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

Abstract

We consider a family of contour dynamics equations depending on a parameter $\al$ with $0<\alpha \le 1$. 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.

26 pages