Validity of Prandtl expansions for steady MHD in the Sobolev framework

TL;DR

This paper proves the nonlinear stability of Prandtl-type shear flows in steady 2D MHD with magnetic effects stabilizing the flow, using Sobolev space analysis and boundary degeneracy management.

Contribution

It establishes the validity of Prandtl expansions for steady MHD in Sobolev spaces without requiring velocity monotonicity, highlighting magnetic stabilization effects.

Findings

Nonlinear stability of shear flows with magnetic fields.

Effective boundary degeneracy management techniques.

Validation of Prandtl expansions in steady MHD.

Abstract

This paper is concerned with the vanishing viscosity and magnetic resistivity limit for the two-dimensional steady incompressible MHD system on the half plane with no-slip boundary condition on velocity field and perfectly conducting wall condition on magnetic field. We prove the nonlinear stability of shear flows of Prandtl type with nondegenerate tangential magnetic field, but without any positivity or monotonicity assumption on the velocity field. It is in sharp contrast to the steady Navier-Stokes equations and reflects the stabilization effect of magnetic field. Unlike the unsteady MHD system, we manage the degeneracy on the boundary caused by no-slip boundary condition and obtain the estimates of solutions by introducing an intrinsic weight function and some good auxiliary functions.