MATHEMATICA BOHEMICA, Vol. 126, No. 2, pp. 457-468 (2001)

A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition

H. Morimoto, H. Fujita

Hiroko Morimoto, Meiji University, Kawasaki 214-8571, Japan, e-mail:; Hiroshi Fujita, Tokai University, Tokyo 151-0063, Japan

Abstract: We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain $\Omega $ under the general outflow condition. Let $T$ be a 2-dimensional straight channel $\Bbb R \times (-1,1)$. We suppose that $\Omega \cap \{x_1 < 0 \}$ is bounded and that $\Omega \cap \{x_1 > -1 \} = T \cap \{x_1 > -1 \}$. Let $V$ be a Poiseuille flow in $T$ and $\mu $ the flux of $V$. We look for a solution which tends to $V$ as $x_1 \rightarrow \infty $. Assuming that the domain and the boundary data are symmetric with respect to the $x_1$-axis, and that the axis intersects every component of the boundary, we have shown the existence of solutions if the flux is small (Morimoto-Fujita [8]). Some improvement will be reported in this note. We also show certain regularity and asymptotic properties of the solutions.

Keywords: stationary Navier-Stokes equations, non-vanishing outflow, 2-dimensional semi-infinite channel, symmetry

Classification (MSC2000): 35Q30, 76D05

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