Existence of Regular Time-Periodic Solutions to Shear-Thinning Fluids

In this note we investigate the existence of time-periodic solutions to the p -Navier–Stokes system in the singular case of p ∈ ( 1 , 2 ) , which describes the flows of an incompressible shear-thinning fluid. In the 3D space-periodic setting and for p ∈ [ 5 3 , 2 ) we prove the existence of a regular time-periodic solution corresponding to a time-periodic force datum that is assumed small in a suitable sense. As a particular case we obtain “regular” steady solutions.