Existence of Compressible Current-Vortex Sheets: Variable Coefficients Linear Analysis

Abstract.We study the initial-boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the Rankine-Hugoniot relations about an unsteady piecewise smooth solution. This solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity (current-vortex sheet). Under some assumptions on the unperturbed flow, we prove an energy a priori estimate for the linearized problem. Since the tangential discontinuity is characteristic, the functional setting is provided by the anisotropic weighted Sobolev space W21,σ. Despite the fact that the constant coefficients linearized problem does not meet the uniform Kreiss-Lopatinskii condition, the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current-vortex sheets. The result of this paper is a necessary step in proving the local-in-time existence of current-vortex sheet solutions of the nonlinear equations of magnetohydrodynamics.

[1]  J. Fejer.  Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids. , 1964 .

[2]  T. Yanagisawa,et al.  The trace theorem on anisotropic Sobolev spaces , 1994 .

[3]  Jean-François Coulombel,et al.  Weak Stability of Nonuniformly Stable Multidimensional Shocks , 2002, SIAM J. Math. Anal..

[4]  Jeffrey Rauch,et al.  Symmetric positive systems with boundary characteristic of constant multiplicity , 1985 .

[5]  A. Sen.  Effect of Compressibility on Kelvin‐Helmholtz Instability in a Plasma , 1964 .

[6]  S. Alinhac,et al.  Existence d'ondes de rarefaction pour des systems quasi‐lineaires hyperboliques multidimensionnels , 1989 .

[7]  V. Vladimirov,et al.  The stability of steady magnetohydrodynamic flows with current-vortex sheets , 2003 .

[8]  Paolo Secchi,et al.  Well-posedness of characteristic symmetric hyperbolic systems , 1996 .

[9]  A. Majda.  The existence of multi-dimensional shock fronts , 1983 .

[10]  Science and Engineering. (Book Reviews: Non-Linear Wave Propagation. With applications to physics and magnetohydrodynamics) , 1964 .

[11]  F. Massey,et al.  Differentiability of solutions to hyperbolic initial-boundary value problems , 1974 .

[12]  A. Majda,et al.  The existence of multidimensional shock fronts , 1983 .

[13]  Tosio Kato,et al.  The Cauchy problem for quasi-linear symmetric hyperbolic systems , 1975 .

[14]  D. Serre,et al.  Generic types and transitions in hyperbolic initial–boundary-value problems , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[15]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[16]  E. Parker.  Dynamical properties of stellar coronas and stellar winds. iii - the dynamics of coronal streamers. , 1964 .

[17]  W. Axford.  The interaction of the solar wind with the interstellar medium , 1972 .

[18]  On an initial boundary value problem for the equations of ideal magneto-hydrodynamics , 1995 .

[19]  Paolo Secchi,et al.  The initial-boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity , 1996, Differential and Integral Equations.

[20]  W. Axford.  NOTE ON A PROBLEM OF MAGNETOHYDRODYNAMIC STABILITY , 1962 .

[21]  R. Phillips,et al.  Local boundary conditions for dissipative symmetric linear differential operators , 1960 .

[22]  A. Blokhin,et al.  Stability of Strong Discontinuities in Fluids and MHD , 2002 .

[23]  A. Blokhin.  Uniqueness of the classical solution of a mixed problem for equations of gas dynamics with boundary conditions on the shock wave , 1982 .

[24]  John W. Miles,et al.  On the stability of a plane vortex sheet with respect to three-dimensional disturbances , 1963, Journal of Fluid Mechanics.

[25]  An estimate of the energy integral of a mixed problem for the equations of gasdynamics with boundary conditions on a shock wave , 1981 .

[26]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[27]  H. Fahr,et al.  Plasma transport across the heliopause , 1986 .

[28]  H. Kreiss.  Initial boundary value problems for hyperbolic systems , 1970 .

[29]  A. Jefrey,et al.  Non-Linear Wave Propagation , 1964 .

[30]  Evgenii Mikhailovich Lifshitz,et al.  ELECTROSTATICS OF CONDUCTORS , 1984 .

[31]  A. Majda,et al.  The stability of multidimensional shock fronts , 1983 .

[32]  A. Matsumura,et al.  The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition , 1991 .

[33]  M. Ruderman,et al.  On the model of the solar wind-interstellar medium interaction with two shock waves , 1976 .

[34]  Y. Trakhinin.  A Complete 2D Stability Analysis of Fast MHD Shocks in an Ideal Gas , 2003 .

[35]  K. Friedrichs.  Symmetric hyperbolic linear differential equations , 1954 .

[36]  T. Shirota,et al.  On the Initial‐Boundary‐Value Problem for the Linearized Equations of Magnetohydrodynamics , 1998 .

[37]  D. Michael.  The stability of a combined current and vortex sheet in a perfectly conducting fluid , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[38]  Paolo Secchi,et al.  Linear symmetric hyperbolic systems with characteristic boundary , 1995 .

[39]  Jean-François Coulombel,et al.  On the transition to instability for compressible vortex sheets , 2004, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[40]  A. Majda.  The stability of multi-dimensional shock fronts , 1983 .