Local Well-posedness of the Free-Boundary Incompressible Magnetohydrodynamics with Surface Tension

TL;DR

This paper establishes the local well-posedness of 3D free-boundary incompressible ideal MHD equations with surface tension, using an approximation approach and energy estimates, addressing the complex coupling of velocity and magnetic fields.

Contribution

It extends previous work by proving local existence and uniqueness for free-boundary MHD with surface tension, overcoming challenges from the strong velocity-magnetic field coupling.

Findings

Proves local well-posedness of 3D free-boundary incompressible MHD with surface tension.

Develops an approximation scheme with artificial viscosity converging to the actual solution.

Establishes a priori energy estimates for the coupled MHD system.

Abstract

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by $\kappa>0$ whose solution converges to that of the MHD equations as $\kappa\to 0$. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.