The existence of uniform attractors for the 3D micropolar equations with nonlinear damping term - Xue-li Song; Yuan-yuan Liu; Xiao-tian Xie (AIMS Mathematics, 2024)
This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When $ β > 3 $, with initial data $ (u_{τ}, ω_{τ})\in V_{1}× V_{2} $ and external forces $ (f_{1}, f_{2})\in \mathcal{H}(f_{1}^{0})× \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}× V_{2})×(\mathcal{H}(f^{0}_{1})× \mathcal{H}(f^{0}_{2})), V_{1}× V_{2}) $-continuity of the family of processes $ \{U_{(f_{1}, f_{2})}(t, τ)\}_{t≥τ} $ is demonstrated. Meanwhile, the $ (V_{1}× V_{2}, H^2(Ω)×H^2(Ω)) $-uniform compactness of $ \{U_{(f_{1}, f_{2})}(t, τ)\}_{t≥τ} $ is proved. Finally, the existence of a $ (V_{1}× V_{2}, V_{1}× V_{2}) $-uniform attractor and a $ (V_{1}× V_{2}, H^{2}(Ω)× H^{2}(Ω)) $-uniform attractor are obtained. Furthermore, the $ (V_{1}× V_{2}, V_{1}× V_{2}) $-uniform attractor and the $ (V_{1}× V_{2}, H^{2}(Ω)× H^{2}(Ω)) $-uniform attractor are verified to be the same.