Global regularity criteria for the Navier-Stokes equations based on one approximate solution

TL;DR

This paper establishes criteria for global regularity of 3D Navier-Stokes solutions based on a single approximate solution, using scale-invariant relations, applicable to various approximation methods.

Contribution

It introduces novel regularity criteria linking approximate solutions to global regularity, extending understanding of Navier-Stokes behavior.

Findings

Global regularity can be inferred from approximate solutions under scale-invariant conditions.

The criteria apply to a broad class of approximate solutions, including mollified solutions.

Two different analytical approaches yield consistent results.

Abstract

Considering the three-dimensional incompressible Navier-Stokes equations on the whole space, we address the question: is it possible to infer global regularity of a mild solution from a single approximate solution? Assuming a relatively simple scale-invariant relation involving the size of the approximate solution, the resolution parameter, and the initial energy, we show that the answer is affirmative for a general class of approximate solutions, including Leray's mollified solutions. Two different treatments leading to essentially the same conclusion are presented.