Stability and exponential decay for the 2D magneto-micropolar equations with partial dissipation

The stability and large-time behavior problem on the magneto-micropolar equations has evoked a considerable interest in recent years. In this paper, we study the stability and exponential decay near magnetic hydrostatic equilibrium to the two-dimensional magneto-micropolar equations with partial dissipation in the domain Ω=T×R. In particular, we takes advantage of the geometry of the domain T×R to divide u into zeroth mode and the nonzero modes, and obey a strong version of the Poincaré's inequality, which plays a crucial role in controlling the nonlinearity. Moreover, we find that the oscillation part of the solution decays exponentially to zero. Finally, our result mathematically verifies that the stabilization effect of a background magnetic field on magneto-micropolar fluids.