Existence of time-periodic strong solutions to the Navier-Stokes equation in the whole space

Abstract

In this paper, the existence of time-periodic strong solutions to the Navier-Stokes equation in $R^n$ is established under a suitable smallness condition on the external force. Our analysis is based on splitting periodic solutions into steady and purely periodic parts. One advantage of this decomposition is the availability of slightly more regularity in time of the purely periodic part. We apply this property to construct time-periodic solutions of the Navier-Stokes equation with information on the classes of their steady and purely periodic parts. It is also shown that the small solution v constructed in our existence theorem is unique within a class of time-periodic, not necessarily small, solutions having the same integrability properties as v.