Geodesic
A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition
Mathematica Bohemica, Tome 126 (2001) no. 2, pp. 457-468

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2001.134017
Classification : 35B40, 35B65, 35Q30, 76D03, 76D05
Keywords: stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry 
Morimoto, H.; Fujita, H. A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition. Mathematica Bohemica, Tome 126 (2001) no. 2, pp. 457-468. doi: 10.21136/MB.2001.134017
@article{10_21136_MB_2001_134017,
     author = {Morimoto, H. and Fujita, H.},
     title = {A remark on the existence of steady {Navier-Stokes} flows in {2D} semi-infinite channel involving the general outflow condition},
     journal = {Mathematica Bohemica},
     pages = {457--468},
     year = {2001},
     volume = {126},
     number = {2},
     doi = {10.21136/MB.2001.134017},
     mrnumber = {1844283},
     zbl = {0981.35049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2001.134017/}
}
TY  - JOUR
AU  - Morimoto, H.
AU  - Fujita, H.
TI  - A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition
JO  - Mathematica Bohemica
PY  - 2001
SP  - 457
EP  - 468
VL  - 126
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2001.134017/
DO  - 10.21136/MB.2001.134017
LA  - en
ID  - 10_21136_MB_2001_134017
ER  - 
%0 Journal Article
%A Morimoto, H.
%A Fujita, H.
%T A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition
%J Mathematica Bohemica
%D 2001
%P 457-468
%V 126
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2001.134017/
%R 10.21136/MB.2001.134017
%G en
%F 10_21136_MB_2001_134017

[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

[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. | DOI | MR

[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 | Zbl

[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

[5] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer, 1994. | Zbl

[6] Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 1969. | MR | Zbl

[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,.

Cité par Sources :