On the local existence and blow-up for generalized SQG patches

We study patch solutions of a family of transport equations given by a parameter $\alpha$, $0< \alpha <2$, with the cases $\alpha =0$ and $\alpha =1$ corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for $H^{2}$ patches in the half-space setting for $0<\alpha< 1/3$, allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of $\alpha$ for which finite time singularities have been shown in \cite{KYZ} and \cite{KRYZ}. Second, we establish that patches remain regular for $0<\alpha<2$ as long as the arc-chord condition and the regularity of order $C^{1+\delta}$ for $\delta>\alpha/2$ are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in \cite{CFMR} and \cite{ScottDritschel} for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in \cite{G} and in \cite{CCCGW}, giving local-wellposedness for patches in $H^{2}$ for $0<\alpha < 1$ and in $H^3$ for $1<\alpha<2$.