Weak solutions for Navier--Stokes equations with initial data in weighted $L^2$ spaces

We show the existence of global weak solutions to the three dimensional Navier–Stokes equations with initial velocity in the weighted spaces $$L^2_{w_\gamma }$$, where $$w_\gamma (x)=(1+\vert x\vert )^{-\gamma }$$ and $$0<\gamma \leqq 2$$, using new energy controls. As an application we give a new proof of the existence of global weak discretely self-similar solutions to the three dimensional Navier–Stokes equations for discretely self-similar initial velocities which are locally square integrable.