Asymptotic behavior of solution of the non-resistive 2D MHD equations on the half space

In this paper, we obtain the global well-posedness and the asymptotic behavior of solution of non-resistive 2D MHD problem on the half space. We overcome the difficulty of zero spectrum gap by building the relationship between half space and the whole space, and get the resolvent estimate for the weak diffusion system. We use the two-tier energy method that couples the boundedness of high-order $(H^3)$ energy to the decay of low-order energy, the latter of which is necessary to control the growth of the highest energy.