Stability of an incompressible plasma–vacuum interface with displacement current in vacuum

We study the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement when the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, to better understand the influence of the electric field in vacuum, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Under the necessary and sufficient stability condition for a planar interface found earlier by Trakhinin, we prove an energy a priori estimate for the linearized constant coefficient problem. The process of derivation of this estimate is based on various methods, including a secondary symmetrization of the vacuum Maxwell equations, the derivation of a hyperbolic evolutionary equation for the interface function, and the construction of a degenerate Kreiss-type symmetrizer for an elliptic-hyperbolic problem for the total pressure.