Validity of Prandtl layer theory for steady magnetohydrodynamics over a moving plate with nonshear outer ideal MHD flows

TL;DR

This paper rigorously validates the boundary layer theory for steady 2D viscous incompressible MHD flows over a moving plate, extending previous shear flow results to nonshear outer flows with convergence rates in Sobolev spaces.

Contribution

It extends the validation of Prandtl boundary layer theory from shear to nonshear outer ideal MHD flows, providing convergence rates in Sobolev spaces.

Findings

Boundary layer expansion is valid for nonshear flows.

Convergence rates are established in Sobolev sense.

Results generalize previous shear flow validations.

Abstract

In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ under the assumption of a moving boundary at $\{Y=0\}$. The validity of the boundary layer expansion and the convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the nonshear flows.