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A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition. (English) Zbl 0981.35049
Summary: 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 $$\Omega \cap \{x_1 < 0 \}$$ 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 proved the existence of solutions if the flux is small (Morimoto-Fujita in 1999). Some improvement is reported in this note. We also show certain regularity and asymptotic properties of the solutions.

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids
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