  We show the existence of global weak solutions of the 3D Navier-Stokes
equations with initial velocity in the weighted spaces L 2 w$\gamma$ , where w
$\gamma$ (x) = (1 + |x|) --$\gamma$ and 0 < $\gamma$ $\le$ 2, using new energy
controls. As application we give a new proof of the existence of global weak
discretely self-similar solutions of the 3D Navier-Stokes equations for
discretely self-similar initial velocities which are locally square
inte-grable.
