Stability and exponential decay for the compressible viscous non-resistive MHD system

Abstract How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R 3 or T 3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T 3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.