TY - JOUR

T1 - Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics

AU - Trakhinin, Yuri

N1 - Publisher Copyright: © 2020, Springer Nature Switzerland AG. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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.

AB - 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.

KW - Current-vortex sheets

KW - Local-in-time existence of discontinuous solutions

KW - Shallow water magnetohydrodynamics

KW - Shock waves

KW - Symmetric hyperbolic system

KW - EXISTENCE

KW - BOUNDARY-VALUE-PROBLEMS

KW - EQUATIONS

KW - SYSTEMS

UR - http://www.scopus.com/inward/record.url?scp=85087202817&partnerID=8YFLogxK

U2 - 10.1007/s00033-020-01344-2

DO - 10.1007/s00033-020-01344-2

M3 - Article

AN - SCOPUS:85087202817

VL - 71

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

SN - 0044-2275

IS - 4

M1 - 118

ER -