Global solution of 3D MHD-Boussinesq system with mixed partial dissipation and thermal damping

In this paper, we consider the initial value problem for the three-dimensional MHD-Boussinesq system with mixed partial dissipation and thermal damping. This system characterizes the dynamics of velocity, magnetic field and temperature perturbations near hydrostatic equilibrium. By exploiting the special structure of this system, we use 'double energy' to establish two desired energy inequalities. Then, we employ the bootstrapping argument to obtain the global stability and the large time behavior of the classical solution.