On 2-D Boussinesq equations for MHD convection with stratification effects

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This paper is concerned with the two-dimensional magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. The first progress on this topic was made independently by Chae and Hou–Li [8,26] where the Boussinesq system with partial constant viscosity is obtained. Recently, Wang–Zhang [45] considered the temperature-dependent viscosity and thermal diffusivity, and Li–Xu [16] generalized the Wang–Zhang's result to the inviscid case with temperature-dependent thermal diffusivity. In this paper, we include the stratification and magnetic effects and consider the full system, in the framework of low regularity. We prove that, without any smallness assumption on the initial data, the full system is globally well-posed. Moreover, by applying the uniformly bounded generalized Oseen operator, time decay estimate of the solution is obtained.