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 false(∂x2x22b1,∂x1x12b2false), and Laplacian thermal diffusivity Δθ; (ii) partial fractional dissipation false(normalΛx22αu1,normalΛx12αu2false), partial magnetic diffusion false(∂x2x22b1,∂x1x12b2false), and Laplacian thermal diffusivity Δθ; (iii) partial fractional magnetic diffusion false(normalΛx22βb1,normalΛx12βb2false), Laplacian thermal diffusivity Δθ, and without Laplacian dissipation Δu (i.e., μ=0), and establish the global regularity for each cases.