The simplified Bardina equation on two-dimensional closed manifolds

This paper we study the viscous simplified Bardina equations on two-dimensional closed manifolds $M$ imbedded in $\mathbb{R}^3$. First we will show that the existence and uniqueness of the weak solutions. Then the existence of a maximal attractor is proved and the upper bound for the global Hausdorff and fractal dimensions of the attractor is obtained. The applications to the two-dimensional sphere ${S}^2$ and the square torus ${T}^2$ will be treated. Finally, we prove the existence of the inertial manifolds in the case ${S}^2$.