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