Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations

In this paper, we consider the two dimensional incompressible anisotropic magneto-micropolar fluid equations with partial mixed velocity dissipations, magnetic diffusion and horizontal vortex viscosity, and analyze the stability near a background magnetic field. At present, major works on the equations of magneto-micropolar fluid mainly focus on the global regularity of the solutions. While the stability of the solutions remains an open problem. This paper concentrates on establishing the stability for the linear and nonlinear system respectively. Two goals have been achieved. The first is to obtain the explicit decay rates for the solution of the linear system in $H^s({\mathbb{R}}^2)$ Sobolev space. The second assesses the nonlinear stability by establishing the a priori estimate and employing bootstrapping arguments. Our results reveal that any perturbations near a background magnetic field is globally stable in Sobolev space $H^2({\mathbb{R}}^2)$.