| Abstract: |
| We obtain the existence 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 and the trajectory attractor are strong, strongly compact, respectively. As a consequence, we obtain the strong equicontinuity of all bounded complete trajectories and the finite strong uniform tracking property that for any fixed accuracy and time length, a finite number of trajectories on the global attractor are able to capture in strong metric all trajectories after sufficiently large time. Our method is based on a new established framework called evolutionary system, whose trajectories are solutions to the nonautonomous 3D NSE. The 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. |
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