Stability of incompressible current-vortex sheets

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

On a Priori Energy Estimates for Characteristic Boundary Value Problems

“…Instead, the linearized problem with general data F and g = 0 admits an a priori estimate with a loss of two derivatives, see [34] for details. Analogous results for incompressible current-vortex sheets are obtained in [4] and [23].…”

Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD

“…In Section 8 we prove the well-posedness of the original linearized problem in conormal Sobolev spaces (see Section 3 for their definition). At last, in Section 9, using as in [12] a current-vorticity-type linearized system, we estimate missing normal derivatives of the perturbations of the velocity and the plasma magnetic field and prove the well-posedness of the linearized problem in Sobolev spaces (more precisely, in weighted Sobolev spaces, see Section 3), as stated in Section 4. 1.1.…”

“…From system (2.21) we can deduce nonhomogeneous equations which are a linearized counterpart of the divergence constraint (1.22) and the "redundant" boundary condition (1.23). More precisely, with reference to [15,Proposition 2] and [12] for the proof, we have the following. …”

“…Current-vorticity-type linearized system: proof of Theorem 4.1. Just as in (6.13), from (2.30a) and (2.31) we can estimate the normal derivatives of the normal components of the velocity and the plasma magnetic field through conormal derivatives and the source term: To prove the existence of missing normal derivatives of W γ (and obtain estimates for them) we use arguments similar to those in [12] for incompressible current-vortex sheets. That is, we write down a current-vorticity-type linearized system which is a linear symmetric hyperbolic system for the linearized vorticity ξ γ = ∇ × U γ and current z γ = ∇ × B γ , where For obtaining this system we rewrite equations (2.30a) and (2.30b) as ∂ t U γ + γU γ + 1…”

“…are satisfied for the basic state (2.2), if they hold at t = 0 (see [15], [12] for the proof). Thus, for the basic state we also require the fulfillment of conditions (2.5) at t = 0.…”