Stabilizing effect of surface tension for the linearized MHD-Maxwell free interface problem

TL;DR

This paper demonstrates that surface tension has a stabilizing effect on the linearized free interface problem in ideal compressible MHD-Maxwell equations, verified through energy estimates without stability assumptions.

Contribution

It provides the first energy a priori estimate for the linearized problem with surface tension, confirming its stabilizing role in MHD-Maxwell free boundary problems.

Findings

Surface tension stabilizes the linearized MHD-Maxwell interface problem.

Energy estimates are derived without stability conditions on the unperturbed flow.

Large vacuum electric fields can cause ill-posedness when surface tension is absent.

Abstract

We consider an interface with surface tension that separates a perfectly conducting inviscid fluid from a vacuum. The fluid flow is governed by the equations of ideal compressible magnetohydrodynamics (MHD), while the electric and magnetic fields in vacuum satisfy the Maxwell equations. With boundary conditions on the interface this forms a nonlinear hyperbolic problem with a characteristic free boundary. For the corresponding linearized problem we derive an energy a priori estimate in a conormal Sobolev space without assuming any stability conditions on the unperturbed flow. This verifies the stabilizing effect of surface tension because, as was shown in [Mandrik and Trakhinin, in Commun. Math. Sci. 12 (2014), 1065-1100], a sufficiently large vacuum electric field can make the linearized problem ill-posed for the case of zero surface tension. The main ingredients in proving the energy estimate are a suitable secondary symmetrization of the Maxwell equations in vacuum and making full use of the boundary regularity enhanced from the surface tension.