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 $H^2({\mathbb{𝕋}}^2)$ without any solutions in $L_t^{\infty }{H}^2$. Moreover, we prove strong critical norm inflation for $C^{\infty }$-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.