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

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 $L^3({I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}R}^3)$, 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 $H^{\frac{1}{2}}({I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}R}^3)$. The scale invariance of these two norms is discussed.