Three-dimensional two-phase Euler equations with general density ratio and interfacial vorticity

Establish a comprehensive mathematical theory for the three-dimensional two-phase free-boundary Euler system with general density ratio and interfacial vorticity.

Background

The paper studies two immiscible, incompressible, inviscid fluids separated by a moving three-dimensional interface, allowing a nonzero interfacial vorticity and a general density ratio. The authors derive an autonomous contour-dynamics formulation, obtain a direct paradifferential reduction of the associated singular Birkhoff–Rott operators, and prove local Sobolev well-posedness for small perturbations in the stable Kelvin–Rayleigh–Taylor regime.

The broader three-dimensional two-phase problem is not resolved by the paper in full generality: the result is restricted to small data and parameters in a strictly stable regime, whereas the system with general density ratio and interfacial vorticity is described as remaining largely open.