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Medientyp: E-Artikel Titel: On the existence of incompressible current–vortex sheets: study of a linearized free boundary value problem Beteiligte: Trakhinin, Yuri Erschienen: Wiley, 2005 Erschienen in: Mathematical Methods in the Applied Sciences, 28 (2005) 8, Seite 917-945 Sprache: Englisch DOI: 10.1002/mma.600 ISSN: 0170-4214; 1099-1476 Entstehung: Anmerkungen: Beschreibung: AbstractWe study the initial boundary value problem resulting from the linearization of the equations of ideal incompressible magnetohydrodynamics and the jump conditions on the hypersurface of tangential discontinuity (current–vortex sheet) about an unsteady piecewise smooth solution. Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the so‐called loss of derivatives in the normal direction to the boundary takes place even for the constant coefficients linearized problem, for the variable coefficients problem and non‐planar current–vortex sheets the natural functional setting is provided by the anisotropic weighted Sobolev space W21,σ. The result of this paper is a necessary step to prove the local in time existence of solutions of the original non‐linear free boundary value problem. The uniqueness of the regular solution of this problem follows already from the a priori estimate we obtain for the linearized problem. Copyright © 2005 John Wiley & Sons, Ltd.