Boundary layer problems in the vanishing viscosity-diffusion\\ limits for the incompressible MHD system

Abstract

In this paper, we study boundary layer problem for the incompressible magneto-hydrodynamical (MHD) system in the presence of physical boundaries with the standard Dirichlet boundary 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 Prandtls type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD system converges strongly to the superposition of the solution to the ideal MHD system with a Prandtls 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 of 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.