  We consider the Navier-Stokes system in three dimensions perturbed by a
transport noise which is sufficiently smooth in space and rough in time. The
existence of a weak solution was proved recently, however, as in the
deterministic setting the question of uniqueness remains a major open problem.
An important feature of systems with uniqueness is the semigroup property
satisfied by their solutions. Without uniqueness, this property cannot hold
generally. We select a system of solutions satisfying the semigroup property
with appropriately shifted rough path. In addition, the selected solutions
respect the well accepted admissibility criterium for physical solutions,
namely, maximization of the energy dissipation. Finally, under suitable
assumptions on the driving rough path, we show that the Navier-Stokes system
generates a measurable random dynamical system. To the best of our knowledge,
this is the first construction of a measurable single-valued random dynamical
system in the state space for an SPDE without uniqueness.
