A non linear estimate on the life span of solutions of the three dimensional Navier-Stokes equations

TL;DR

This paper derives improved lower bounds on the lifespan of regular solutions to the 3D Navier-Stokes equations by incorporating nonlinear initial data norms and scale-invariant energy estimates, advancing understanding of solution longevity.

Contribution

It introduces novel lower bounds for solution lifespan that depend on nonlinear functions of initial data, surpassing classical fixed point estimates.

Findings

Bounds significantly improve previous estimates

Use of scale-invariant energy estimates is effective

Examples demonstrate the bounds' significance

Abstract

The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, whichinvolve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.