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

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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.