Stability and long time asymptotic for the 2D anisotropic micropolar Rayleigh–Bénard problem with horizontal dissipation

In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh–Bénard problem with horizontal dissipation. Based on the energy method and the bootstrapping argument, global classical solutions are proved under the assumptions of small initial data. Moreover, time-decay rates of the oscillation part of global classical solutions are obtained by combining the energy method and Poincaré type inequality.