Global Well-posedness of two-dimensional Magnetohydrodynamic Flows with Partial Dissipation and Magnetic Diffusion

In this paper, we establish the global well-posedness and conditional global well-posedness of incompressible magnetohydrodynamic (MHD) flows with mixed partial dissipation and magnetic diffusion in two dimensions. This work is inspired by the recent pioneer work [C. Cao and J. Wu, Adv. Math., 226 (2011), pp. 1803--1822] on whether classical solutions of the two-dimensional incompressible MHD equations without full dissipation and magnetic diffusion terms can develop finite-time singularity.