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Mathematics > Analysis of PDEs

arXiv:1812.08675 (math)
[Submitted on 20 Dec 2018 (v1), last revised 25 Dec 2019 (this version, v2)]

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

Authors:Yuri Trakhinin
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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 coefficients 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.
Comments: 17 pages
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn); Plasma Physics (physics.plasm-ph)
MSC classes: 35M33, 35L45, 35Q35, 76B03, 76W05
Cite as: arXiv:1812.08675 [math.AP]
  (or arXiv:1812.08675v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1812.08675

Submission history

From: Yuri Trakhinin [view email]
[v1] Thu, 20 Dec 2018 16:24:26 UTC (21 KB)
[v2] Wed, 25 Dec 2019 11:54:37 UTC (25 KB)
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