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We study the motion of charged liquid drop in three dimensions where theequations of motions are given by the Euler equations with free boundary withan electric field. This is a well-known problem in physics going back to thefamous work by Rayleigh. Due to experiments and numerical simulations oneexpects the charged drop to form conical singularities called Taylor cones,which we interpret as singularities of the flow. In this paper, we study thewell-posedness, regularity and the formation of singularities of the solution.Our main theorem roughly states that if the flow remains C^{1,\alpha}-regularin shape and the velocity remains Lipschitz-continuous, then the flow remainssmooth, i.e., C^{\infty} in time and space, assuming that the initial data issmooth. Due to the appearance of Taylor cones we expect theC^{1,\alpha}-regularity assumption to be optimal, while theLipschitz-regularity assumption on the velocity is standard in the classicaltheory of the Euler equations. We also quantify the C^{\infty}-estimate viahigh order energy estimates. This result is new also for the Euler equationswith free boundary without the electric field. We point out that we do notconsider the problem of existence in this paper. It will be studied inforthcoming work.

A priori estimates for the motion of charged liquid drop: A dynamic approach via free boundary Euler equations