Steady super-Alfv\'enic MHD shocks with aligned fields in two-dimensional almost flat nozzles

Abstract

The Lorentz force induced by the magnetic field in MHD flow introduces a fundamental difference from pure gas dynamics by facilitating the anisotropic propagation of small disturbances, thus the type of steady MHD equations depends on not only the Mach number but also the Alfv\'en number. In the super-Alfv\'enic case, we derive an admissible condition for the locations of transonic shock fronts in terms of the nozzle wall profile and the exit total pressure (the kinetic plus magnetic pressure). Starting from this initial approximation, a nonlinear existence of super-Alfv\'enic transonic shock solution to steady MHD equations is established. Our admissible condition is slightly different from that first introduced by Fang-Xin in [Comm. Pure Appl. Math., 74 (2021), pp. 1493-1544], and because our formulation is based on the deformation-curl decomposition of the steady MHD equations, our admissible condition has the advantage that a direct generalization to three dimensional case is available at least at the level of the initial approximation of the shock position.