Global well-posedness and decay of the 2D incompressible MHD equations with horizontal magnetic diffusion

This paper concerns two-dimensional incompressible magnetohydrodynamic (MHD) equations with damping only in the vertical component of velocity equations and horizontal diffusion in magnetic equations. If the magnetic field is not taken into consideration the system is reduced to Euler-like equations with an extra Riesz transform-type term. The global well-posedness of Euler-like equations remains an open problem in the whole plane R-2. When coupled with the magnetic field, the global well-posedness and the stability for the MHD system in R-2 have yet to be settled too. This paper here focuses on the space domain TxR, with T being a 1D periodic box. We establish the global well-posedness of the 2D anisotropic MHD system. In addition, the algebraic decay rate in the H-2-setting has also been obtained. We solve this by decomposing the physical quantity into the horizontal average and its corresponding oscillation portion, establishing strong Poincare-type inequalities and some anisotropic inequalities and combining the symmetry conditions imposed on the initial data.