Affiliations: [a] Faculty of Information Technology, Department of Mathematics, Thuyloi University, Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Vietnam | [b] Faculty of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Ha Noi, Vietnam
Abstract: The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere S2, the square torus T2 and the three-torus T3. In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both S2 and T2. Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on T2 to study the lower bound for attractor’s dimensions in the case of T3.
