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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.

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