On the long-time behaviour of solutions to unforced evolution Navier–Stokes equations under Navier boundary conditions

We study the asymptotic behaviour of the solutions to Navier–Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest. The paper draws its main motivation from celebrated results by Foias and Saut (1984) under Dirichlet conditions; here the choice of the boundary conditions requires carefully considering the geometry of the domain $\Omega$, due to the possible lack of the Poincaré inequality in presence of symmetries. In non-axially symmetric domains we show the validity of the Foias–Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias–Saut result holds for initial data belonging to one of the invariants.