Skip to main content

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

  • Chapter
Instability in Models Connected with Fluid Flows I

Part of the book series: International Mathematical Series ((IMAT,volume 6))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.

    Google Scholar 

  2. V. I. Arnold, Small denominators. I. Mappings of the circumference onto itself, Am. Math. Soc. Transl. Ser. 2 46 (1965), p. 213-284.

    Google Scholar 

  3. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer, 1997.

    Google Scholar 

  4. A. Babin, A. Mahalov, and B. Nicolaenko, Global splitting, integrability and regularity of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, Eur. J. Mech. B 15 (1996), 291-300.

    MATH  MathSciNet  Google Scholar 

  5. A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity and integrability of 3D Euler and Navier–Stokes equations for uniformly rotating fluids, Asymptotic Anal. 15 (1997), 103–150.

    MATH  MathSciNet  Google Scholar 

  6. A. Babin, A. Mahalov, and B. Nicolaenko, Global regularity of 3D rotating Navier–Stokes equations for resonant domains, Indiana Univ. Math. J. 48 (1999), no. 3, 1133-1176.

    Google Scholar 

  7. A. Babin, A. Mahalov, and B. Nicolaenko, 3D Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J. 50 (2001), 1-35.

    MATH  MathSciNet  Google Scholar 

  8. J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys. 94 (1984), 61-66.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. S. Besicovitch, Almost Periodic Functions, Dover, New York, 1954.

    Google Scholar 

  10. N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach Sci. Publ., New York, 1961.

    Google Scholar 

  11. J. P. Bourguignon and H. Brezis, Remark on the Euler equations, J. Func. Anal. 15 (1974), 341-363.

    Article  MATH  MathSciNet  Google Scholar 

  12. Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, Intermittency in the joint cascade of energy and helicity, Phys. Rev. Letters 90 (2003), p. 214503.

    Article  Google Scholar 

  13. C. Corduneanu, Almost Periodic Functions, Wiley-Interscience, New York, 1968.

    MATH  Google Scholar 

  14. J. Deng, T. Y. Hou, and X. Yu, Geometric properties and nonblowup of 3D incompressible Euler flow, Commun. Partial Differ. Equations 30 (2005), no. 3, 225-243.

    Google Scholar 

  15. R. J. DiPerna and P. L. Lions, Ordinary differential equations, Sobolev spaces and transport theory, Invent. Math. 98 (1989), 511-547.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. L. Fefferman, Existence and smoothness of the Navier–Stokes equations, In: The Millennium Prize Problems, Clay Math. Inst., Cambridge, MA (2006), pp. 57-67.

    Google Scholar 

  17. U. Frisch, Turbulence: the Legacy of A. N. Kolmogolov, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  18. E. Frolova, A. Mahalov, and B. Nicolaenko, Restricted interactions and global regularity of 3D rapidly rotating Navier–Stokes equations in cylindrical domains, J. Math. Sci., New York. [To appear]

    Google Scholar 

  19. E. B. Gledzer, System of hydrodynamic type admitting two quadratic integrals of motion, Sov. Phys. Dokl. 18 (1973), 216-217.

    MATH  Google Scholar 

  20. E. B. Gledzer, F. V. Dolzhanskij, and A. M. Obukhov, Systems of Hydrodynamic Type and Their Application [in Russian], Nauka, Moscow, 1981.

    Google Scholar 

  21. F. Golse, A. Mahalov, and B. Nicolaenko, Infinite dimensional systems of coupled rigid bodies asymptotic to the 3D Euler equations. [In preparation]

    Google Scholar 

  22. J. Guckenheimer and A. Mahalov, Resonant triad interaction in symmetric systems, Physica D 54 (1992), 267-310.

    Article  MATH  MathSciNet  Google Scholar 

  23. T. Y. Hou and R. Li, Dynamic depletion of vortex stretching and non-blowup of the 3D incompressible Euler equations, J. Nonlinear Sci. 16 (2006), 639–664.

    Article  MATH  MathSciNet  Google Scholar 

  24. T. Kato, Nonstationary flows of viscous and ideal fluids inR3, J. Func. Anal. 9 (1972), 296-305.

    Article  MATH  Google Scholar 

  25. R. M. Kerr, Evidence for a singularity of the three dimensional, incompressible Euler equations, Phys. Fluids 5 (1993), no. 7, 1725-1746.

    Google Scholar 

  26. M. Lesieur, Turbulence in Fluids, 2nd edition, Kluwer, Dortrecht, 1990.

    Google Scholar 

  27. P. L. Lions, Mathematical Topics in Fluid Mechanics: Incompressible Models, Vol 1, Oxford University Press, Oxford, 1998.

    MATH  Google Scholar 

  28. A. Mahalov, The instability of rotating fluid columns subjected to a weak external Coriolis force, Phys. Fluids A 5 (1993), no. 4, 891-900.

    Google Scholar 

  29. A. Mahalov, B. Nicolaenko, C. Bardos, and F. Golse, Non blow-up of the 3D Euler equations for a class of three-dimensional initial data in cylindrical domains, Methods Appl. Anal. 11 (2004), no. 4, 605-634.

    Google Scholar 

  30. S. V. Manakov, Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976), no. 4, 328-329.

    Google Scholar 

  31. J. J. Moreau, Une methode de cinematique fonctionelle en hydrodynamicque [in French], C. R. Acad. Sci. Paris 249 (1959), 2156-2158

    MATH  MathSciNet  Google Scholar 

  32. J. J. Moreau, Constantes d’un ilôt tourbillonaire en fluide parfait barotrope [in French], C. R. Acad. Sci. Paris 252 (1961), 2810-2812

    MATH  MathSciNet  Google Scholar 

  33. H. K. Moffatt, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 106 (1969), 117-129.

    Article  Google Scholar 

  34. H. Poincaré, Sur la précession des corps déformables [in French], Bull. Astronomique 27 (1910), 321-356.

    Google Scholar 

  35. S. L. Sobolev, On one new problem in mathematical physics [in Russian], Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), no. 1, 3–50.

    Google Scholar 

  36. S. M. Visik, On invariant characteristics of quadratically nonlinear systems of cascade type, Sov. Math. Dokl. 17 (1976), 895-899.

    MATH  Google Scholar 

  37. J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interactions of Waves in Plasmas, Pergamon, Oxford, 1977.

    Google Scholar 

  38. V. I. Yudovich, Non-stationary flow of an ideal incompressible liquid, U.S.S.R. Comput. Math. Math. Phys. 3 (1963), 1407-1456.

    Google Scholar 

  39. V. I. Yudovich, Uniqueness theorem for the basic nonstationary problem in the dynamics of an ideal incompressible fluid, Math. Res. Lett. 2 (1995), 27-38.

    MATH  MathSciNet  Google Scholar 

  40. V. E. Zakharov and S. V. Manakov, Resonant interactions of wave packets in nonlinear media, Sov. Phys. JETP Lett. 18 (1973), 243-245.

    Google Scholar 

  41. V. E. Zakharov and S. V. Manakov, The theory of resonance interaction of wave packets in nonlinear media, Sov. Phys. JETP 42 (1976), 842-850.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Golse, F., Mahalov, A., Nicolaenko, B. (2008). Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains. In: Bardos, C., Fursikov, A. (eds) Instability in Models Connected with Fluid Flows I. International Mathematical Series, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75217-4_7

Download citation

Publish with us

Policies and ethics