Abstract We consider the critical dissipative surface quasi-geostrophic equation on R^2 or 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^, p for 1p2 and 1 2p.
Guo et al. (Sat,) studied this question.