Time periodic Navier-Stokes flow with nonhomogeneous boundary condition

It is known that the Navier-Stokes initial boundary value problem for non-homogeneous boundary condition has a unique local solution (e.g., O. A. Ladyzhenskaya\cite{lady}). Nevertheless, it seems to the author that there is no results for the periodic problem with non-homogeneous boundary condition satisfying the general outflow condition. We consider the periodic problem for the Navier-Stokes equations in a two dimensional bounded domain. In case of a symmetric domain, we obtain a periodic weak solution for symmetric boundary values satisfying only the general outflow condition