Free boundary problem for two-dimensional ElectroHydroDynamic Equations with a gravity field

TL;DR

This paper investigates the free boundary problem in two-dimensional electrohydrodynamics with gravity, revealing how electric field decay rates influence the formation of singularities like corners or cusps on the fluid interface.

Contribution

It characterizes the singular profiles of the free interface near stagnation points based on the electric field decay rate, extending understanding of the Stokes conjecture in two-phase flows.

Findings

Fast decay of electric field leads to Stokes corner singularity.

Critical decay rate allows asymmetric corner formation.

Slow decay results in cusp singularity, destroying corner structures.

Abstract

This paper studies a two-phase free boundary problem governed by the ElectroHydroDynamic equations, which describes a perfectly conducting, incompressible, irrotational fluid with gravity, surrounded by a dielectric gas. The interface separating fluid and gas is referred to as the free boundary. It is known that the free surface remains smooth away from the stagnation points, where the relative velocity of the incompressible fluid vanishes. In the presence of gravity, the Stokes conjecture, proved by Varvaruca and Weiss [Acta. Math. 206, 363-403, (2011)], implies that the corner type singularity will occur in the one-phase incompressible fluid. It is natural to ask whether this conjecture still holds in the two-phase flow problem. As a consequence, the primary objective of this work is to characterize the possible singular profiles of the free interface near the stagnation points in the presence of an electric field. Our main result is the discovery of a critical decay rate of the electric field near the stagnation points which indicates the classification of the singular profiles of the free surface. More precisely, we showed that when the decay rate of the electric field is faster than the critical decay rate, its negligible effect implies that the singular profile must be the well-known Stokes corner. When the electric field decays as the critical decay rate, the symmetry of the corner region may be broken, giving rise to either a Stokes corner or an asymmetric corner as the possible singular profile. If the decay rate is slower than the critical decay rate, the electric field dominates and completely destroys the corner structure, resulting in a cusp singularity. The analysis of these singularities relies on variational principles and geometric methods. Key technical tools include a Weiss-type monotonicity formula, a frequency formula, and a concentration-compactness argument.