  In this paper, we analyze a tamed 3D Navier-Stokes equation in uniform
$C^2$-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 simple applications, some well known
results for the classical Navier-Stokes equations in unbounded domains are
covered.
