# Article

 Title: A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition (English) Author: Morimoto, H. Author: Fujita, H. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 126 Issue: 2 Year: 2001 Pages: 457-468 Summary lang: English . Category: math . 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 $\Omega \cap \lbrace x_1 < 0 \rbrace$ 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. (English) Keyword: stationary Navier-Stokes equations Keyword: non-vanishing outflow Keyword: 2-dimensional semi-infinite channel Keyword: symmetry MSC: 35B40 MSC: 35B65 MSC: 35Q30 MSC: 76D03 MSC: 76D05 idZBL: Zbl 0981.35049 idMR: MR1844283 DOI: 10.21136/MB.2001.134017 . Date available: 2009-09-24T21:52:42Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134017 . Reference: [1] Amick, C. J.: Steady solutions of the Navier-Stokes equations for certain unbounded channels and pipes.Ann. Scuola Norm. Sup. Pisa 4 (1977), 473–513. MR 0510120 Reference: [2] Amick, C. J.: Properties of steady Navier-Stokes solutions for certain unbounded channel and pipes.Nonlinear Analysis, Theory, Methods & Applications, Vol. 2 (1978), 689–720. MR 0512162, 10.1016/0362-546X(78)90014-7 Reference: [3] Fujita, H.: On the existence and regularity of the steady-state solutions of the Navier-Stokes equation.J. Fac. Sci., Univ. Tokyo, Sec. I 9 (1961), 59–102. Zbl 0111.38502, MR 0132307 Reference: [4] Fujita, H.: On stationary solutions to Navier-Stokes equations in symmetric plane domains under general out-flow condition.Proceedings of International Conference on Navier-Stokes Equations, Theory and Numerical Methods, June 1997, Varenna Italy, Pitman Reseach Notes in Mathematics 388, pp. 16–30. MR 1773581 Reference: [5] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations.Springer, 1994. Zbl 0949.35005 Reference: [6] Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow.Gordon and Breach, New York, 1969. Zbl 0184.52603, MR 0254401 Reference: [7] Morimoto, H., Fujita, H.: A remark on existence of steady Navier-Stokes flows in a certain two dimensional infinite tube.Technical Reports Dept. Math., Math-Meiji 99-02, Meiji Univ. Reference: [8] Morimoto, H., Fujita, H.: On stationary Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition.NSEC7, Ferrara, Italy,. .

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