A regularity criterion for the Navier-Stokes equations via two entries of the velocity Hessian tensor

TL;DR

This paper establishes a new criterion for the smoothness of solutions to the Navier-Stokes equations, based on only two specific entries of the velocity Hessian tensor, simplifying the conditions needed for regularity.

Contribution

It introduces a novel regularity criterion for Navier-Stokes solutions that depends solely on two entries of the velocity Hessian tensor, advancing understanding of solution smoothness.

Findings

Provides a sufficient condition for smoothness based on two Hessian entries

Simplifies previous regularity criteria for Navier-Stokes equations

Enhances theoretical understanding of solution regularity in fluid dynamics

Abstract

We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\R^3$, and provide a sufficient condition to ensure the smoothness of the solution. It involves only two entries of the velocity Hessian.