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 and horizontal diffusion magnetic equations. If field is not taken into consideration system reduced to Euler-like an extra Riesz transform-type term. The global well-posedness remains open problem whole plane R2. When coupled field, stability for MHD R2 have yet be settled too. here focuses on space domain T×R, T being a 1D periodic box. We establish 2D anisotropic system. In addition, algebraic decay rate H2-setting has also been obtained. solve this by decomposing physical quantity average its corresponding oscillation portion, establishing strong Poincaré-type inequalities some combining symmetry conditions imposed initial data.