Time periodic solutions of the Navier-Stokes equations under general outflow condition in a two dimensional symmetric channel

Abstract

In this paper we will prove that there exists a time periodic solution of the Navier-Stokes equations with the inhomogeneous boundary condition for infinite symmetric channels in $\R^2$. In two and three dimensional more generalized infinite channels (than treated in this paper) H.~Beir\~ao~da Veiga \cite{Beirao} proved that there exists time periodic solutions of the Navier-Stokes equations with the homogeneous boundary condition under a small time periodic flux. G.~P.~Galdi and A.~M.~Robertson \cite{GalRob} obtained time-periodic Poiseuille flow in a straight channel with a smooth cross section. C.~J.~Amick \cite{Amick2} proved that in two and three dimensional unbounded channels there exists solutions of the stationary Navier-Stokes equations with the nonhomogenous boundary condition. H.~Morimoto and H.~Fujita \cite{Morimoto1} and H.~Morimoto \cite{Morimoto2} proved that in a two dimensional certain unbounded symmetric channel there exists symmetric solutions of the stationary Navier-Stokes equations with a special symmetric Dirichlet boundary condition. T-P.~Kobayashi \cite{Kobayashi3} demonstrated that for two and three dimensional infinite channels time periodic solutions of the Navier-Stokes equations exist under the same condition as C.~J.~Amick \cite{Amick2}. In this paper using the condition of H.~Morimoto and H.~Fujita \cite{Morimoto1} and H.~Morimoto \cite{Morimoto2}, we obtain time priodic solutions.