Time periodic problem of the Navier-Stokes equations in an exterior domain with periodically moving boundary

In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partialΩ(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spac