A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition

H. Morimoto; H. Fujita

Mathematica Bohemica (2001)

  • Volume: 126, Issue: 2, page 457-468
  • ISSN: 0862-7959

Abstract

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We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain Ω under the general outflow condition. Let T be a 2-dimensional straight channel ℝ × ( - 1 , 1 ) . We suppose that Ω ∩ { x 1 < 0 } is bounded and that Ω ∩ { x 1 > - 1 } = T ∩ { x 1 > - 1 } . Let V be a Poiseuille flow in T and μ the flux of V . We look for a solution which tends to V as x 1 → ∞ . 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.

How to cite

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Morimoto, H., and Fujita, H.. "A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition." Mathematica Bohemica 126.2 (2001): 457-468. <http://eudml.org/doc/248832>.

@article{Morimoto2001,
abstract = {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.},
author = {Morimoto, H., Fujita, H.},
journal = {Mathematica Bohemica},
keywords = {stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry; stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry},
language = {eng},
number = {2},
pages = {457-468},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition},
url = {http://eudml.org/doc/248832},
volume = {126},
year = {2001},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 126
IS - 2
SP - 457
EP - 468
AB - 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.
LA - eng
KW - stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry; stationary Navier-Stokes equations; non-vanishing outflow; 2-dimensional semi-infinite channel; symmetry
UR - http://eudml.org/doc/248832
ER -

References

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  2. 10.1016/0362-546X(78)90014-7, Nonlinear Analysis, Theory, Methods & Applications, Vol. 2 (1978), 689–720. (1978) MR0512162DOI10.1016/0362-546X(78)90014-7
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  4. 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. MR1773581
  5. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer, 1994. (1994) Zbl0949.35005
  6. The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. (1969) Zbl0184.52603MR0254401
  7. 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. On stationary Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition, NSEC7, Ferrara, Italy,. 

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