Finite time singularity of a vortex patch model in the half plane

The question of global regularity v.s. finite time blow-up remains open for many fluid equations. In this talk, I will discuss a family of equations which interpolate between the 2D Euler equation and the surface quasi-geostrophic (SQG) equation. We focus on the patch dynamics for this family of equation in the half-plane, and obtain the following results: For the 2D Euler patch model, the patches remain globally regular even if they initially touch the boundary of the half-plane; whereas for the family of equations that are slightly more singular than the 2D Euler equation, the patches can develop a finite-time singularity. This talk is based on a joint work with A. Kiselev, L. Ryzhik and A. Zlatos.