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

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.