Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity

TL;DR

This paper establishes a new regularity criterion for the 3D Navier-Stokes equations based on the divergence of the velocity direction, linking it to global solution regularity through standard analytical methods.

Contribution

It introduces a novel regularity criterion involving the divergence of the velocity direction, differing from previous vorticity-based conditions.

Findings

Control of Div(u/|u|) in L_t^p(L_x^q) ensures global regularity.

The criterion is analogous to vorticity direction conditions but focuses on velocity.

The proof uses standard methods relating divergence of velocity direction to energy growth.

Abstract

In this short note, we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation, and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of $\Div (u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. But in this case the condition is not on the vorticity, but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.