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We show that the classical Cauchy problem for the incompressible 3dNavier-Stokes equations with $(-1)$-homogeneous initial data has a globalscale-invariant solution which is smooth for positive times. Our main technicaltools are local-in-space regularity estimates near the initial time, which areof independent interest.

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions