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

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