Smoothness criteria for Navier-Stokes equations in terms of regularity along the steam lines

TL;DR

This paper establishes a new regularity criterion for Navier-Stokes solutions based on the second derivative of the velocity magnitude along stream lines, advancing understanding of fluid flow regularity conditions.

Contribution

It introduces a novel regularity criterion involving the second derivative of |u| along stream lines for weak solutions of Navier-Stokes equations.

Findings

Regular solutions are guaranteed under specific second derivative constraints.

The criterion links flow regularity to geometric properties along stream lines.

Provides a new perspective on fluid flow smoothness based on line-based derivatives.

Abstract

This article is devoted to a regularity criteria for solutions of the Navier-Stokes equations in terms of regularity along the stream lines. More precisely, we prove that a suitable weak solution for the Navier-Stokes equations is regular under some constraint on the second derivative of |u| along the stream lines.