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

Abstract

AbstractWe 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‐i‐time existence, uniqueness of solutions 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 the MHD boundary layer. This justifies the physical understanding that the magnetic field has a stabilizing effect on MHD boundary layer in rigorous mathematics. © 2018 Wiley Periodicals, Inc.