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 H1.5 + δ Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the H2 + δ space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in H2 + 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.
All Science Journal Classification (ASJC) codes
- Control and Optimization
- Applied Mathematics