# A priori Estimates for the Incompressible Free-Boundary Magnetohydrodynamics Equations with Surface Tension

## TL;DR

This paper establishes a priori estimates for 3D incompressible free-boundary MHD equations with surface tension, reducing regularity requirements and providing the first such results for ideal free-boundary MHD.

## Contribution

It introduces novel a priori estimates for free-boundary MHD with surface tension, eliminating the need for extra regularity on the flow map.

## Key findings

- First a priori estimates for free-boundary MHD with surface tension
- Reduction of regularity requirements for the flow map
- Establishment of estimates in Lagrangian coordinates with H^{3.5} regularity

## Abstract

We consider the three-dimensional incompressible free-boundary magnetohydrodynamics (MHD) equations in a bounded domain with surface tension on the boundary. We establish a priori estimate for solutions in the Lagrangian coordinates with $H^{3.5}$ regularity. To the best of our knowledge, this is the first result focusing on the incompressible ideal free-boundary MHD equations with surface tension. It is worth pointing out that the $1/2$-extra spatial regularity for the flow map $\eta$ is no longer required in this manuscript thanks to the presence of the surface tension on the boundary.