Strong ill-posedness for SQG in critical Sobolev spaces

Abstract

We prove that the inviscid surface quasigeostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data H 2 ‫ޔ(‬ 2 ) without any solutions in L ∞ t H 2 .Moreover, we prove strong critical norm inflation for C ∞ -smooth data.Our proof is robust and extends to give similar ill-posedness results for the family of modified SQG equations which interpolate the SQG with the two-dimensional incompressible Euler equations.