Homogenization of trajectory statistical solutions for the 3D incompressible magneto-micropolar fluids
Hujun Yang, Xiaoling Han, Xuan Wang, Caidi Zhao · Discrete and Continuous Dynamical Systems - S · 2022
This article investigates the 3D incompressible magneto-micropolar fluids containing rapidly oscillating external force and angular momentum with respect to spatial variables. Under suitable assumptions on the oscillating external force and angular momentum, the existence of the trajectory attractor $ {\mathcal{A}^{\rm tr}_\epsilon} $ and trajectory statistical solution $ \mu_{\epsilon, w_\epsilon} $ is established, as well as the convergence of $ {\mathcal{A}^{\rm tr}_\epsilon} $ to the trajectory attractor $ {\mathcal{A}^{\rm tr}_0} $ of the homogenized fluids. Then we prove the homogenization of trajectory statistical solution by establishing that $ \mu_{\epsilon, w_\epsilon} $ converges to the trajectory statistical solution $ \mu_{0, w} $ of the homogenized fluids as $ \epsilon\rightarrow 0^+ $. Our results reveal that the trajectory statistical information obtained from the 3D incompressible magneto-micropolar fluids with rapidly oscillating external force and angular momentum has certain homogenization effect with respect to spatial variables.