The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term

This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ \beta > 3 $, with initial data $ (u_{\tau}, \omega_{\tau})\in V_{1}\times V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})\times \mathcal{H}(f_{2}^{0}) $, some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the $ ((V_{1}\times V_{2})\times(\mathcal{H}(f^{0}_{1})\times \mathcal{H}(f^{0}_{2})), V_{1}\times V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is demonstrated. Meanwhile, the $ (V_{1}\times V_{2}, \mathbf{H}^2(\Omega)\times\mathbf{H}^2(\Omega)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, \tau)\}_{t\geq\tau} $ is proved. Finally, the existence of a $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and a $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}\times V_{2}, V_{1}\times V_{2}) $-uniform attractor and the $ (V_{1}\times V_{2}, \mathbf{H}^{2}(\Omega)\times \mathbf{H}^{2}(\Omega)) $-uniform attractor are verified to be the same.