TY - JOUR

T1 - Local Existence of MHD Contact Discontinuities

AU - Morando, Alessandro

AU - Trakhinin, Yuri

AU - Trebeschi, Paola

PY - 2018/5/1

Y1 - 2018/5/1

N2 - We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional planar flows provided that the Rayleigh–Taylor sign condition [ ∂p/ ∂N] < 0 on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis (Morando et al. in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh–Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser iteration.

AB - We prove the local-in-time existence of solutions with a contact discontinuity of the equations of ideal compressible magnetohydrodynamics (MHD) for two dimensional planar flows provided that the Rayleigh–Taylor sign condition [ ∂p/ ∂N] < 0 on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity. MHD contact discontinuities are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. This paper is a natural completion of our previous analysis (Morando et al. in J Differ Equ 258:2531–2571, 2015) where the well-posedness in Sobolev spaces of the linearized problem was proved under the Rayleigh–Taylor sign condition satisfied at each point of the unperturbed discontinuity. The proof of the resolution of the nonlinear problem given in the present paper follows from a suitable tame a priori estimate in Sobolev spaces for the linearized equations and a Nash–Moser iteration.

KW - COMPRESSIBLE EULER EQUATIONS

KW - IDEAL MAGNETO-HYDRODYNAMICS

KW - VACUUM INTERFACE PROBLEM

KW - BOUNDARY VALUE-PROBLEM

KW - CURRENT-VORTEX SHEETS

KW - WELL-POSEDNESS

KW - MAGNETOHYDRODYNAMICS

UR - http://www.scopus.com/inward/record.url?scp=85035324221&partnerID=8YFLogxK

U2 - 10.1007/s00205-017-1203-3

DO - 10.1007/s00205-017-1203-3

M3 - Article

AN - SCOPUS:85035324221

VL - 228

SP - 691

EP - 742

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -