Local Existence of MHD Contact Discontinuities

Alessandro Morando, Yuri Trakhinin, Paola Trebeschi

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)691-742
Number of pages52
JournalArchive for Rational Mechanics and Analysis
Volume228
Issue number2
DOIs
Publication statusPublished - 1 May 2018

Keywords

  • COMPRESSIBLE EULER EQUATIONS
  • IDEAL MAGNETO-HYDRODYNAMICS
  • VACUUM INTERFACE PROBLEM
  • BOUNDARY VALUE-PROBLEM
  • CURRENT-VORTEX SHEETS
  • WELL-POSEDNESS
  • MAGNETOHYDRODYNAMICS

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