# Article

Full entry | PDF   (0.5 MB)
Keywords:
equations of Navier-Stokes type; stationary case; exterior problem in 2D
Summary:
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 $\lim_{|x| \to \infty} |x|^{1/3} |u (x)| = 0$ uniformly. For slow flows the latter condition can be replaced by $\lim_{|x| \to \infty} |u (x)| = 0$ uniformly. In particular, these results hold for the classical Navier-Stokes case.
References:
[BF] Bildhauer M., Fuchs M.: Variational integrals of splitting type: higher integrability under general growth conditions. Ann. Mat. Pura Appl. 188 (2009), 467–496. DOI 10.1007/s10231-008-0085-2 | MR 2512159 | Zbl 1181.49035
[Fu] 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.
[FuSe] Fuchs M., Seregin G.A.: Variational methods for problems from plasticity theory and for generalized Newtonian fluids. Lecture Notes in Mathematics, 1749, Springer, Berlin-Heidelberg-New York, 2000. DOI 10.1007/BFb0103751 | MR 1810507 | Zbl 0964.76003
[FuZha] Fuchs M., Zhang G.: Liouville theorems for entire local minimizers of energies defined on the class $L \log L$ and for entire solutions of the stationary Prandtl-Eyring fluid model. Calc. Var. 44 (2012), no. 1–2, 271–295. DOI 10.1007/s00526-011-0434-7 | MR 2898779
[Ga1] 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. MR 1284205
[Ga2] 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. MR 1284206 | Zbl 0949.35005
[Ga3] 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. MR 1770187 | Zbl 0948.35097
[GM] Giaquinta M., Modica G.: Nonlinear systems of the type of stationary Navier-Stokes system. J. Reine Angew. Math. 330 (1982), 173–214. MR 0641818
[La] Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1969. MR 0254401 | Zbl 0184.52603
[MNRR] Málek J., Nečas J., Rokyta M., Růžička M.: Weak and Measure Valued Solutions to Evolutionary PDEs. Chapman & Hall, London, 1996. MR 1409366 | Zbl 0851.35002

Partner of