Journal of Mathematical Fluid Mechanics

, Volume 17, Issue 3, pp 513–532 | Cite as

Infinite Energy Solutions for Dissipative Euler Equations in \({\mathbb{R}^2}\)

Open Access
Article

Abstract

We study the system of Euler equations with the so-called Ekman damping in the whole 2D space. The global well-posedness and dissipativity for the weak infinite energy solutions of this problem in the uniformly local spaces is verified based on the further development of the weighted energy theory for the Navier–Stokes and Euler type problems. In addition, the existence of weak locally compact global attractor is proved and some extra compactness of this attractor is obtained.

Mathematics Subject Classification

35Q30 35Q35 

Keywords

Euler equations Ekman damping infinite energy solutions weighted energy estimates unbounded domains 

References

  1. 1.
    Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. Nauka, Moscow (1989) (North Holland, Amsterdam 1992)Google Scholar
  2. 2.
    Ball J.: Global attractors for damped semilinear wave equations. Partial Differ. Equ. Appl. Discrete Contin. Dyn. Syst. 10(1-2), 31–52 (2004)CrossRefGoogle Scholar
  3. 3.
    Barcilon V., Constantin P., Titi E.S.: Existence of solutions to the Stommel–Charney model of the gulf stream. SIAM J. Math. Anal. 19(6), 1355–1364 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bardos C., Titi E.S.: Euler equations for incompressible ideal fluids. Russian Math. Surv. 62(3), 409–451 (2007)MathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Bessaih H., Flandoli F.: Weak attractor for a dissipative Euler equation. J. Dynam. Differ. Equ. 12(4), 713–732 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brull S., Pareschi L.: Dissipative hydrodynamic models for the diffusion of impurities in a gas. Appl. Math. Lett. 19, 516–521 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Chepyzhov V., Vishik M., Zelik S.: Strong trajectory attractors for dissipative Euler equations. J. Math. Pures Appl. (9) 96(4), 395–407 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    DiPerna R., Lions P.: Ordinary differential equations, Sobolev spaces and transport theory. Invent. Math. 98, 511–547 (1989)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Efendiev M., Miranville A., Zelik S.: Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proc. Roy. Soc. Edinb. Sect., A 134(2), 271–315 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Efendiev M., Zelik S.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Commun. Pure Appl. Math. 54(6), 625–688 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gallay, T., Slijepcevic, S.: Uniform boundedness and long-time asymptotics for the two-dimensional Navier–Stokes equations in an infinite cylinder. J. Math. Fluid Mech. 17, 23–46 (2015)Google Scholar
  13. 13.
    Gallay, T.: Infinite energy solutions of the two-dimensional Navier–Stokes equations, preprint arXiv:1411.5156 (2014)
  14. 14.
    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)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Ilyin, A.A.: The Euler equations with dissipation, Mat. Sb. 182 (12), 1729–1739 (1991) [Sb. Math. 74 (2), 475–485 (1993)]Google Scholar
  16. 16.
    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)MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    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)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ioffe A.: On lower semicontinuity of integral functionals I. SIAM J. Control Optim. 15, 521–538 (1977)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kelliher, D., Filho, M., Lopes, H.: Serfati solutions to the 2D Euler equations on exterior domains, preprint, arXiv:1401.2655 (2014)
  20. 20.
    Lemarie-Rieusset, P.: Recent developments in the Navier–Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, vol. 431. Chapman & Hall/CRC, Boca Raton, FL (2002)Google Scholar
  21. 21.
    Lions, P.-L.: Mathematical topics in fluid mechanics. vol. 1. Incompressible models, Oxford Lecture Ser. Math. Appl., vol. 3, Clarendon Press, Oxford (1996)Google Scholar
  22. 22.
    Lions J.L.: Quelques Méthodes de Résolutions des Problèmes aux Limites Non linéaires. Dunod et Gauthier-Villars, Paris (1969)Google Scholar
  23. 23.
    Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam (2008)Google Scholar
  24. 24.
    Moise I., Rosa R., Wang X.: Attractors for non-compact semigroups via energy equations. Nonlinearity 11(5), 1369–1393 (1998)MathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Pedlosky J.: Geophysical Fluid Dynamics. Springer, New York (1979)CrossRefGoogle Scholar
  26. 26.
    Pennant J., Zelik S.: Global well-posedness in uniformly local spaces for the Cahn–Hilliard equation in \({\mathbb{R}^3}\). Commun. Pure Appl. Anal. 12(1), 461–480 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Robertson, A., Robertson, W.: Topological vector spaces. Reprint of the second edition. Cambridge Tracts in Mathematics, vol. 53. Cambridge University Press, Cambridge-New York (1980)Google Scholar
  28. 28.
    Sawada O., Taniuchi Y.: A remark on L -solutions to the 2D Navier–Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)MathSciNetCrossRefADSGoogle Scholar
  29. 29.
    Serfati P.: Solutions C 1 en temps, n-log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris Sér. I Math. 320(5), 555–558 (1995)MathSciNetGoogle Scholar
  30. 30.
    Temam R.: Navier–Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam New York-Oxford (1977)Google Scholar
  31. 31.
    Yudovich V.I.: Non-stationary flow of an ideal incompressible fluid. Zh. Vychisl. Mat. Mat. Fiz. 3, 1032–1066 (1963)Google Scholar
  32. 32.
    Yudovich V.I.: Some bounds for solutions of elliptic equations. Mat. Sb. (N.S.) 59, 229–244 (1962)MathSciNetGoogle Scholar
  33. 33.
    Zelik S.: Spatially nondecaying solutions of the 2D Navier–Stokes equation in a strip. Glasg. Math. J. 49(3), 525–588 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zelik, S.: Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II, 255–327, Int. Math. Ser. (N. Y.), 7, Springer, New York (2008)Google Scholar
  35. 35.
    Zelik S.: Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Commun. Pure Appl. Math. 56(5), 584–637 (2003)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zelik S.: Infinite energy solutions for damped Navier–Stokes equations in \({\mathbb{R}^2}\). J. Math. Fluid Mech. 15, 717–745 (2013)MathSciNetCrossRefADSGoogle Scholar

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© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems RASMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.Department of MathematicsUniversity of SurreyGuildfordUK

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