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In this paper, we extend some results proved in previous references forthree-dimensional Navier-Stokes equations. We show that when the norm of thevelocity field is small enough in $L^3({I\!\!R}^3)$, then a global smoothsolution of the Navier-Stokes equations is ensured. We show that a similarresult 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 isdiscussed.

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