We consider the critical dissipative surface quasi-geostrophic (SQG) equation on $\mathbb{ R}^2$ or $\mathbb{ T}^2$. Despite global regularity of the equation, we show that the data-to-solution map at the critical level $H^1$ is not uniformly bounded. We construct solutions that experience $H^1$ norm inflation from smooth, compactly supported initial data with large $H^1$ norm. We also demonstrate small-data norm inflation in supercritical Sobolev spaces $W^{ \beta,p}$ for $1<p<2$ and $1\le\beta<\tfrac{ 2}{ p}$.
Categories: math.AP