Global regularity problem of two‐dimensional magnetic Bénard fluid equations

Abstract

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)αu, partial magnetic diffusion , and Laplacian thermal diffusivity Δθ; (ii) partial fractional dissipation , partial magnetic diffusion , and Laplacian thermal diffusivity Δθ; (iii) partial fractional magnetic diffusion , Laplacian thermal diffusivity Δθ, and without Laplacian dissipation Δu (i.e., ), and establish the global regularity for each cases.