Time-periodic solutions to the Navier--Stokes equations on the whole space including the two-dimensional case

Let us consider the incompressible Navier–Stokes equations with the time-periodic external forces in the whole space ${\mathbb{R}}^n$ with $n\ge 2$ and investigate the existence and non-existence of time-periodic solutions. In the higher dimensional case $n\ge 3$, we construct a unique small solution for given small time-periodic force in the scaling critical spaces of Besov type and prove its stability under small perturbations. In contrast, for the two-dimensional case $n=2$, the time-periodic solvability of the Navier–Stokes equations has been long standing open. It is the central work of this paper that we have now succeeded in solving this issue negatively by providing examples of small external forces such that each of them does not generate time-periodic solutions.