Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

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Abstract

We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [25] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-vortex sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-vortex sheets.

Original languageEnglish
Article number118
Number of pages13
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume71
Issue number4
DOIs
Publication statusPublished - 1 Jul 2020

Keywords

  • Current-vortex sheets
  • Local-in-time existence of discontinuous solutions
  • Shallow water magnetohydrodynamics
  • Shock waves
  • Symmetric hyperbolic system
  • EXISTENCE
  • BOUNDARY-VALUE-PROBLEMS
  • EQUATIONS
  • SYSTEMS

OECD FOS+WOS

  • 1.01 MATHEMATICS

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