Validity of Prandtl expansions for steady MHD in the Sobolev framework

Abstract

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