Singularities of Euler flow? Not out of the blue!

Journal of Statistical Physics

Volumn 113, Issue 5-6, 2003, Pages 761-781

  Frisch, U ^a,b   Matsumoto, T ^a,b,c   Bec, J ^a,b  

^a UNIVERSITÉ CÔTE D'AZUR   (France)

^b LOS ALAMOS NATIONAL LABORATORY   (United States)

^c KYOTO UNIVERSITY   (Japan)

Author keywords

Blow up; Complex singularities; Euler equation; Turbulence

Indexed keywords

EID: 3543151404     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1027308602344     Document Type: Conference Paper

References (59)

1. L. Onsager, Statistical hydrodynamics, Nuovo Cimento 6:279-287 (Suppl. Ser. IX) (1949).
2. U. Frisch, Fully developed turbulence and singularities, in Les Houches 1981: Chaotic Behavior of Deterministic Systems, G. Iooss, R. Hellemann, and R. Stora, eds. (North-Holland, Amsterdam, 1983).
3. S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15:1-89 (1943).
4. U. Frisch, A. Pouquet, P.-L. Sulem, and M. Meneguzzi, The dynamics of two-dimensional ideal MHD, J. Méc Théor. Appliqu. (Paris), 191-216 (Special issue on two-dimensional turbulence) (1983).
5. M.-E. Brachet, M. Meneguzzi, A. Vincent, H. Politano, and P.-L. Sulem, Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows, Phys. Fluids 4:2845-2854 (1992).
6. U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
7. P. Constantin, An Eulerian-Lagrangian approach for incompressible fluids: Local theory, J. Amer. Math. Soc. 14:263-277 (2000).
8. A. J. Majda and A. L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2001).
9. W. Wolibner, Un théorème sur l'existence du mouvement plan d'un fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z 37:698-726 (1933).
10. E. Hölder, Über die unbeschränkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegretzen inkompressiblen Flüssigkeit, Math. Z 37:727-738 (1933).
11. L. Lichtenstein, Über einige Existenzätze der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze, Math. Z 23:89-154; 309-316 (1925).
12. J. T. Beale, T. Kato, and A. J. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94:61-66 (1985).
13. C. Fefferman, Existence and smoothness of the Navier-Stokes equations, www.claymath.org/prizeproblems/navierstokes.htm (2000).
14. S. A. Orszag, Statistical theory of turbulence, in Les Houches 1973: Fluid Dynamics, R. Balian and J. L. Peube, eds. (Gordon & Breach, New York, 1977).
15. M.-E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch, Small-scale structure of the Taylor-Green vortex, J. Fluid. Mech. 130:411-452 (1983).
16. S. Tanveer and C. G. Speziale, Singularities of the Euler equation and hydrodynamic stability, Phys. Fluids A 5:1456-1465 (1993).
17. A. Bhattarcharjee, C. S. Ng, and Xiaogang Wang, Finite-time singularity and Kolmogorov spectrum in a symmetric three-dimensional spiral model, Phys. Rev. E 52:5110-5123 (1995).
18. C. Bardos, S. Benachour, and M. Zerner, Analyticité des solutions periodiques de l'équation d'Euler en deux dimensions, C. R. Acad. Sci. Paris A 282:995-998 (1976).
19. C. Goulaouic and M. S. Baouendi, Cauchy problem for analytic pseudo-differential operators, Comm. Partial Differential Equations 1:135-189 (1976).
20. S. Benachour, Analyticité des solutions de l'équation d'Euler en trois dimensions, C. R. Acad. Sci. Paris A 283:107-110 (1976).
21. 3, Ann. Scuola Norm. Sup. Pisa 4:647-687 (1977).
22. S. Benachour, Analyticity of solutions of Euler equations, Arch. Rat. Mech. Anal. 71:271-299 (1979).
23. p, J. Funct. Anal. 152:447-466 (1998).
24. H. A. Rose and P.-L. Sulem, Fully developed turbulence and statistical mechanics, J. Phys. France 39:441-484 (1978).
25. C. Sulem, P.-L. Sulem, and H. Frisch, Tracing complex singularities with spectral methods, J. Comput. Phys. 50:138-161 (1983).
26. G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable: Theory and Technique (McGraw-Hill, New York, 1966).
27. U. Frisch and R. Morf, Intermittency in nonlinear dynamics and singularities at complex times, Phys. Rev. A 23:2673-2705 (1981).
28. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods (SIAM, Philadelphia, 1977).
29. R. Grauer and T. C. Sideris, Numerical computation of 3D incompressible ideal fluids with swirl, Phys. Rev. Lett. 67:3511-3514 (1991).
30. Finite time singularities in ideal fluids with swirl, Physica D 88:116-132 (1995).
31. A. Pumir and E. D. Siggia, Development of singular solutions to the axisymmetric Euler equations, Phys. Fluids A 4:1472-1491 (1992).
32. W. E and C.-W. Shu, Small-scale structures in Boussinesq convection, Phys. Fluids 6:49 (1994).
33. R. E. Caflisch, N. Ercolani, and G. Steele, Geometry of Singularities for the Steady Boussinesq Equations, UCLA, CAM Report 95-22 (1995).
34. R. Krasny, A study of singularity formation in a vortex sheet by the point-vortex approximation, J. Fluid Mech. 167:65-93 (1986).
35. U. Frisch, The analytic structure of turbulent flows, in Proceed. Chaos and Statistical Methods, Sept. 1983, Kyoto, Y. Kuramoto, ed. (Springer, 1984), pp. 211-220.
36. D. Bessis and J.-D. Fournier, Pole condensation and the Riemann surface associated with a shock in Burgers' equation, J. Phys. Lett. (Paris) 45:L833-L841 (1984).
37. D. Bessis and J.-D. Fournier, Complex singularities and the Riemann surface for the Burgers equation, in Nonlinear Physics, Proceedings of the International Conference, Shangai, April 24-30, 1989, Gu Chaohao, Li Yishen, and Tu Guizhan, eds. (Springer, Berlin, 1990), pp. 252-257.
38. E. Behr, J. Necas, and H. Wu, On blowup of solutions for Euler equations, Math. Model. and Numer. Anal. 35:229-238 (2001).
39. J. B. Bell and D. L. Marcus, Vorticity intensification and transition to turbulence in the three-dimensional Euler equations, Comm. Math. Phys. 147:371-394 (1992).
40. M.-E. Brachet, Direct simulation of three-dimensional turbulence in the Taylor-Green vortex, Fluid Dyn. Res. 8:1-8 (1991).
41. R. E. Caflisch and G. C. Papanicolaou, eds., Singularities in fluids, plasmas and optics, NATO Series C, Vol. 404 (Kluwer, Dordrecht, 1993).
42. P. Constantin, Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Comm. Math. Phys. 104:311-329 (1986).
43. P. Constantin, Geometric statistics in turbulence, SIAM Rev. 36:73-98 (1994).
44. P. Constantin, C. Fefferman, and A. J. Majda, Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations 21:559-571 (1996).
45. C. R. Doering and J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations (Cambridge University Press, Cambridge, 1995).
46. J. D. Gibbon and M. Heritage, Angular dependence and growth of vorticity in the three-dimensional Euler equations, Phys. Fluids 9:901-909 (1997).
47. R. Grauer, C. Marliani, and K. Germaschewski, Adaptive mesh refinement for singular solutions of the incompressible Euler equations, Phys. Rev. Lett. 80:4177-4180 (1998).
48. 3, J. Funct. Anal. 9:296-305 (1972).
49. R. M. Kerr, Evidence for a singularity of the three-dimensional incompressible Euler equations, Phys. Fluids A 5:1725-1746 (1993).
50. J. Gibbon, B. Galanti, and R. M. Kerr, Stretching and compression of vorticity in the 3-D Euler equations, in Turbulence Structure and Vortex Dynamics, J. C. R. Hunt and J. C. Vassilicos, eds. (Cambridge University Press, Cambridge, 2000).
51. S. Kida, Three-dimensional periodic flows with high-symmetry, J. Phys. Soc. Japan 54:2132-2136 (1985).
52. A. J. Majda, Vorticity, turbulence and acoustics in fluid flow, SIAM Rev. 33:349-388 (1991).
53. C. Marchioro and M. Pulvirenti, The Mathematical Theory of Incompressible Nonviscous Fluids (Springer, New York, 1991).
54. D. W. Moore, The spontaneous appearance of a singularity in the shape of an evolving vortex sheet, Proc. Roy. Soc. A 365:105-119 (1979).
55. K. Ohkitani and J. D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier-Stokes flows, Phys. Fluids 12:3181-3194 (2000).
56. R. B. Pelz and O. N. Boratav, On a possible Euler singularity during transition in a high-symmetry flow, in Small-Scale Structures in Three-Dimensional Hydro and Magneto Hydrodynamic Turbulence, M. Meneguzzi, A. Pouquet, and P.-L. Sulem, eds., Lecture Notes in Physics, Vol. 462 (Springer, Berlin, 1995), pp. 25-32.
57. R. B. Pelz, Symmetry and the hydrodynamic blow-up problem, J. Fluid Mech. 444:299-320 (2001).
58. G. Ponce, Remark on a paper by J. T. Beale, T. Kato, and A. Majda, Comm. Math. Phys. 98:349-353 (1985).
59. G. I. Taylor and A. E. Green, Mechanism of the production of small eddies from large ones, Proc. Roy. Soc. A 158:499-521 (1937).