A Remark on the Existence of Steady Navier-Stokes Flows in a Certain Two-Dimensional Infinite Channel

Abstract

We consider the steady Navier-Stokes equations $$ \left\{ \begin{array}{@{}l@{\hspace{2pt}}ll} (\mathbf{u}\cdot\nabla)\mathbf{u} & =\nu \Delta\mathbf{u} -\nabla p & \text{in}~\Omega\,,\\ \text{div}\,\mathbf{u} & =0 & \text{in} ~\Omega\,, \end{array} \right. $$ in a 2-dimensional unbounded multiply-connected domain $\Omega$ contained in an infinite straight channel $\mathbf{R}\times(-1,1)$, under general outflow condition. We look for a solution which tends to a Poiseuille flow at infinity. In this note, we shall show the existence of solution to this problem under the assumption of symmetry with respect to the axis for the domain and the boundary value, and for small Poiseuille flow. We do not assume that the boundary value is small. The regularity and the asymptotic behavior of the solution are also discussed.