Abstract :
We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex
sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28
(2005) 917–945] where an energy a priori estimate for the linearized free boundary value
problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics
was proved for a part of the whole stability domain found a long time ago in [S.I.
Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium,
Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem
of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we
derive an a priori estimate in the whole stability domain. The crucial point in deriving this
estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem
for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss’
type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition,
i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of
derivatives in the energy estimate. The result of this paper is a necessary step to prove
the local-in-time existence of stable nonplanar incompressible current-vortex sheets by
a suitable Nash–Moser type iteration scheme.
