Well-Posedness for Low Regularity Solutions to the g-SQG Equation with Regular Level Sets

We show that the generalized SQG equation on the plane is locally well-posed in spaces of low regularity solutions (essentially H\"older continuous with H\"older exponents depending on the equation parameter $\alpha\in(0,\frac 12)$) that have $H^2$ level sets (i.e., with $L^2$ curvatures). Moreover, for $\alpha\le\frac 16$ and initial data satisfying some additional hypotheses we show that the corresponding solutions can stop existing only when their level sets lose $H^2$-regularity, and hence not just due to level set collisions or "pile ups".