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In this paper, we we study boundary layer problems for the incompressible MHDsystems in the presence of physical boundaries with the standard Dirichletoundary conditions with small generic viscosity and diffusion coefficients. Weidentify a non-trivial class of initial data for which we can establish theuniform stability of the Prandtl's type boundary layers and prove rigorouslythat the solutions to the viscous and diffusive incompressible MHD systemsconverges strongly to the superposition of the solution to the ideal MHDsystems with a Prandtl's type boundary layer corrector. One of the maindifficulties is to deal with the effect of the difference between viscosity anddiffusion coefficients and to control the singular boundary layers resultingfrom the Dirichlet boundary conditions for both the viscosity and the magneticfields. One key derivation here is that for the class of initial data weidentify here, there exist cancelations between the boundary layers of thevelocity field and that of the magnetic fields so that one can use an elaborateenergy method to take advantage this special structure. In addition, in thecase of fixed positive viscosity, we also establish the stability of diffusiveboundary layer for the magnetic field and convergence of solutions in the limitof zero magnetic diffusion for general initial data.

Boundary Layer Problems in the Viscosity-Diffusion Vanishing Limits for the Incompressible MHD Systems