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 R-n, n >= 3, with the external forces of non -divergence form. Firstly, we consider the existence of time periodic solutions in BC(R; L-n,L-infinity. (R-n)) assuming the smallness of external forces in BC(R; L-1 (R-3)) and BC(R; (L-n/3,L-infinity (R-n)) in the case n >= 4. Next, we show that the mild solution above becomes a strong solution in the topology of L-n,L-infinity (R-n) with a natural condition of the external force, derived from the strong solvability of the inhomogeneous Stokes equations in L" (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. (C) 2017 Elsevier Inc. All rights reserved.