A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations

We construct a Poiseuille type ow which is a bounded entire solution of the nonstationary Navier-Stokes and the Stokes equations in a half space with non-slip boundary condition. Our result in particular implies that there is a nontrivial solution for the Liouville problem under the non-slip boundary condition. A review for cases of the whole space and a slip boundary condition is included