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

TL;DR

This paper proves that for all global Leray solutions of the 3D incompressible Navier-Stokes equations, the supremum norm of the velocity decays at a specific rate over time, using elementary Fourier and energy methods.

Contribution

It establishes new decay rates for Leray solutions and their difference from Stokes approximations using elementary techniques.

Findings

t^{3/4}|| u(.,t) ||_{sup} --> 0 as t --> infty

t || u(.,t) - v(.,t) ||_{sup} --> 0 as t --> infty

Decay rates are proven using standard Fourier and energy methods.

Abstract

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.