Incompressible inviscid resistive MHD surface waves in 2D

We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field.
Comment: The Dirichlet boundary condition $B=\bar B$ imposed for the magnetic field is not proper, and it should be replaced by the jump condition where the magnetic field in vacuum is involved, that is, one should consider the full plasma-vacuum interface model when considering the resistive MHD