On the vanishing dissipation limit for the incompressible MHD equations on bounded domains

Abstract

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD as the dissipation tends to zero.