^{2,\beta }$ Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for every $W^{2,p}$ Sobolev space unless $p=2$.