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

Finite time singularity formation for the modified SQG patch equation