# On Convergence of Arbitrary *D*-Solution of Steady Navier–Stokes System in 2*D* Exterior Domains

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## Abstract

We study solutions to stationary Navier–Stokes system in a two dimensional exterior domain. We prove that any such solution with a finite Dirichlet integral converges to a constant vector at infinity uniformly. No additional conditions (on symmetry or smallness, etc.) are assumed. In the proofs we develop the ideas of the classical papers of Gilbarg and Weinberger (Ann Sc Norm Pisa (4) 5:381–404, 1978) and Amick (Acta Math 161:71–130, 1988).

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## Notes

### Acknowledgements

The authors are grateful to the anonymous referee for many useful remarks and suggestions. M. Korobkov was partially supported by the Ministry of Education and Science of the Russian Federation (Grant 14.Z50.31.0037) and by the Russian Federation for Basic Research (Project Numbers 18-01-00649 and 17-01-00875). The research of K. Pileckas was funded by the Grant No. S-MIP-17-68 from the Research Council of Lithuania.

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### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Amick C.J.: On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math.
**161**, 71–130 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Amick C.J.: On the asymptotic form of Navier–Stokes flow past a body in the plane. J. Differ. Equ.
**91**, 149–167 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.Engelking R.: General Topology, 2nd edn. Sigma Series in Pure Mathematics, Vol. 6. Heldermann Verlag. Berlin (1989)Google Scholar
- 4.Finn R., Smith D.R.: On the stationary solutions of the Navier–Stokes equations in two dimensions. Arch. Ration. Mech. Anal.
**25**, 26–39 (1967)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Fujita H.: On the existence and regularity of the steady-state solutions of the Navier–Stokes equation. J. Fac. Sci. Univ. Tokyo(1A)
**9**, 59–102 (1961)zbMATHGoogle Scholar - 6.Galdi, G.P.:
*An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady–State Problems*. Springer, 2011Google Scholar - 7.Galdi G.P., Sohr H.: On the asymptotic structure of plane steady flow of a viscous fluid in exterior domains. Arch. Ration. Mech. Anal.
**131**, 101–119 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Gilbarg D., Weinberger H.F.: Asymptotic properties of Leray’s solution of the stationary two–dimensional Navier–Stokes equations. Russ. Math. Surv.
**29**, 109–123 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Gilbarg D., Weinberger H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Sc. Norm. Pisa (4)
**5**, 381–404 (1978)MathSciNetzbMATHGoogle Scholar - 10.Korobkov M.V., Pileckas K., Russo R.: The existence of a solution with finite Dirichlet integral for the steady Navier–Stokes equations in a plane exterior symmetric domain. J. Math. Pures Appl.
**101**, 257–274 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Korobkov, M.V., Pileckas, K., Russo, R.: On the steady Navier–Stokes equations in 2D exterior domains, arXiv:1711.02400
- 12.Ladyzhenskaia, O.A.:
*The Mathematical Theory of Viscous Incompressible Fluid*. Gordon and Breach, 1969Google Scholar - 13.Leray J.: Étude de diverses équations intégrales non linéaire et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl.
**12**, 1–82 (1933)zbMATHGoogle Scholar - 14.Malý J., Swanson D., Ziemer W.P.: The Coarea formula for Sobolev mappings. Trans. AMS
**355**(2), 477–492 (2002)CrossRefzbMATHGoogle Scholar - 15.Pileckas K., Russo R.: On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem. Math. Ann.
**352**, 643–658 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Russo A.: A note on the two–dimensional steady-state Navier–Stokes problem. J. Math. Fluid Mech.,
**11**, 407–414 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 17.Sazonov, L.I.: On the asymptotic behavior of the solution of the two-dimensional stationary problem of the flow past a body far from it. (Russian)
*Mat. Zametki***65**, 246–253, 1999; translation in*Math. Notes***65**, 246–253, 1999Google Scholar