  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.
