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

We consider a family of weights which permit to generalize the Leray procedure to obtain weak suitable solutions of the 3D incompressible Navier–Stokes equations with initial data in weighted $$L^2$$ spaces. Our principal result concerns the existence of regular global solutions when the initial velocity is an axisymmetric vector field without swirl such that both the initial velocity and its vorticity belong to $$L^2 ( (1+ r^2)^{-\frac{\gamma }{2}} dx ) $$ , with $$r= \sqrt{x_1^2 + x_2^2}$$ and $$\gamma \in (0, 2) $$ .