Time periodic strong solutions to the incompressible Navier-Stokes equations with external forces of non-divergence form

We discuss the time periodic problem to the incompressible Navier-Stokes equations on the whole space Rn, n≥3, with the external forces of non-divergence form. Firstly, we consider the existence of time periodic solutions in BC(R;Ln,∞(Rn)) assuming the smallness of external forces in BC(R;L1(R3)) and BC(R;Ln3,∞(Rn)) in the case n≥4. Next, we show that the mild solution above becomes a strong solution in the topology of Ln,∞(Rn) with a natural condition of the external force, derived from the strong solvability of the inhomogeneous Stokes equations in Ln,∞(Rn). For this aim, we re-construct a strong solvability of an abstract evolution equation where the associated semigroup is not strongly continuous at t=0.