Local existence of solutions to the 2D MHD boundary layer equations without monotonicity in Sobolev space

In this work, we investigated the local existence of the solutions to the 2D magnetohydrodynamic (MHD) boundary layer equations on the half plane by energy methods in weighted Sobolev space. Compared to the existence of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays an important role, we used the initial tangential magnetic field with a lower bound delta> 0 instead of the monotonicity condition of the tangential velocity.