The α-SQG patch problem is illposed in C2,β and W2,p

We consider the patch problem for the (Formula presented.) -(surface quasi-geostrophic) SQG system with the values (Formula presented.) and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint (Formula presented.) Hölder spaces, as well as in (Formula presented.), (Formula presented.) spaces. In stark contrast to the Euler case, we prove that for (Formula presented.), the (Formula presented.) -SQG patch problem is strongly illposed in every (Formula presented.) Hölder space with (Formula presented.). Moreover, in a suitable range of regularity, the same strong illposedness holds for every (Formula presented.) Sobolev space unless (Formula presented.).