## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 39, Number 3 (2010), 291-316.

### 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.

#### Article information

**Source**

Hokkaido Math. J., Volume 39, Number 3 (2010), 291-316.

**Dates**

First available in Project Euclid: 29 October 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1288357971

**Digital Object Identifier**

doi:10.14492/hokmj/1288357971

**Mathematical Reviews number (MathSciNet)**

MR2743826

**Zentralblatt MATH identifier**

05829564

**Subjects**

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]

Secondary: 76D05: Navier-Stokes equations [See also 35Q30]

**Keywords**

time periodic solutions of the Navier-Stokes equations general outflow condition stationary symmetric Navier-Stokes flow symmetry 2-D infinite channels the poiseuille velocity

#### Citation

KOBAYASHI, Teppei. Time periodic solutions of the Navier-Stokes equations under general outflow condition in a two dimensional symmetric channel. Hokkaido Math. J. 39 (2010), no. 3, 291--316. doi:10.14492/hokmj/1288357971. https://projecteuclid.org/euclid.hokmj/1288357971