Liouville-type theorems for steady Navier-Stokes system in a slab with Navier boundary conditions

In this paper, the Liouville-type theorems for the steady Navier-Stokes system in a slab supplemented with Navier boundary conditions are investigated. Specifically, we prove that any bounded smooth solution must be zero if either the swirl or radial velocity is axisymmetric, or $r{u}^r$ decays to zero as r tends to infinity. When the velocity is not big in $L^{\infty }$-space, the general three-dimensional steady Navier-Stokes flow in a slab with the Navier boundary conditions must be a Poiseuille type flow. The key idea of the proof is to establish the Saint-Venant type estimates that characterize the growth of Dirichlet integral of nontrivial solutions.