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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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