Global solutions to the non-local navier-stokes equations

This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato's strategy and this context, with initial conditions in the divergence-free Lebesgue space L-d(sigma) (R-d). Temporal decay at 0 and infinity are obtained for the solution and its gradient.