On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition

We prove a local-in-time existence of solutions result for the two dimensional incompressible Euler equations with a moving boundary, with no surface tension, under the Rayleigh–Taylor stability condition. The main feature of the result is a local regularity assumption on the initial vorticity, namely H^{1.5 + δ} Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the H^{2 + δ} space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in H^{2 + δ}. The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D.