Global regularity of the 2D anisotropic magnetic Bénard system with vertical dissipation

This paper studies the Cauchy problem to the two-dimensional (2D) anisotropic magnetic Bénard system with partial dissipation. The system considered in this paper has only vertical dissipation. We establish here the global regularity of classical solutions to this system. In proving the global regularity of classical solutions, the main difficulties are the absence of the horizontal dissipation and the presence of the buoyancy force term θe2 as well as the Rayleigh-Bénard convection term u2. To overcome them, we establish two new Gronwall type inequalities, which may be of independent interest. In addition, the global regularity of the 2D magnetic Bénard system with vertical dissipation, vertical magnetic diffusion and horizontal thermal diffusivity also holds true. This settles the global regularity issue unsolved in the previous works.