Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

Abstract

We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent \gamma >1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-\epsilon estimates to the regular solutions. At the second level, we show global existence of weak solutions to the system with artificial pressure by sending \epsilon to 0 and deriving uniform-in-\delta estimates. Then global weak solution to the original system is constructed by vanishing \delta. The key issue in the limit passage is the strong convergence of approximate sequence of the density and magnetic field. This is accomplished by following the technique developed in [15, 26] and using the new technique of variable reduction developed by Vasseur et al. [33] in order to handle the cross terms.