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

Abstract

We consider the patch problem for the α-(surface quasi-geostrophic) SQG system with the values α = 0 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 C2,β Hölder spaces, as well as in W2,p, (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 W2,p Sobolev space unless p = 2.