  This paper focuses on the 2D compressible magnetohydrodynamic (MHD) equations
without magnetic diffusion in a periodic domain. We present a systematic
approach to establishing the global existence of smooth solutions when the
initial data is close to a background magnetic field. In addition, stability
and large-time decay rates are also obtained. When there is no magnetic
diffusion, the magnetic field and the density are governed by forced transport
equations and the problem considered here is difficult. This paper implements
several key observations and ideas to maximize the enhanced dissipation due to
hidden structures and interactions. In particular, the weak smoothing and
stabilization generated by the background magnetic field and the extra
regularization in the divergence part of the velocity field are fully
exploited. Compared with the previous works, this paper appears to be the first
to investigate such system on bounded domains and the first to solve this
problem by pure energy estimates, which help reduce the complexity in other
approaches. In addition, this paper combines the well-posedness with the
precise large-time behavior, a strategy that can be extended to higher
dimensions.
