On the Kneser property of the three-dimensional Navier-Stokes equations with damping

In this paper, we study the connectedness and compactness of the attainability set of weak solutions to the three-dimensional Navier--Stokes equations with damping. Depending on the value of the parameter \b{eta}, which controls the damping term, we establish these results with respect to either the weak or the strong topology of the phase space. In the latter case, we also prove that the global attractor is connected. Additionally, we establish results concerning the regularity of the global attractor and provide a new proof of its existence for strong solutions.