On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection

Abstract

This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state ( u ¯ = 0 , B ¯ = 0 , θ ¯ = θ 0 ( y ) ) with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if a + ≥ a − , and d 2 d y 2 κ ( θ 0 ( y ) ) ≤ 0 or 0 d 2 d y 2 κ ( θ 0 ( y ) ) ≤ β 0 , with β 0 > 0 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 θ 0 ( y ) on the upper and lower boundary.

Keywords

• Boussinesq-MHD system
• asymptotic stability
• nonlinear instability