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

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 flow with the no-slip boundary condition of velocity and perfectly conducting walls for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space–time uniform norm L∞(H1). In addition, 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.