On the supnorm form of Leray's problem for the incompressible Navier-Stokes equations

We show that t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty for all (global) Leray solutions of the incompressible Navier-Stokes equations in R3. It is also shown that t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty, where v(.,t) is the Stokes approximation, as well as other fundamental results. In spite of the difficulty of these questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.