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Keywords:
stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry
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.
References:
[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
[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
[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. MR 0132307 | Zbl 0111.38502
[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
[5] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer, 1994. Zbl 0949.35005
[6] Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969. MR 0254401 | Zbl 0184.52603
[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.
[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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