Existence for the $\al$-patch model and the QG sharp front in Sobolev spaces

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