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A priori Estimates for 3D Incompressible Current-Vortex Sheets

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Abstract

We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.

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Authors and Affiliations

  1. CNRS, Université Lille 1 and Team Project SIMPAF of INRIA Lille Nord Europe, Laboratoire Paul Painlevé (UMR CNRS 8524), Bâtiment M2, Cité Scientifique, 59655, Villeneuve D’Ascq Cedex, France

    J. -F. Coulombel

  2. Dipartimento di Matematica, Facoltà di Ingegneria, Università di Brescia, Via Valotti, 9, 25133, Brescia, Italy

    A. Morando, P. Secchi & P. Trebeschi

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  1. J. -F. Coulombel
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  2. A. Morando
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  3. P. Secchi
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  4. P. Trebeschi
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Correspondence to P. Secchi.

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Communicated by P. Constantin

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Coulombel, J.F., Morando, A., Secchi, P. et al. A priori Estimates for 3D Incompressible Current-Vortex Sheets. Commun. Math. Phys. 311, 247–275 (2012). https://doi.org/10.1007/s00220-011-1340-8

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  • DOI: https://doi.org/10.1007/s00220-011-1340-8

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