Infinite-Energy 2D Statistical Solutions to the Equations of Incompressible Fluids

Open Access
Article

Abstract

We develop the concept of an infinite-energy statistical solution to the Navier–Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier–Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by making a vanishing viscosity argument.

Keywords

Statistical solutions Navier–Stokes equations Euler equations 

Mathematics Subject Classification (2000)

76D06 76D05 

Notes

Acknowledgments

The author was supported in part by NSF grant DMS-0705586 during the period of this work.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Ben-Artzi M.: Global solutions of two-dimensional Navier-Stokes and Euler equations. Arch. Ration. Mech. Anal. 128(4), 329–358 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chemin J.-Y.: A remark on the inviscid limit for two-dimensional incompressible fluids. Comm. Partial Differ. Equat. 21(11–12), 1771–1779 (1996)MATHMathSciNetGoogle Scholar
  3. 3.
    Chemin, J.-Y.: Perfect Incompressible Fluids, vol. 14 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1998) (Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie)Google Scholar
  4. 4.
    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)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Foiaş C.: Statistical study of Navier-Stokes equations. I, II. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972)MathSciNetGoogle Scholar
  6. 6.
    Foiaş C.: Statistical study of Navier-Stokes equations. I, II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973)MATHGoogle Scholar
  7. 7.
    Foias C., Manley O., Rosa R., Temam R.: Navier-Stokes Equations and Turbulence, vol. 83 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2001)Google Scholar
  8. 8.
    Iftimie D., Kelliher J.P.: Remarks on the vanishing obstacle limit for a 3d viscous incompressible fluid. Proc. Am. Math. Soc. 137(2), 685–694 (2009)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kelliher J.P.: The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity. Math. Res. Lett. 11(4), 519–528 (2004)MATHMathSciNetGoogle Scholar
  10. 10.
    Kelliher J.P.: Expanding domain limit for incompressible fluids in the plane. Comm. Math. Phys. 278(3), 753–773 (2008)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kelliher, J.P., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Vanishing viscosity limit for an expanding domain in space. Ann. Inst. Henri Poincaré (C). doi: 10.1016/j.anihpc.2009.07.007
  12. 12.
    Kim, N.: Large friction limit and the inviscid limit of 2-D Navier-Stokes equations under Navier friction condition. SIAM Math. Anal., to appearGoogle Scholar
  13. 13.
    Temam, R.: Navier-Stokes Equations, Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI (2001) (Reprint of the 1984 edn)Google Scholar
  14. 14.
    Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3:1032–1066 (1963) (Russian)Google Scholar
  15. 15.
    Yudovich V.I.: Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid. Math. Res. Lett. 2(1), 27–38 (1995)MATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaRiversideUSA

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