Global regularity problem of two‐dimensional magnetic Bénard fluid equations

In the paper, we devote to broadening the current global regularity results for the two‐dimensional magnetic Bénard fluid equations. We study three cases: (i) fractional Laplacian dissipation (‐ Δ) α u , partial magnetic diffusion ( ∂x 2x 22b 1 , ∂x 1x 12b 2 ) , and Laplacian thermal diffusivity Δ θ ; (ii) partial fractional dissipation ( Λx 22 αu 1 , Λx 12 αu 2 ) , partial magnetic diffusion ( ∂x 2x 22b 1 , ∂x 1x 12b 2 ) , and Laplacian thermal diffusivity Δ θ ; (iii) partial fractional magnetic diffusion ( Λx 22 βb 1 , Λx 12 βb 2 ) , Laplacian thermal diffusivity Δ θ , and without Laplacian dissipation Δ u (i.e., μ = 0 ), and establish the global regularity for each cases.