Communications in Mathematical Physics

, Volume 269, Issue 3, pp 693–713 | Cite as

On the Global Evolution of Vortex Filaments, Blobs, and Small Loops in 3D Ideal Flows



We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable Hamiltonian functions. The approximate models we analyze (essentially discrete and continuous vortex filaments and vortex loops) are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high Reynolds number. Finally, we apply our strategy to discrete models for filaments used in numerical methods.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams R.A. (1975) Sobolev spaces. Pure and Applied Mathematics, Vol. 65. New York-London, Academic PressGoogle Scholar
  2. 2.
    Beale J.T., Kato T., Majda A. (1984) Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94(1):61–66MATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Bell J., Markus D. (1992) Vorticity intensification and the transition to turbulence in the three-dimensional Euler equation. Commun. Math. Phys. 147(2):371–394MATHCrossRefADSGoogle Scholar
  4. 4.
    Berselli L.C., Bessaih H. (2002) Some results for the line vortex equation. Nonlinearity 15(6):1729–1746MATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Bessaih, H., Flandoli, F.: A mean field result with application to 3D vortex filaments. In: Probabilistic methods in fluids, River Edge, NJ: World Sci. Publishing, 2003, pp. 22–34Google Scholar
  6. 6.
    Bessaih H., Gubinelli M., Russo F.(2005) The evolution of a random vortex filament. Ann. Prob. 33(5):1825–1855MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Buttke, T.F.: Velicity methods: Lagrangian numerical methods which preserve the Hamiltonian structure of incompressible fluid flow. In: Vortex flows and related numerical methods (Grenoble, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 395, Kluwer Acad. Publ., Dordrecht. 1993, pp. 39–57Google Scholar
  8. 8.
    Buttke T.F. (1988) Numerical study of superfluid turbulence in the self-induction approximation. J. Comput. Phys. 76, 301MATHCrossRefADSGoogle Scholar
  9. 9.
    Chorin A.J. (1994) Vorticity and turbulence. New York, Springer-VerlagMATHGoogle Scholar
  10. 10.
    Constantin P. (1994) Geometric statistics in turbulence. SIAM Rev. 36(1):73–98MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Constantin, P.: Near identity transformations for the Navier-Stokes equations. In: Handbook of mathematical fluid dynamics, Vol. II, Amsterdam: North-Holland, 2003, pp. 117–141Google Scholar
  12. 12.
    Constantin P., Majda A.J., Tabak E. (1994) Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7(6):1495–1533MATHMathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Córdoba A., Córdoba D. (2004) A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249(3):511–528MATHCrossRefADSGoogle Scholar
  14. 14.
    Córdoba A., Córdoba D., Fefferman C.L., Fontelos M.A. (2004) A geometrical constraint for capillary jet breakup. Adv. Math. 187(1):228–239MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Cottet G.-H., Koumoutsakos P.D. (2002) Vortex methods, theory and practice. Cambridge, Cambridge, Univ. PressMATHGoogle Scholar
  16. 16.
    Flandoli F. (2002) A probabilistic description of small scale structures in 3D fluids. Ann. Inst. H. Poincaré Probab. Statist. 38(2):207–228MATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Flandoli F., Gubinelli M. (2002) Gibbs ensembles of vortex filaments. Probab. Theory Related Fields 22(3):317–340MathSciNetCrossRefGoogle Scholar
  18. 18.
    Frisch U. (1995) Turbulence. The legacy of A.N. Kolmogorov. Cambridge, Cambridge Univ PressMATHGoogle Scholar
  19. 19.
    Gallavotti, G.: Foundations of fluid dynamics. Translated from the Italian. Texts and Monographs in Physics. Berlin: Springer-Verlag, 2002Google Scholar
  20. 20.
    Hasimoto H. (1972) A soliton on a vortex filament. J. Fluid. Mech. 51, 477–485MATHCrossRefADSGoogle Scholar
  21. 21.
    Helmholtz H. (1885) Uber integrale der hydrodynamischen gleichungen welche den Wirbelbewegungen entsprechen. Crelle J. 55, 25Google Scholar
  22. 22.
    Holm D.D. (2003) Rasetti-Regge Dirac bracket formulation of Lagrangian fluid dynamics of vortex filaments. Nonlinear waves: computation and theory, II (Athens, GA, 2001). Math. Comput. Simulation 62(1-2):53–63MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lord Kelvin (Sir William Thomson),: On vortex motion. Trans. Royal Soc. Edin. 25, 217–260 (1869)Google Scholar
  24. 24.
    Klein R., Majda A.J. (1991) Self-stretching of a perturbed vortex filament. I. The asymptotic equation for deviations from a straight line. Phys. D 49(3):323–352MATHMathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Klein R., Majda A.J. (1991) Self-stretching of a perturbed vortex filament. II. Structure of solutions. Phys. D 53(2–4):267–294MATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Lions P.L. (1997) On Euler equations and statistical physics. Scuola Normale Superiore, PisaMATHGoogle Scholar
  27. 27.
    Lions P.L., Majda A.J. (2000) Equilibrium statistical theory for nearly parallel vortex filaments. Comm. Pure Appl. Math. 53(1):76–142MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Lyons T.J. (1998) Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14(2):215–310MATHMathSciNetGoogle Scholar
  29. 29.
    Lyons T.J., Qian Z. (2002) System control and rough paths. Oxford Mathematical Monographs. Oxford, Oxford Univ. PressGoogle Scholar
  30. 30.
    Marsden, J.E., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Order in chaos (Los Alamos, N.M., 1982). Phys. D 7(1-3), 305–323 (1983)Google Scholar
  31. 31.
    Moore D.W. (1972) Finite amplitude waves on aircraft trailing vortices. Aero. Quarterly 23, 307–314Google Scholar
  32. 32.
    Roberts P. (1972) A Hamiltonian theory for weakly interacting vortices. Mathematika 19, 169–179MATHCrossRefGoogle Scholar
  33. 33.
    Rosenhead L. (1930) The spread of vorticity in the wake behind a cylinder. Proc. Royal Soc. 127, 590–612MATHADSGoogle Scholar
  34. 34.
    Saffman P.G. (1992) Vortex dynamics. Cambridge, Cambridge Univ. PressMATHGoogle Scholar
  35. 35.
    Osedelets V.I. (1988) On a new way of writing the Navier-Stokes equation: the Hamiltonian formalism. Russ. Math. Surv. 44, 210–211CrossRefGoogle Scholar
  36. 36.
    Vincent A., Meneguzzi M. (1991) The spatial structure and the statistical properties of homogeneous turbulence. J. Fluid. Mech. 225(1):1–25MATHCrossRefADSGoogle Scholar
  37. 37.
    Wolibner W. (1933) Un théoreme sur l’existence du mouvement plan d’un fluide parfait homogène incompressible, pendant un temps infiniment longue. Math. Z. 37, 698–726MATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Young L.C. (1936) An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. Comput. Math. & Math. Phys. 3, 1407–1456 (1963) (Russian)Google Scholar
  40. 40.
    Zhou H. (1997) On the motion of a slender vortex filament. Phys. Fluids 9, 970–981MathSciNetCrossRefADSMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Dipartimento di Matematica Applicata “U.Dini”Università di PisaPisaItaly

Personalised recommendations