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

TL;DR

This paper investigates the stability of a plasma-vacuum interface in ideal incompressible magnetohydrodynamics, incorporating the displacement current in vacuum to understand electric field effects, and establishes an energy estimate under a known stability condition.

Contribution

It extends the classical plasma-vacuum interface analysis by including displacement current, providing a new energy a priori estimate under the stability condition.

Findings

Proved an energy a priori estimate for the linearized problem.

Developed a secondary symmetrization of Maxwell equations.

Constructed a Kreiss-type symmetrizer for the elliptic-hyperbolic problem.

Abstract

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 in [Trakhinin Y. arXiv:1812.08675], 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.