Vanishing Viscosity Limit of Short Wave-Long Wave Interactions in Planar Magnetohydrodynamics

Abstract

We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schr\"{o}dinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an aurora-type phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters, to a solution of the limit decoupled system involving the compressible Euler equations and a nonlinear Schr\"{o}dinger equation. The vanishing viscosity limit serves to justify the SW-LW interactions in the limit equations as, in this setting, the SW-LW interactions cannot be defined in a straightforward way, due to the possible occurrence of vacuum.