Well-Posedness and Finite Time Singularity for Touching g-SQG Patches on the Plane

We prove local well-posedness as well as singularity formation for the g-SQG patch model on the plane (so on a domain without a boundary), with $\alpha \in (0,\frac{1}{6}]$ and patches being allowed to touch each other. We do this by bypassing any auxiliary contour equations and tracking patch boundary curves directly instead of their parametrizations. In our results, which are sharp in terms of $\alpha$, the patch boundaries have $L^2$ curvatures and a singularity occurs when at least one of these $L^2$-norms blows up in finite time.