Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

On violent instability of a plasma-vacuum interface for an incompressible plasma flow and a nonzero displacement current in vacuum

Pages: 321 – 337



Yuri Trakhinin (Sobolev Institute of Mathematics, Novosibirsk, Russia; and Department of Mechanics and Mathematics, Novosibirsk State University, Novosibirsk, Russia)


In the classical statement of the plasma-vacuum interface problem in ideal magnetohydrodynamics (MHD) one neglects the displacement current in the vacuum region that gives the div-curl system of pre-Maxwell dynamics for the vacuum magnetic field. For understanding the influence of the vacuum electric field on the evolution of a plasma-vacuum interface we do not neglect the displacement current and consider the full Maxwell equations in vacuum. For the case of an incompressible plasma flow, by constructing an Hadamard-type ill-posedness example for the constant coefficient linearized problem we find a necessary and sufficient condition for the violent instability of a planar plasma-vacuum interface. In particular, we prove that as soon as the unperturbed plasma and vacuum magnetic fields are collinear, any nonzero unperturbed vacuum electric field makes the planar interface violently unstable. This shows the necessity of the corresponding non-collinearity condition for well-posedness and a crucial role of the vacuum electric field in the evolution of a plasma-vacuum interface.


ideal incompressible magnetohydrodynamics, plasma-vacuum interface, free boundary problem, ill-posedness

2010 Mathematics Subject Classification

35L45, 35M33, 35Q35, 76B03, 76W05

Received 20 December 2018

Accepted 27 September 2019

Published 20 June 2022