Liouville-type theorems for steady solutions to the Navier-Stokes system in a slab

Liouville-type theorems for the steady incompressible Navier-Stokes system are investigated for solutions in a three-dimensional (3-D) slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru<^>r$ is bounded, and that general 3-D solutions must be Poiseuille flows when the velocity is not big in $L<^>\infty$ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or $ru<^>r$ decays to zero as $r$ tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of the Dirichlet integral of non-trivial solutions.