Boundary layer problem and zero viscosity-diffusion limit of the incompressible magnetohydrodynamic system with no-slip boundary conditions

The purpose of this paper is to study the boundary layer problem and zero viscosity-diffusion limit of the initial boundary value problem for the incompressible viscous and diffusive magnetohydrodynamic (MHD) system with (no-slip characteristic) Dirichlet boundary conditions and to prove that the corresponding Prandtl's type boundary layer are stable with respect to small viscosity-diffusion coefficients. The main difficulty here comes from Dirichlet boundary conditions for the velocity and magnetic field. Firstly, we identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously the solution of incompressible viscous-diffusion MHD system converges strongly to the sum of the solution to the ideal MHD system and the approximating solution to Prandtl's type boundary layer equation by using the elaborate energy methods and the special structure of the solution to ideal MHD system with the initial data we identify here, which yields that there exists the cancellation between the boundary layer of the velocity and that of the magnetic field. Secondly, for general initial data, we obtain zero viscosity-diffusion limit of the incompressible viscous and diffusive MHD system with the different horizontal and vertical viscosities and magnetic diffusions, when they go to zero with the different speeds, and, we prove rigorously the convergence of the incompressible viscous and diffusion MHD system to the ideal MHD system or the anisotropic MHD system by constructing the exact boundary layers and using the elaborate energy methods. We also mention that these results obtained here should be the first rigorous ones on the stability of Prandtl's type boundary layer for the incompressible viscous and diffusion MHD system with no-slip characteristic boundary condition.