*preprint*

**Inserted:** 22 nov 2021

**Year:** 2021

**Abstract:**

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
expects 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, regularity and the formation of singularities of the solution.
Our main theorem roughly states that if the flow remains C^{{1,\alpha}}-regular
in shape and the velocity remains Lipschitz-continuous, then the flow remains
smooth, i.e., C^{{\infty}} in time and space, assuming that the initial data is
smooth. Due to the appearance of Taylor cones we expect the
C^{{1,\alpha}}-regularity assumption to be optimal, while the
Lipschitz-regularity assumption on the velocity is standard in the classical
theory of the Euler equations. We also quantify the C^{{\infty}}-estimate via
high order energy estimates. This result is new also for the Euler equations
with free boundary without the electric field. We point out that we do not
consider the problem of existence in this paper. It will be studied in
forthcoming work.