Local well-posedness of solutions to 2D mixed Prandtl equations in Sobolev space without monotonicity and lower bound

In this paper, we investigate two-dimensional Prandtl–Shercliff regime equations on the half plane and prove the local existence and uniqueness of solutions for any initial datum by using the classical energy methods in Sobolev space. Compared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, this monotonicity condition is not needed for 2D mixed Prandtl equations. Besides, compared with the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, this lower bound condition is also not needed for 2D mixed Prandtl equations. In other words, we need neither the monotonicity condition of the tangential velocity nor the initial tangential magnetic field has a lower bound and for any initial datum in this paper. As far as we have learned, this is the first result of $2D$ mixed Prandtl–Shercliff regime equations in Sobolev space.