Title: Boundary Layer Problems in the Viscosity-Diffusion Vanishing Limits for the Incompressible MHD Systems,
Abstract: In this paper, we we study boundary layer problems for the incompressible MHD
systems in the presence of physical boundaries with the standard Dirichlet
oundary conditions with small generic viscosity and diffusion coefficients. 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
that the solutions to the viscous and diffusive incompressible MHD systems
converges strongly to the superposition of the solution to the ideal MHD
systems with a Prandtl's type boundary layer corrector. One of the main
difficulties is to deal with the effect of the difference between viscosity and
diffusion coefficients and to control the singular boundary layers resulting
from the Dirichlet boundary conditions for both the viscosity and the magnetic
fields. One key derivation here is that for the class of initial data we
identify here, there exist cancelations between the boundary layers of the
velocity field and that of the magnetic fields so that one can use an elaborate
energy method to take advantage this special structure. In addition, in the
case of fixed positive viscosity, we also establish the stability of diffusive
boundary layer for the magnetic field and convergence of solutions in the limit
of zero magnetic diffusion for general initial data.