A tamed 3D Navier-Stokes equation in uniform C²-domains

In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform C²-domains (not necessarily bounded), which obeys the scaling invariance principle, and prove the existence and uniqueness of strong solutions to this tamed equation. In particular, if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, the existence of a global attractor for the tamed equation in bounded domains is also proved. As a simple application, we obtain that the set of all initial values for which the classical Navier-Stokes equation admits a bounded strong solution is open in H².