Periodicity and Inertial Manifolds for Navier-Stokes Flows on $d$-Spheres

Consider a $d$-sphere $\mathbb{S}^d$ and denote by $Γ(T\mathbb{S}^d)$ the set of all vector fields on $\mathbb{S}^d$. We study the Navier-Stokes equations (NSE): $$ \partial_t u + \nabla_u u + grad π= \mathbfΔ u + \operatorname{div} f(\cdot, t);\, \operatorname{div} u=0,$$ for the vector field $u(