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

Morimoto, H. ; Fujita, H.

Mathematica Bohemica, Tome 126 (2001) no. 2, pp. 457-468.

Résumé

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 $\mathbb{R} \times (-1,1)$. We suppose that $\Omega \cap \lbrace x_1 0 \rbrace $ is bounded and that $\Omega \cap \lbrace x_1 > -1 \rbrace = T \cap \lbrace x_1 > -1 \rbrace $. 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.

Détail

DOI : 10.21136/MB.2001.134017

Classification : 35B40, 35B65, 35Q30, 76D03, 76D05
Keywords: stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry