Zero Surface Tension Limit of the Free-Boundary Problem in Incompressible Magnetohydrodynamics

TL;DR

This paper proves that solutions to free-boundary incompressible ideal MHD equations with surface tension converge to solutions without surface tension under the Rayleigh-Taylor sign condition, extending previous results.

Contribution

It establishes the zero surface tension limit for free-boundary incompressible ideal MHD equations using advanced analytical techniques and boundary structures.

Findings

Convergence of solutions as surface tension approaches zero

Validation of the Rayleigh-Taylor sign condition for stability

Extension of previous theoretical frameworks in MHD

Abstract

We show that the solution of the free-boundary incompressible ideal magnetohydrodynamic (MHD) equations with surface tension converges to that of the free-boundary incompressible ideal MHD equations without surface tension given the Rayleigh-Taylor sign condition holds true initially. This result is a continuation of the authors' previous works [17,32,16]. Our proof is based on the combination of the techniques developed in our previous works [17,32,16], Alinhac good unknowns, and a crucial anti-symmetric structure on the boundary.