Finite time singularity formation for the modified SQG patch equation

It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars which interpolates between these two equations, via a parameter $\alpha\in[0,\frac 12]$ appearing in the kernels of their Biot-Savart laws. The values $\alpha=0$ and $\alpha=\frac 12$ correspond to the 2D Euler and SQG cases, respectively. We prove global in time regularity for the 2D Euler patch model, even if the patches initially touch the boundary of the half-plane. On the other hand, for any sufficiently small $\alpha>0$, we exhibit initial data which lead to a singularity in finite time. Thus, these results show a phase transition in the behavior of solutions to these equations, and provide a rigorous foundation for classifying the 2D Euler equations as critical.