  In this paper, we are concerned with the magnetic effect on the Sobolev
solvability of boundary layer equations for the 2D incompressible MHD system
without resistivity. The MHD boundary layer is described by the Prandtl type
equations derived from the incompressible viscous MHD system without
resistivity under the no-slip boundary condition on the velocity. Assuming that
the initial tangential magnetic field does not degenerate, a local-in-time
well-posedness in Sobolev spaces is proved without the monotonicity condition
on the velocity field. Moreover, we show that if the tangential magnetic field
shear layer is degenerate at one point, then the linearized MHD boundary layer
system around the shear layer profile is ill-posed in the Sobolev settings
provided that the initial velocity shear flow is non-degenerately critical at
the same point.
