Global well-posedness and long time behavior of 2D MHD equations with partial dissipation in half space

Abstract

In this paper, we obtain the low order global well-posedness and the asymptotic behavior of solution of 2D MHD problem with partial dissipation in half space with non-slip boundary condition. When magnetic field equal zero, the system be reduced to partial dissipation Navier-Stokes equation, so this result also implies the stabilizing effects of magnetic field in electrically conducting fluids. We use the resolvent estimate method to obtain the long time behavior for the solution of weak diffusion system, which is not necessary to prove global well-posedness.