The simplified Bardina equations on two-dimensional closed manifolds

In this paper we study the viscous simplified Bardina equation on the two-dimensional closed manifold $M$ which is embedded in $\mathbb{R}^3$. First, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation on $M$. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the attractor. We also prove the existence of an inertial manifold for the equation on the two-dimensional sphere ${S}^2$.