  We obtain the existence and the structure of the weak uniform (with respect
to the initial time) global attractor and construct a trajectory attractor for
the 3D Navier-Stokes equations (NSE) with a fixed time-dependent force
satisfying a translation boundedness condition. Moreover, we show that if the
force is normal and every complete bounded solution is strongly continuous,
then the uniform global attractor is strong, strongly compact, and solutions
converge strongly toward the trajectory attractor. Our method is based on
taking a closure of the autonomous evolutionary system without uniqueness,
whose trajectories are solutions to the nonautonomous 3D NSE. The established
framework is general and can also be applied to other nonautonomous dissipative
partial differential equations for which the uniqueness of solutions might not
hold. It is not known whether previous frameworks can also be applied in such
cases as we indicate in open problems related to the question of uniqueness of
the Leray-Hopf weak solutions.
