Some elementary estimates for the Navier-Stokes system.

In this paper we study the incompressible Navier-Stokes equations in ${mathbb {T}}^3=[0,1]^3$ with periodic boundary conditions. We show that a weak solution of the Navier-Stokes equations that is small in $L^{infty }(0,T;Phi (2))$ is also smooth. We also give elementary proofs of some classical regularity results for the Navier-Stokes equations involving the Sobolev space $H^{frac {1}{2}}({mathbb {T}}^3)$. [ABSTRACT FROM AUTHOR]