Journal of Mathematical Fluid Mechanics

, Volume 15, Issue 4, pp 717–745 | Cite as

Infinite Energy Solutions for Damped Navier–Stokes Equations in \({\mathbb{R}^2}\)

  • Sergey ZelikEmail author


We study the so-called damped Navier–Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier–Stokes type problems. Note that any divergent free vector field \({u_0 \in L^\infty(\mathbb{R}^2)}\) is allowed and no assumptions on the spatial decay of solutions as \({|x| \to \infty}\) are posed. In addition, applying the developed theory to the case of the classical Navier–Stokes problem in \({\mathbb{R}^2}\), we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.

Mathematics Subject Classification (2010)

35Q30 35Q35 


Navier–Stokes equations infinite-energy solutions weighted energy estimates unbounded domains 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Afendikov A., Mielke A.: Dynamical properties of spatially non-decaying 2D Navier–Stokes flows with Kolmogorov forcing in an infinite strip. J. Math. Fluid Mech. 7, 51–67 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989) (North Holland, Amsterdam, 1992)Google Scholar
  3. 3.
    Brull S., Pareschi L.: Dissipative hydrodynamic models for the diffusion of impurities in a gas. Appl. Math. Lett. 19, 516–521 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chepyzhov V.V., Vishik M.I.: Trajectory attractors for dissipative 2D Euler and Navier–Stokes equations. Russian J. Math. Phys. 15(2), 156–170 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Chepyzhov V., Vishik M., Zelik S.: Strong trajectory attractors for dissipative Euler equations. J. Math. Pures Appl., (9) 96(4), 395–407 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantin P., Ramos F.: Inviscid limit for damped and driven incompressible Navier–Stokes equations in \({\mathbb{R}^2}\). Comm. Math. Phys. 275(2), 529–551 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Efendiev M., Miranville A., Zelik S.: Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proc. Roy. Soc. Edinburgh Sect. A 134(2), 271–315 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Efendiev M., Zelik S.: . Comm. Pure Appl. Math. 54(6), 625–688 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fursikov A., Gunzburger M., Hou L.: Inhomogeneous boundary value problems for the three-dimensional evolutionary Navier–Stokes equations. J. Math. Fluid Mech. 4, 45–75 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Fursikov, A.: Flow of a viscous incompressible fluid around a body: boundary value problems and work minimization for a fluid. (Russian) Sovrem. Mat. Fundam. Napravl. 37, 83–130 (2010); translation in J. Math. Sci. (N. Y.) 180 (2012), no. 6, 763–816Google Scholar
  11. 11.
    Galdi G., Maremonti P., Zhou Y.: On the Navier Stokes problem in exterior domains with non decaying initial data. J. Math. Fluid Mech. 14, 633–652 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Giga Y., Matsui S., Sawada O.: Global existence of two-dimensional Navier–Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech. 3, 302–315 (2001)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Ilyin, A.A.: The Euler equations with dissipation. Mat. Sb. 182(12), 1729–1739 (1991)[Sb. Math. 74(2), 475–485 (1993)]Google Scholar
  14. 14.
    Ilyin A. A., Titi E. S.: Sharp estimates for the number of degrees of freedom of the damped-driven 2D Navier–Stokes equations. J. Nonlinear Sci. 16(3), 233–253 (2006)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Ilyin A.A., Miranville A., Titi E.S.: Small viscosity sharp estimates for the global attractor of the 2D damped-driven Navier–Stokes equations. Commun. Math. Sci. 2(3), 403–426 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lemarie-Rieusset, P.: Recent developments in the Navier–Stokes problem. In: Research Notes in Mathematics, vol. 431, Chapman & Hall, Boca Raton (2002)Google Scholar
  17. 17.
    Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of differential equations: evolutionary equations. Vol. IV, Handb. Differ. Equ., pp. 103–200. Elsevier, Amsterdam (2008)Google Scholar
  18. 18.
    Pedlosky J.: Geophysical Fluid Dynamics. Springer, New York (1979)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pennant, J., Zelik, S.: Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in \({\mathbb{R}^3}\). Comm. Pure Appl. Anal. 12(1), 461–480 (2013)Google Scholar
  20. 20.
    Sawada J., Taniuchi Y.: A remark on L -solutions to the 2D Navier–Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Temam, R.: Navier–Stokes equations, theory and numerical analysis. North-Holland, Amsterdam (1977)Google Scholar
  22. 22.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. In: Applied Mathematics Series, 2nd edn. Springer, Berlin (1988) (2nd edn., New York, 1997)Google Scholar
  23. 23.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)Google Scholar
  24. 24.
    Vishik, M.I., Fursikov, A.V.: Matematicheskie zadachi statisticheskoi gidromekhaniki. (Russian) [Mathematical problems of statistical hydromechanics] “Nauka”, Moscow, (1980) [English translation: Mathematical problems of statistical hydromechanics. Kluwer Academic publishers, Dorrend, Boston, London (1988)]Google Scholar
  25. 25.
    Zelik S.: Spatially nondecaying solutions of the 2D Navier–Stokes equation in a strip. Glasg. Math. J. 49(3), 525–588 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zelik, S. Weak spatially nondecaying solutions of 3D Navier–Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II. Int. Math. Ser. (N.Y.) 7, 255–327 (2008)Google Scholar
  27. 27.
    Zelik S.: Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm. Pure Appl. Math. 56(5), 584–637 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SurreyGuildfordUK

Personalised recommendations