Global-in-$x$ stability of Prandtl layer expansions for steady magnetohydrodynamics flows over a moving plate

TL;DR

This paper proves the global-in-$x$ Sobolev stability of Prandtl layer expansions for 2-D steady incompressible MHD flows over a moving plate, allowing for degeneracy in the tangential magnetic field, which differs from previous unsteady and steady cases.

Contribution

It establishes the first Sobolev stability result for steady MHD flows with degeneracy in the tangential magnetic field, extending the understanding of Prandtl layer behavior.

Findings

Proves global-in-$x$ Sobolev stability of Prandtl layers in steady MHD flows.

Allows degeneracy of the tangential magnetic field in the stability analysis.

Differs from prior unsteady and steady case results by handling magnetic field degeneracy.

Abstract

In this paper, we obtain the global-in-$x$ Sobolev stability of Prandtl layer expansions for 2-D steady incompressible MHD flows with shear outer ideal MHD flows $(1,0,\sigma,0)$ ($\sigma\geq 0$) on a moving plate. It is worth noticing that in the Sobolev sense, the degeneracy of the tangential magnetic field in our result is allowed, which is much different from both the unsteady case in [25,26,27] and the steady case in [3,4].