One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics

Article history: Received 5 June 2009 Revised 2 March 2010 Available online 12 August 2010 Extending investigations of Barker, Humpherys, Lafitte, Rudd, and Zumbrun for compressible gas dynamics and Freistühler and Trakhinin for compressible magnetohydrodynamics, we study by a combination of asymptotic ODE estimates and numerical Evans function computations the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a γ -law gas with γ ∈ [1,3]. Other γ -values may be treated similarly, but were not checked numerically. Depending on magnetic field strength, these shocks may be of fast Lax, intermediate (overcompressive), or slow Lax type; however, the shock layer is independent of magnetic field, consisting of a purely gas-dynamical profile. In each case, our results indicate stability. Interesting features of the analysis are the need to renormalize the Evans function in order to pass continuously across parameter values where the shock changes type or toward the large-amplitude limit at frequency λ = 0 and the systematic use of winding number computations on Riemann surfaces. © 2010 Elsevier Inc. All rights reserved. * Corresponding author. E-mail addresses: [email protected] (B. Barker), [email protected] (J. Humpherys), [email protected] (K. Zumbrun). 1 Research of B. Barker was partially supported under NSF grants number DMS-0607721 and DMS-0300487. 2 Research of J. Humpherys was partially supported under NSF grant DMS-0607721 and DMS-CAREER-0847074. 3 Research of K. Zumbrun was partially supported under NSF grants number DMS-0070765 and DMS-0300487. 0022-0396/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2010.07.019 2176 B. Barker et al. / J. Differential Equations 249 (2010) 2175–2213