Verification of Prandtl boundary layer ansatz for the steady electrically conducting fluids with a moving physical boundary

In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ with a moving flat boundary $\{Y=0\}$. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the same order in term of the reciprocal of a small parameter $\epsilon$, the tangential component of the magnetic field does not degenerate near the boundary and the ratio of the strength of tangential component of magnetic field and tangential component of velocity is suitably small. And the error terms are estimated in $L^\infty$ sense.