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A remark on the existence of steady Navier-Stokes flows in a certain two-dimensional infinite channel. (English) Zbl 1042.35049
The steady Navier-Stokes equations $\begin{cases} (u\cdot\nabla)u= \nu\Delta u-\nabla p\quad &\text{in }\Omega,\\ \text{div\,}u= 0\quad &\text{in }\Omega\end{cases}\tag{1}$ are considered in a two-dimensional unbounded multiply-connected domain $$\Omega$$ contained in an infinite straight channel $$T= \mathbb{R}\times (-1,1)$$.
The boundary condition is as follows $\begin{cases} u= \beta\quad &\text{on }\partial\Omega,\\ u\to\mu U\quad &\text{as }| x_1|\to\infty,\text{ in }\Omega,\end{cases}\tag{2}$ where $$\beta$$ is a given function on $$\partial\Omega= \bigcup^N_{i=0} \Gamma_i$$ compactly supported, $$U$$ is the Poiseuille flow in $$T: U= {3\over 4}(1- x^2_2,0)$$ and $$\mu$$ is a constant.
For the boundary value $$\beta$$, the general outflow condition $\int_{\partial\Omega} \beta\cdot n\,d\sigma= \sum^N_{i=0} \int_{\Gamma_i} \beta\cdot n\,d\sigma= 0$ ($$n$$ is the unit outward normal vector to $$\partial\Omega$$) is supposed.
The existence of solution to (1) and (2) under the assumption of symmetry with respect to the $$x_1$$-axis for the domain and the boundary value $$\beta$$ and for small $$|\mu|$$ (without smallness assumption on $$\beta$$) is shown.
The regularity and the asymptotic behaviour of the solution are also discussed.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
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