| Abstract: |
| We show that for any fixed accuracy and time length T, a finite number of T-time length pieces of the complete bounded solutions on the global attractor are capable of uniformly approximating all Leray-Hopf weak solutions within the accuracy in the natural strong metric after sufficiently large time when the 3D Navier-Stokes equations is with a fixed normal force and every complete bounded solution is strongly continuous. Moreover, we obtain the strong equicontinuity of all the complete bounded solutions on the global attractor. These results follow by proving the existence of a strongly compact strong trajectory attractor for such a system. The notion of a (weak) trajectory attractor was previously constructed for a family of auxiliary systems including the originally considered one. We developed a framework called evolutionary system, with which a (weak) trajectory attractor can be actually defined for the original system nearly ten years ago. Very recently, the theory of trajectory attractors is further developed in the natural strong metric for our purpose. 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. |
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