Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial-boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion. To overcome the difficulty caused by the vanishing viscosities, we first establish the elliptic system for Ux and by, which are estimated by ∇ × ux and ∇ × by, respectively. Then, we establish the global estimates for ∇ × u and ∇ × b.