Strong illposedness for SQG in critical Sobolev spaces

Abstract

We prove that the inviscid surface quasi-geostrophic (SQG) equations are strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^{2}(\bbT^2)$ 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 two-dimensional incompressible Euler equations.

35 pages

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Cited by in corpus (1)

• The $\alpha$-SQG patch problem is illposed in $C^{2,\beta }$ and $W^{2,p}$