Validity of Prandtl Expansions for Steady MHD in the Sobolev Framework

This paper concerns the vanishing viscosity and magnetic resistivity limit for two-dimensional steady incompressible MHD system on half-plane with no-slip boundary conditions velocity fields perfectly conducting wall fields. We prove nonlinear stability of shear flows Prandtl type nondegenerate tangential but without any positivity or monotonicity assumption It contrasts sharply Navier–Stokes equations reflects stabilization effect The main aims in analysis are to design an intrinsic weight function treat incompatibility natural multiplier establish critical Hardy-type inequalities properly weighted estimates that lead almost optimal convergence rate.