(untitled) (Journal of Mathematical Study, 2019)
In this paper, we study the global regularity issue of two dimensional incompressible magnetic Bénard equations with partial dissipation and magnetic diffusion. It remains open whether the smooth initial data produce solutions that are globally regular in time for all values of the parameters involved in the equations. We present conditional global regularity of the solutions. Moreover, we prove the global regularity for the slightly regularized system.