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We consider a family of weights which permit to generalize the Lerayprocedure to obtain weak suitable solutions of the 3D incom-pressibleNavier-Stokes equations with initial data in weighted L 2 spaces. Our principalresult concerns the existence of regular global solutions when the initialvelocity is an axisymmetric vector field without swirl such that both theinitial velocity and its vorticity belong to L 2 ((1 + r 2) -- $\gamma$ 2 dx),with r = x 2 1 + x 2 2 and $\gamma$ $\in$ (0, 2).

Weighted energy estimates for the incompressible Navier-Stokes equations and applications to axisymmetric solutions without swirl