On Periodic Solutions of the Incompressible Navier-Stokes Equations on Non-compact Riemannian Manifolds

In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier-Stokes equations on the noncompact manifolds with negative Ricci curvature tensor. In our strategy, we combine the dispersive and smoothing estimates for Stokes semigroups and Massera-type theorem to establish the existence and uniqueness of the time periodic mild solution to Stokes equation on Riemannian manifolds. Then using fixed point arguments, we can pass to semilinear equations to obtain the existence and uniqueness of the periodic solution to the imcompressible Navier-Stokes equations under the action of a periodic external force. The stability of the solution is also proved by using the cone inequality.