MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory

We study the well-posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl type equations that are derived from the incompressible MHD system with non-slip boundary condition on the velocity and perfectly conducting condition on the magnetic field. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-in-time existence, uniqueness of solution for the nonlinear MHD boundary layer equations. Compared with the well-posedness theory of the classical Prandtl equations for which the monotonicity condition of the tangential velocity plays a crucial role, this monotonicity condition is not needed for MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics.