Contents Online
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
DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n2.a2
Author
Abstract
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.
Keywords
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