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

Abstract

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 $$H^{1.5+\delta }$$H1.5+� Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the $$H^{2+\delta }$$H2+� space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in $$H^{2+\delta }$$H2+�. 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.