On the steady Prandtl type equations with magnetic effects arising from 2D incompressible MHD equations in a half plane

In this paper, we study a steady Prandtl type boundary layer system of equations, which is derived from the two-dimensional incompressible magnetohydrodynamics equations without magnetic diffusion under the no-slip boundary condition. By the classical von Mises transformation, this Prandtl type system is reduced to a single second order quasilinear parabolic equation, in which the coefficient depends on the strength of magnetic field. For different ratios of the magnetic field and the velocity field, we prove the global existence of solutions to the boundary layer system and the non-existence of global smooth solutions, respectively.