The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations

The attainability set of the weak solutions of the three-dimensional Navier-Stokes equations which satisfy an energy inequality is shown to be a weakly compact and weakly connected subset of the space H, i.e. the Kneser property holds in the weak topology for such weak solutions. The proof of weak connectedness uses the strong connectedness of the attainability set of the weak solutions of the globally modified Navier-Stokes equations, which is first proved. The weak connectedness of the weak global attractor of the three-dimensional Navier-Stokes equations is also established.