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 $((V_1\times V_2)\times (\mathcal{H}(f_1^0)\times \mathcal{H}(f_2^0)),V_1\times V_2)$-continuity of the family of processes $\{U_{(f_1,f_2)}(t,\tau ){\}}_{t\ge \tau }$ is demonstrated. Meanwhile, the $(V_1\times V_2,{\mathbf{H}}^2(\mathrm{\Omega })\times {\mathbf{H}}^2(\mathrm{\Omega }))$-uniform compactness of $\{U_{(f_1,f_2)}(t,\tau ){\}}_{t\ge \tau }$ is proved. Finally, the existence of a $(V_1\times V_2,V_1\times V_2)$-uniform attractor and a $(V_1\times V_2,{\mathbf{H}}^2(\mathrm{\Omega })\times {\mathbf{H}}^2(\mathrm{\Omega }))$-uniform attractor are obtained. Furthermore, the $(V_1\times V_2,V_1\times V_2)$-uniform attractor and the $(V_1\times V_2,{\mathbf{H}}^2(\mathrm{\Omega })\times {\mathbf{H}}^2(\mathrm{\Omega }))$-uniform attractor are verified to be the same.