On the nonlinear stability and instability of the boussinesq system for magnetohydrodynamics convection

This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if a+ < a, this equilibrium state is nonlinearly unstable. Here, a+ and a-are the values of the equilibrium temperature θ0(y) on the upper and lower boundary.