Global Well-posedness of Free Interface Problems for the incompressible Inviscid Resistive MHD

TL;DR

This paper proves the global well-posedness and exponential decay of solutions for the plasma-vacuum interface problem in incompressible resistive MHD with surface tension, highlighting the stabilizing role of magnetic fields.

Contribution

It establishes the first global well-posedness results for this free interface problem without irrotational assumptions, revealing a damping mechanism induced by resistivity and magnetic fields.

Findings

Solutions decay to equilibrium almost exponentially.

Magnetic fields have a strong stabilizing effect.

Damping structure for vorticity due to resistivity and magnetic field.

Abstract

We consider the plasma-vacuum interface problem in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field. The lower plasma region is governed by the incompressible inviscid and resistive MHD, the upper vacuum region is governed by the pre-Maxwell equations, and the effect of surface tension is taken into account on the free interface. The global well-posedness of the problem, supplemented with physical boundary conditions, around the equilibrium is established, and the solution is shown to decay to the equilibrium almost exponentially. Our results reveal the strong stabilizing effect of the magnetic field as the global well-posedness of the free-boundary incompressible Euler equations, without the irrotational assumption, around the equilibrium is unknown. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field. A similar global well-posedness for the plasma-plasma interface problem is obtained, where the vacuum is replaced by another plasma.