On instability and stability of the incompressible micropolar fluids in a bounded domain

In this paper, we investigate the instability and stability of three-dimensional incompressible micropolar fluids in a bounded domain. Under the instability condition, we establish the linear and nonlinear instability around the steady-state solution. For linearized problem, we construct an exponential growth solution in H2 by a modified variational method. Based on the linear instability and the Gronwall-type global estimates of perturbation problem, taking the micro-rotation viscosity μr to be suitably small such that N<32 in (2.7), we show the equilibrium state is nonlinear unstable in the sense of Hadamard. On the other hand, when the steady density is lighter with increasing height, we show the steady state is linearly globally stable. In particular, if ρ̄′ is a constant, the steady state is nonlinearly asymptotically stable.