  In this paper, we investigate the well-posedness theory for the MHD boundary
layer system in two-dimensional space. The boundary layer equations are
governed by the Prandtl type equations that are derived from the full
incompressible MHD system with non-slip boundary condition on the velocity,
perfectly conducting condition on the magnetic field, and Dirichlet boundary
condition on the temperature when the viscosity coefficient depends on the
temperature. To derive the Prandtl type boundary layer system, we require all
the hydrodynamic Reynolds numbers, magnetic Reynolds numbers and Nusselt
numbers tend to infinity at the same rate. Under the assumption that the
initial tangential magnetic field is not zero, one applies the energy methods
to establish the local-in-time existence and uniqueness of solution for the MHD
boundary layer equations without the necessity of monotonicity condition.
