The $\hat{I}\pm$-SQG patch problem is illposed in $C^{2,{\hat{I}}^2}$ and $W^{2,p}$

Abstract

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

v2: minor revision, to appear in CPAM