On the role of $L^3$ and $H^{\frac{1}{2}}$ norms in hydrodynamics

TL;DR

This paper investigates how smallness in $L^3$ and $H^{1/2}$ norms of the velocity field guarantees global smooth solutions for 3D Navier-Stokes equations, extending previous results and discussing scale invariance.

Contribution

It extends existing results by establishing smallness conditions in $L^3$ and $H^{1/2}$ norms that ensure global regularity of solutions.

Findings

Small $L^3$ norm implies global smooth solutions.

Small $H^{1/2}$ norm also guarantees regularity.

Discussion of scale invariance of these norms.

Abstract

In this paper, we extend some results proved in previous references for three-dimensional Navier-Stokes equations. We show that when the norm of the velocity field is small enough in $L^3({I\!\!R}^3)$, then a global smooth solution of the Navier-Stokes equations is ensured. We show that a similar result holds when the norm of the velocity field is small enough in $H^{\frac{1}{2}}({I\!\!R}^3)$. The scale invariance of these two norms is discussed.