On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity

Michael Bildhauer; Martin Fuchs

Commentationes Mathematicae Universitatis Carolinae (2012)

  • Volume: 53, Issue: 2, page 221-236
  • ISSN: 0010-2628

Abstract

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On the complement of the unit disk B we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field u is equal to zero provided u | ∂ B = 0 and lim | x | → ∞ | x | 1 / 3 | u ( x ) | = 0 uniformly. For slow flows the latter condition can be replaced by lim | x | → ∞ | u ( x ) | = 0 uniformly. In particular, these results hold for the classical Navier-Stokes case.

How to cite

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Bildhauer, Michael, and Fuchs, Martin. "On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity." Commentationes Mathematicae Universitatis Carolinae 53.2 (2012): 221-236. <http://eudml.org/doc/246524>.

@article{Bildhauer2012,
abstract = {On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided $u|_\{\partial B\} = 0$ and $\lim _\{|x| \rightarrow \infty \} |x|^\{1/3\} |u (x)| = 0$ uniformly. For slow flows the latter condition can be replaced by $\lim _\{|x| \rightarrow \infty \} |u (x)| = 0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.},
author = {Bildhauer, Michael, Fuchs, Martin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {equations of Navier-Stokes type; stationary case; exterior problem in 2D; power-law fluids; Navier-Stokes equation; stationary case; exterior problem in 2D},
language = {eng},
number = {2},
pages = {221-236},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity},
url = {http://eudml.org/doc/246524},
volume = {53},
year = {2012},
}

TY - JOUR
AU - Bildhauer, Michael
AU - Fuchs, Martin
TI - On the exterior problem in 2D for stationary flows of fluids with shear dependent viscosity
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2012
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 53
IS - 2
SP - 221
EP - 236
AB - On the complement of the unit disk $B$ we consider solutions of the equations describing the stationary flow of an incompressible fluid with shear dependent viscosity. We show that the velocity field $u$ is equal to zero provided $u|_{\partial B} = 0$ and $\lim _{|x| \rightarrow \infty } |x|^{1/3} |u (x)| = 0$ uniformly. For slow flows the latter condition can be replaced by $\lim _{|x| \rightarrow \infty } |u (x)| = 0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.
LA - eng
KW - equations of Navier-Stokes type; stationary case; exterior problem in 2D; power-law fluids; Navier-Stokes equation; stationary case; exterior problem in 2D
UR - http://eudml.org/doc/246524
ER -

References

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  1. Bildhauer M., Fuchs M., 10.1007/s10231-008-0085-2, Ann. Mat. Pura Appl. 188 (2009), 467–496. Zbl1181.49035MR2512159DOI10.1007/s10231-008-0085-2
  2. Fuchs M., Liouville theorems for stationary flows of shear thickening fluids in the plane, J. Math. Fluid Mech. DOI 10.1007/s00021-011-0070-1. 
  3. Fuchs M., Seregin G.A., 10.1007/BFb0103751, Lecture Notes in Mathematics, 1749, Springer, Berlin-Heidelberg-New York, 2000. Zbl0964.76003MR1810507DOI10.1007/BFb0103751
  4. Fuchs M., Zhang G., 10.1007/s00526-011-0434-7, Calc. Var. 44 (2012), no. 1–2, 271–295. MR2898779DOI10.1007/s00526-011-0434-7
  5. Galdi G., An Introduction to the Mathematical Theory of the Navier-Stokes Equations Vol. I, Springer Tracts in Natural Philosophy, 38, Springer, Berlin-Heidelberg-New York, 1994. MR1284205
  6. Galdi G., An Introduction to the Mathematical Theory of the Navier-Stokes Equations Vol. II, Springer Tracts in Natural Philosophy, 39, Springer, Berlin-Heidelberg-New York, 1994. Zbl0949.35005MR1284206
  7. Galdi G., On the existence of symmetric steady-state solutions to the plane exterior Navier-Stokes problem for arbitrary large Reynolds number, Advances in Fluid Dynamics, Quad. Mat., 4, Aracne, Rome, (1999), 1–25. Zbl0948.35097MR1770187
  8. Giaquinta M., Modica G., Nonlinear systems of the type of stationary Navier-Stokes system, J. Reine Angew. Math. 330 (1982), 173–214. MR0641818
  9. Ladyzhenskaya O.A., The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, 1969. Zbl0184.52603MR0254401
  10. Málek J., Nečas J., Rokyta M., Růžička M., Weak and Measure Valued Solutions to Evolutionary PDEs, Chapman & Hall, London, 1996. Zbl0851.35002MR1409366

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