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

Abstract

Consider a $d$-sphere $\mathbb{S}^d$ and denote by $\Gamma(T\mathbb{S}^d)$ the set of all vector fields on $\mathbb{S}^d$. We study the Navier-Stokes equations (NSE): $$ \partial_t u + \nabla_u u + grad \pi = \mathbf{\Delta} u + \operatorname{div} f(\cdot, t);\, \operatorname{div} u=0,$$ for the vector field $u(\cdot, t)\in \Gamma(T\mathbb{S}^d)$, where $\mathbf{\Delta}$ denotes the Ebin-Marsden Laplace operator defined by $\mathbf{\Delta} u= \operatorname{div} (\nabla u + \nabla u^t)^{\sharp}$, and $\operatorname{div} f(\cdot, t)$ is the periodic external force. We investigate the Navier-Stokes equations in the framework of $L^p$-spaces over vector fields on $\mathbb{S}^d$ and prove the existence and uniqueness of a periodic solution to such equations. Moreover, exploiting the distribution of eingenvalues of Ebin-Marsden Laplace operator we show the existence of an inertial manifold for solutions around that solution.