Singularities in axisymmetric free boundaries for ElectroHydroDynamic equations

TL;DR

This paper investigates singularities in axisymmetric free boundary problems for ElectroHydroDynamic equations, demonstrating conditions under which the fluid or gas regions develop cusp singularities, and introduces a potentially new convex conical air-phase solution.

Contribution

It establishes conditions leading to cusp singularities in ElectroHydroDynamic free boundaries and presents a novel convex conical air-phase solution.

Findings

Either the fluid or gas region forms a cusp asymptotically.

Conditions on velocity or electric field lead to singularities.

A new convex conical air-phase solution is identified.

Abstract

We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new.