The α$\alpha$‐SQG patch problem is illposed in C2,β$C^{2,\beta }$ and W2,p$W^{2,p}$

Abstract

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