Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics Yuri Trakhinin 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. Mathematics Subject Classification. 35L40, 35L65, 35L67, 76L05, 76W05. Keywords. Shallow water magnetohydrodynamics, Symmetric hyperbolic system, Local-in-time existence of discontinuous solutions, Shock waves, Current-vortex sheets.
1. Introduction The equations of shallow water magnetohydrodynamics (SMHD) were proposed by Gilman [11] for studying the global dynamics of the solar tachocline which is a thin transition layer between the Sun’s radiative interior and the differentially rotating outer convective zone. The SMHD equations are derived in [11] from the equations of ideal incompressible magnetohydrodynamics under the influence of gravity by depth averaging and assuming that the pressure is hydrostatic and the depth of the layer of a perfectly conducting fluid is small enough. The SMHD system is important not only for astrophysical applications like the solar tachocline (see, e.g., [9,11,32]) but may also be used for modelling conducting shallow water fluids in laboratory and industrial environments (further references about applications of SMHD can be found, for example, in [15]). The equations of SMHD [11] read: dh + hdiv v = 0, (1) dt dv − (B · ∇)B + g∇h = 0, (2) dt dB − (B · ∇)v = 0, (3) dt where h is the height of a conducting fluid, v = (v1 , v2 ) ∈ R2 is the fluid velocity, B = (B1 , B2 ) ∈ R2 is the magnetic field, the constant g > 0 is the gravitational acceleration, d/dt = ∂t + (v · ∇) is the material This work was supported by the Russian Science Foundation under grant No. 20-11-20036. 0123456789().: V,-vol
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derivative, ∂t = ∂/∂t is the time derivative, ∇ = (∂1 , ∂2 ), ∂i = ∂/∂xi (i = 1, 2), x = (x1 , x2 ), and x1 and x2 are spatial coordinates. System (1)–(3) is a closed system for the unknown U = U (t, x) = (h, v, B) ∈ R5 and can be written as the following quasilinear system: A0 (U )∂t U + A1 (U )∂1 U + A2 (U )∂2 U = 0, where A0 = block d
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