Global regularity criterion for the 3D Navier-Stokes equations with large data

TL;DR

This paper establishes a new global regularity criterion for strong solutions to the 3D Navier-Stokes equations with large initial data and external force, based on the integrability of the velocity gradient.

Contribution

It introduces a novel regularity condition involving the gradient of the velocity in certain Lebesgue spaces, extending previous criteria for global existence.

Findings

Global existence of strong solutions under the new criterion

Applicable to large initial data and non-zero external force

Condition involves the gradient of velocity in L^α space

Abstract

In this paper, we study the global regularity of strong solution to the Cauchy problem of 3D incompressible Navier-Stokes equations with large data and non-zero force. We prove that the strong solution exists globally for $\nabla u\in L^{\alpha}\left( 0,T,L^{2}\left( \Omega\right) \right) $ with $2\leq\alpha\leq4$.