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

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.