Abstract
The Kneser theorem for ordinary differential equations without uniqueness saysthat the attainability set is compact and connected at each instant of time.We establish corresponding results for the attainability set of weak solutionsfor the 3D Navier-Stokes equations satisfying an energy inequality. First, wepresent a simplified proof of our earlier result with respect to the weaktopology in the space $H$. Then we prove that this result also holds withrespect to the strong topology on $H$ provided that the weak solutionssatisfying the weak version of the energy inequality are continuous. Finally,using these results, we show the connectedness of the global attractor of afamily of setvalued semiflows generated by the weak solutions of the NSEsatisfying suitable properties.
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