A conditional no-splash criterion for two-fluid Euler--MHD interface

Abstract

We study a two-dimensional incompressible two-fluid system: an inviscid, irrotational Euler phase and a viscous, conducting MHD phase, separated by a free interface. We prove a conditional quantitative no-splash criterion for smooth solutions in a specified admissible class. In addition to a bound on the trace of the Euler velocity, an initial bound on the magnetic trace and bounds on the bulk vorticity, this class imposes bounds on the interface Biot--Savart potential and its tangential derivative, fixed exponential vertical localization, and the normal derivative of the bulk vorticity at the interface together with one tangential derivative of it. It also requires a Biot--Savart representation of the Euler velocity up to a bounded-gradient harmonic part and a positive initial chord-arc bound. Under these assumptions the chord-arc functional has a uniform positive lower bound on $[0,T]$, depending only on the interval length, the decay width and the a priori bound, after an enlargement that absorbs the surface tension and the densities. The proof combines a maximum-principle analysis of the interface-vorticity equation, in which the magnetic field enters through the Lorentz term and the pressure jump, with a geometric reduction in the style of Fefferman--Ionescu--Lie.