The existence and regularity of time-periodic solutions to the three-dimensional Navier-Stokes equations in the whole space

The existence, uniqueness and regularity of time-periodic solutions to the Navier-Stokes equations in the three-dimensional whole space are investigated. We consider the Navier-Stokes equations with a non-zero drift term corresponding to the physical model of a fluid flow around a body that moves with a non-zero constant velocity. The existence of a strong time-periodic solution is shown for small time-periodic data. It is further shown that this solution is unique in a large class of weak solutions that can be considered physically reasonable. Finally, we establish regularity properties for any strong solution regardless of its size.