SpringerLink
Log in

A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier–Stokes equations in 2D

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|−α|u(x)| < ∞ as |x| → ∞ for some α ∈ [0, 1/7).† Bibliography: 10 titles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach (1969).

  2. G. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. I, Springer, Berlin etc. (1994).

    Book  Google Scholar 

  3. G. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. II, Springer, Berlin etc. (1994).

    Book  Google Scholar 

  4. D. Gilbarg and H. F. Weinberger, “Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral,” Ann. S.N.S. Pisa (4), 5, 381–404 (1978).

    MathSciNet  MATH  Google Scholar 

  5. G. Koch, Liouville Theorem for 2D Navier–Stokes Equations. Preprint.

  6. G. Koch, N. Nadirashvili, G. Seregin, and V. Sverǎk, “Liouville theorems for the Navier-Stokes equations and applications,” Acta Math. 203, 83–105 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Fuchs, “Liouville theorems for stationary flows of shear thickening fluids in the plane,” J. Math. Fluid Mech. DOI 10.1007/s00021-011-0070-1.

  8. M. Fuchs and G. Zhang, “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. DOI 10.1007/s00526-011-0434-7.

  9. M. Giaquinta and G. Modica, “Nonlinear systems of the type of stationary Navier–Stokes system,” J. Reine Angew. Math. 330, 173–214 (1982).

    MathSciNet  MATH  Google Scholar 

  10. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin etc. (1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität des Saarlandes, Fachbereich 6.1 Mathematik, Postfach 15 11 50, D–66041, Saarbrücken, Germany

    M. Fuchs

  2. University of Jyväskylä, P.O.Box 35(MaD), FI-40014, Jyväskylä, Finland

    X. Zhong

Authors
  1. M. Fuchs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Fuchs.

Additional information

† Actually, the case α < 1/3 is sufficient (cf. Note Added in Proof below).

Translated from Problems in Mathematical Analysis 60, September 2011, pp. 111–118.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fuchs, M., Zhong, X. A note on a Liouville type result of Gilbarg and Weinberger for the stationary Navier–Stokes equations in 2D . J Math Sci 178, 695–703 (2011). https://doi.org/10.1007/s10958-011-0578-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-011-0578-1

Keywords

Search

Navigation