Stability of the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation

Abstract

In this paper, we investigate the initial value problem for the 2D anisotropic micropolar Rayleigh–Bénard problem with one direction dissipation. But there are only vertical kinematic viscosity, horizontal angular viscosity, and diffusivity; some nonlinear terms cannot be controlled. In addition to the classical energy functional, we introduce the energy functional to help us dominate the nonlinear parts and make the energy closed. Then based on the energy method and the bootstrapping argument, global smooth solutions are proved under the assumptions of small initial data.