Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and Regularity

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].