Abstract
We analyze the linear stability of rectilinear compressible current-vortex sheets in two-dimensional isentropic magnetohydrodynamics, which is a free boundary problem with the boundary being characteristic. In the case when the magnitude of the magnetic field has no jump on the current-vortex sheets, we find a necessary and sufficient condition of linear stability for the rectilinear current-vortex sheets, showing that magnetic fields exert a stabilization effect on compressible vortex sheets. In addition, a loss of regularity with respect to the source terms, both in the interior domain and on the boundary, occurs in a priori estimates of solutions to the linearized problem for a rectilinear current-vortex sheet, as the Kreiss–Lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces. In this study, the construction of symmetrizers for a reduced differential system, which has poles at which the Kreiss–Lopatinskii condition may fail simultaneously, plays a crucial role in the a priori estimates.
Similar content being viewed by others
References
Artola M., Majda A.: Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes. Phys. D 28, 253–281 (1987)
Chen G.Q., Wang Y.G.: Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics. Arch. Ration. Mech. Anal. 187, 369–408 (2008)
Chen, G.Q., Wang, Y.G.: Characteristic discontinuities and free boundary problems for hyperbolic conservation laws. Nonlinear Partial Differential Equations (Eds. Holden H. and Karlsen K.) The Abel Symposium 2010, pp. 53–81, Springer, Berlin, 2012
Coulombel J.F., Secchi P.: The stability of compressible vortex sheets in two space dimensions. Indiana Univ. Math. J. 53, 941–1012 (2004)
Coulombel J.F., Secchi P.: On the transition to instability for compressible vortex sheets. Proc. R. Soc. Edinb. A 134, 885–892 (2004)
Coulombel J.F., Secchi P.: Nonlinear compressible vortex sheets in two space dimensions. Ann. Sci. Ec. Norm. Super. 41, 85–139 (2008)
Kreiss H.O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970)
Lax P.D., Phillips R.S.: Local boundary conditions for dissipative symmetric linear differential operators. Comm. Pure Appl. Math. 13, 427–455 (1960)
Majda A.: The stability of multidimensional shock fronts. Mem. Am. Math. Soc. 275, 1–95 (1983)
Majda A.: The existence of multidimensional shock fronts. Mem. Am. Math. Soc. 281, 1–93 (1983)
Majda A., Osher S.: Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary. Commun. Pure Appl. Math. 28, 607–675 (1975)
Miles J.W.: On the reflection of sound at an interface of relative motion. J. Acoust. Soc. Am. 29, 226–228 (1957)
Miles J.W.: On the disturbed motion of a plane vortex sheet. J. Fluid Mech. 4, 538–552 (1958)
Serre, D.: Systems of Conservation Laws, vol. 2. Cambridge University Press, London, 1999
Trakhinin Y.: On existence of compressible current-vortex sheets: variable coefficients linear analysis. Arch. Ration. Mech. Anal. 177, 331–366 (2005)
Trakhinin Y.: The existence of current-vortex sheets in ideal compressible magnetohydrodynamics. Arch. Ration. Mech. Anal. 191, 245–310 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. E
Rights and permissions
About this article
Cite this article
Wang, YG., Yu, F. Stabilization Effect of Magnetic Fields on Two-Dimensional Compressible Current-Vortex Sheets. Arch Rational Mech Anal 208, 341–389 (2013). https://doi.org/10.1007/s00205-012-0601-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-012-0601-9