Existence and Stability of Compressible and Incompressible Current-Vortex Sheets

  • Yuri Trakhinin
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


Recent author’s results in the investigation of current-vortex sheets (MHD tangential discontinuities) are surveyed. A sufficient condition for the neutral stability of planar compressible current-vortex sheets is first found for a general case of the unperturbed flow. In astrophysical applications, this condition can be considered as the sufficient condition for the stability of the heliopause, which is modelled by an ideal compressible current-vortex sheet and caused by the interaction of the supersonic solar wind plasma with the local interstellar medium (in some sense, the heliopause is the boundary of the solar system). The linear variable coefficients problem for nonplanar compressible current-vortex sheets is studied as well. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces. The a priori estimate deduced for this problem is a necessary step to prove the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of ideal compressible MHD. Analogous results are obtained for incompressible current-vortex sheets. In the incompressibility limit the sufficient stability condition found for compressible current-vortex sheets describes exactly the half of the whole parameter domain of linear stability of planar discontinuities in ideal incompressible MHD.


Current Sheet Linear Stability Tangential Discontinuity Incompressibility Limit Astrophysical Application 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Alinhac S. Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Differential Equations 14, 173–230.Google Scholar
  2. [2]
    Axford W.I. Note on a problem of magnetohydrodynamic stability. Can. J. Phys. 40 (1962), 654–655.zbMATHGoogle Scholar
  3. [3]
    Baranov V.B., Krasnobaev K.V., Kulikovsky A.G. A model of interaction of the solar wind with the interstellar medium. Sov. Phys. Dokl. 15 (1970), 791–793.Google Scholar
  4. [4]
    Blokhin A., Trakhinin Yu. Stability of strong discontinuities in fluids and MHD. In: Handbook of mathematical fluid dynamics, vol. 1, S. Friedlander and D. Serre, eds., North-Holland, Amsterdam, 2002, pp. 545–652.CrossRefGoogle Scholar
  5. [5]
    Casella E., Secchi P., Trebeschi P. Non-homogeneous linear symmetric hyperbolic systems with characteristic boundary. Preprint, 2005.Google Scholar
  6. [6]
    Coulombel J.-F. Weakly stable multidimensional shocks. Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 401–443.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Coulombel J.-F. Well-posedness of hyperbolic initial boundary value problems. J. Math. Pures Appl. (9) 84 (2005), 786–818.zbMATHMathSciNetGoogle Scholar
  8. [8]
    Coulombel J.-F., Secchi P. The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J. 53 (2004), 941–1012.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Florinski V., Pogorelov N.V., Zank G.P., Wood B.E., Cox D.P. On the possibility of a strong magnetic field in the local interstellar medium. Astrophys. J, 604 (2004), 700–706.CrossRefGoogle Scholar
  10. [10]
    Francheteau J., Métivier G. Existence de chocs faibles pour des systèmes quasilinéaires hyperboliques multidimensionnels. Astérisque, no. 268, Soc. Math. France, Paris, 2000.Google Scholar
  11. [11]
    Jeffrey A., Taniuti T. Non-linear wave propagation. With applications to physics and magnetohydrodynamics. Academic Press, New York, London, 1964.zbMATHGoogle Scholar
  12. [12]
    Kreiss H.-O. Initial boundary value problems for hyperbolic systems. Commun. Pure and Appl. Math. 23 (1970), 277–296.CrossRefMathSciNetGoogle Scholar
  13. [13]
    Kulikovsky A.G., Lyubimov G.A. Magnetohydrodynamics. Addison-Wesley, Massachusets, 1965.Google Scholar
  14. [14]
    Ladyzhenskaya O.A. The boundary value problems of mathematical physics. Springer-Verlag, New York, 1985.zbMATHGoogle Scholar
  15. [15]
    Li D. Rarefaction and shock waves for multi-dimensional hyperbolic conservation laws. Commun. Partial Differ. Equations 16 (1991), 425–450.CrossRefzbMATHGoogle Scholar
  16. [16]
    Majda A. The stability of multi-dimensional shock fronts. Mem. Amer.Math. Soc. 41(275), 1983.Google Scholar
  17. [17]
    Michael D.H. The stability of a combined current and vortex sheet in a perfectly conducting fluid. Proc. Cambridge Philos. Soc. 51 (1955), 528–532.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Parker E.N. Dynamical properties of stellar coronas and stellar winds. III. The dynamics of coronal streamers. Astrophys. J. 139 (1964), 690–709.CrossRefMathSciNetGoogle Scholar
  19. [19]
    Ruderman M.S., Fahr H.J. The effect of magnetic fields on the macroscopic instability of the heliopause. II. Inclusion of solar wind magnetic fields. Astron. Astrophys. 299 (1995), 258–266.Google Scholar
  20. [20]
    Secchi P. Linear symmetric hyperbolic systems with characteristic boundary. Math. Methods Appl. Sci. 18 (1995), 855–870.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Secchi P. Some properties of anisotropic Sobolev spaces. Arch. Math. 75 (2000), 207–216.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Syrovatskij S.I. The stability of tangential discontinuities in a magnetohydrodynamic medium. Z. Eksperim. Teoret. Fiz. 24 (1953), 622–629 (in Russian).Google Scholar
  23. [23]
    Trakhinin Yu. On existence of compressible current-vortex sheets: variable coefficients linear analysis. Arch. Rational Mech. Anal., to appear.Google Scholar
  24. [24]
    Trakhinin Yu. On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem. Math. Methods Appl. Sci. 28 (2005), 917–945.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Trakhinin Yu. Dissipative symmetrizers of hyperbolic problems and their applications to shock waves and characteristic discontinuities, submitted.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Yuri Trakhinin
    • 1
    • 2
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of MathematicsUniversity of HullHullUK

Personalised recommendations