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We study the well-posedness theory for the MHD boundary layer. The boundarylayer equations are governed by the Prandtl type equations that are derivedfrom the incompressible MHD system with non-slip boundary condition on thevelocity and perfectly conducting condition on the magnetic field. Under theassumption that the initial tangential magnetic field is not zero, we establishthe local-in-time existence, uniqueness of solution for the nonlinear MHDboundary layer equations. Compared with the well-posedness theory of theclassical Prandtl equations for which the monotonicity condition of thetangential velocity plays a crucial role, this monotonicity condition is notneeded for MHD boundary layer. This justifies the physical understanding thatthe magnetic field has a stabilizing effect on MHD boundary layer in rigorousmathematics.

MHD boundary layers theory in Sobolev spaces without monotonicity. I. well-posedness theory