Norm Inflation For The Critical SQG Equation

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 $1\le \beta <\frac{2}{p}$.