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

Abstract 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.