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 \begin{document}$ L^\sigma_d(\mathbb{R}^d) $\end{document} . Temporal decay at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document} are obtained for the solution and its gradient.