In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier–Stokes equations on the non-compact 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.
The works of the first two authors and the last author were
partially supported by Vietnam Institute for Advance Study in Mathematics (VIASM). The work of
the last author is partly supported by by the Project of Vietnam Ministry of Education and
Training under Project B2022-BKA-06. This work is financially supported by Vietnam National
Foundation for Science and Technology Development (NAFOSTED) under Grant
"On Periodic Solutions of the Incompressible Navier–Stokes Equations on Non-compact Riemannian Manifolds." Taiwanese J. Math. Advance Publication 1 - 27, 2021. https://doi.org/10.11650/tjm/211205