Regularity criteria for the two-and-half-dimensional magnetic Bénard system with partial dissipation, magnetic diffusion, and thermal diffusivity

Abstract This paper concentrates on the global regularity of classical solution to the 212 $2\frac{1}{2}$D magnetic Bénard system with partial dissipation, magnetic diffusion, and thermal diffusivity (i.e., horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity; vertical dissipation, vertical magnetic diffusion, and vertical thermal diffusivity). For the 212 $2\frac{1}{2}$D magnetic Bénard system with full dissipation, magnetic diffusion, and thermal diffusivity, the global existence and uniqueness can be obtained by the standard energy method. However, can the classical solution for the 212 $2\frac{1}{2}$D incompressible magnetic Bénard system still keep its global regularity when losing some partial dissipation, magnetic diffusion, and thermal diffusivity terms? We will give a rigorous proof to the global regularity for the 212 $2\frac{1}{2}$D magnetic Bénard system with horizontal and vertical dissipation, magnetic diffusion, and thermal diffusivity respectively in this paper. Furthermore, we also show that any possible finite time blow-up can be controlled by the L∞ $L^{\infty }$-norm of the vertical velocity and magnetic components, not include the temperature component (see Theorems 1.1 and 1.2). The results extend the recent work by Cheng and Du (J. Math. Fluid Mech. 17:769–797, 2015), and generalize the recent works by Regmi (Math. Methods Appl. Sci. 40:1497–1504, 2017), and Ma and Zhang (Bound. Value Probl. 2018:79, 2018).