A Priori Estimates for the Motion of Charged Liquid Drop: A Dynamic Approach via Free Boundary Euler Equations

We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work by Rayleigh. Due to experiments and numerical simulations one may expect the charged drop to form conical singularities called Taylor cones, which we interpret as singularities of the flow. In this paper, we study the well-posedness of the problem and regularity of the solution. Our main theorem is a criterion which roughly states that if the flow remains \(C^{1,\alpha }\) $C^{1,\alpha }$ -regular in shape and the velocity remains Lipschitz-continuous, then the flow remains smooth, i.e., \(C^\infty \) $C^{\infty }$ in time and space, assuming that the initial data is smooth. Our main focus is on the regularity of the shape of the drop. Indeed, due to the appearance of Taylor cones, which are singularities with Lipschitz-regularity, we expect the \(C^{1,\alpha }\) $C^{1,\alpha }$ -regularity assumption to be optimal. We also quantify the \(C^\infty \) $C^{\infty }$ -regularity via high order energy estimates which, in particular, implies the well-posedness of the problem.