Nonlinear stability and existence of two-dimensional compressible current-vortex sheets

Autor: Morando, Alessandro, Secchi, Paolo, Trebeschi, Paola, Yuan, Difan
Rok vydání: 2023
Předmět:
Druh dokumentu: Working Paper
DOI: 10.1007/s00205-023-01865-w
Popis: We are concerned with nonlinear stability and existence of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash-Moser iteration, provided the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin-Helmholtz instabilities.
Comment: The paper has been accepted for publication in "Archive for Rational Mechanics and Analysis"
Databáze: arXiv
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