  In this paper, we prove the existence and uniqueness of a smooth solution to
a tamed 3D Navier-Stokes equation in the whole space. In particular, if there
exists a bounded smooth solution to the classical 3D Navier-Stokes equation,
then this solution satisfies our tamed equation. Moreover, using this
renormalized equation we can give a new construction for a suitable weak
solution of the classical 3D Navier-Stokes equation introduced in [Scheffer:
Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 1977] and
[Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak solutions of
the Navier-Stokes equations. Comm. Pure Appl. Math., 1982].
