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

Abstract

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.

Keywords

• uniform attractor
• 3d micropolar equation
• nonlinear damping term