Existence for the α-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter α with 0<α⩽1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α=1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.