Stability of the boundary layer expansion for the 3D plane parallel MHD flow

TL;DR

This paper proves the mathematical validity of the Prandtl boundary layer theory for 3D viscous incompressible MHD flows, demonstrating convergence under various norms and highlighting magnetic fields' stabilizing effects.

Contribution

It establishes the convergence of boundary layer expansions for 3D MHD flows with no-slip and conducting wall conditions, including higher-order expansions and effects of magnetic fields.

Findings

Convergence of boundary layer expansion in Sobolev norms including $L^ abla(H^1)$.

Magnetic fields have a stabilizing effect on the flow.

Higher-order boundary layer expansions are validated.

Abstract

In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic (MHD) flow with no-slip boundary condition of velocity and perfectly conducting wall for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm $L^\infty(H^1)$. In addition, the similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the higher-order expansion is also considered.