Regularity criteria for the 3D magnetic Bénard equations without thermal diffusion in terms of pressure

Abstract

In this paper, the authors obtain some new blow‐up criteria for the smooth solutions to the three‐dimensional magnetic Bénard equations without thermal diffusion in terms of pressure. We prove that if with 0 < r ≤ 1, then the strong solutions (u, b, θ) to the magnetic Bénard equations can be extended beyond time . Meanwhile, we also show that provided that with 0 < r ≤ 1, the solutions (u, b, θ) can also be extended smoothly beyond . Finally, we also obtain the regularity criteria on Morrey space, multiplier space, BMO space, and Besov space by imposing some growth conditions only on the pressure field.