Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

TL;DR

This paper establishes a new criterion based on a single component of the velocity gradient tensor that guarantees the global regularity of solutions to the 3D Navier-Stokes equations.

Contribution

It introduces a novel regularity criterion depending on only one entry of the velocity gradient tensor for the 3D Navier-Stokes equations.

Findings

Provides a sufficient condition for global regularity

Applicable to both whole space and periodic boundary conditions

Advances understanding of conditions preventing singularities

Abstract

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions.