Skip to main content
SpringerLink
Log in
Book cover

Turbulent Cascades II pp 71–81Cite as

The Exact Solution to the 3D Vortex Compressible Euler Equation and the Clay Millennium Problem Generalization

  • Conference paper
  • First Online:

  • 540 Accesses

Part of the book series: ERCOFTAC Series ((ERCO,volume 26))

Abstract

The general exact solution of the Cauchy problem to the 3D Euler vortex equation for compressible flow in unbound space is obtained. This solution has singularity at finite time and coincides with the vortex solution of the 3D Hopf equation for particles motion by inertia. A closed description of the evolution of enstrophy and all higher moments for the corresponding vortex field is established, giving an exact solution to the problems of closure in the theory of turbulence. On the base of this solution the smooth solution of the Navier-Stokes 3D equation for viscous compressible medium is obtained taking into account the effective viscosity and representation for the pressure field, which follows from the integral entropy balance equation, not from the medium equation of state. The above provides a positive solution for the Clay Millennium Problem (www.claymath.org) just in the case of its generalization on the Navier-Stokes equation for the compressible medium, for which an absence of smooth solutions on finite time interval has been a priori assumed before.

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

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 EPUB and 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

Learn about institutional subscriptions

References

  1. Euler, L.: Principes generaux du movement des fluids. Acad. Roy. Sci. et des Belles-Lettres de Berlin, Memoires 11, 217–273 (1757); also Physica D, 237, 1825–1839 (2008)

    Google Scholar 

  2. Darrigol, O., Frisch, U.: Physica D 237, 1855 (2008)

    Article  MathSciNet  Google Scholar 

  3. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, Oxford (1987)

    Google Scholar 

  4. Monin, A.S., Yaglom, A.M.: Statistical Hydromechanics, Theory of Turbulence, vol. 1, S.-Petersburg (1992)

    Google Scholar 

  5. Fefferman, C.L.: Existence and smoothness of the Navier Stokes equation. In: The Millennium Prize Problems, pp. 57–67. Clay Mathematics Institute, Cambridge, MA (2006). http://claymath.org/millennium/Navier-Stokes-Equation

  6. Hopf, G.E.: Comm. Pure. Appl. Math. 1, 303 (1948)

    Google Scholar 

  7. Arnold, V.I., Shandarin, S.F., Zeldovich, Ya.B.: Geophys. Astrophys. Fluid Dyn. 20, 111 (1982)

    Google Scholar 

  8. Shandarin, S.F.: Large Scale Structures of the Universe, p. 273. Audouze, J., Peletan, M.C., Szalay, A. (ed.) Kluwer Acad., Dordrecht (1988)

    Google Scholar 

  9. Shandarin, S.F., Zeldovich, Ya.B.: Rev. Mod. Phys. 61, 185 (1989)

    Article  MathSciNet  Google Scholar 

  10. Barrow, J.D., Saich, P.: Class. Quantum Grav. 10, 79 (1993)

    Article  MathSciNet  Google Scholar 

  11. Glasner, A., Livne, E., Meerson, B.: Phys. Rev. Lett. 78, 2112 (1997)

    Article  Google Scholar 

  12. Efrati, E., Livne, E., Meerson, B.: Phys. Rev. Lett. 94, 088001 (2005)

    Google Scholar 

  13. Kolvin, I., Livne, E., Meerson, B.: Phys. Rev. E 82, 021302 (2010)

    Google Scholar 

  14. Kuznetsov, E.A.: Physica D 184, 266 (2003)

    Google Scholar 

  15. Chefranov, S.G.: Sov. Phys. Dokl. 36(4), 286 (1991)

    Google Scholar 

  16. Chefranov, S.G., Chefranov, A.S.: Exact time-dependent solution to the three-dimensional Euler-Helmholtz and Riemann-Hopf equations for vortex flow of compressible medium and one of the Millennium prize problems. arXiv:1703.07239 v3 [physics.flu-dyn] 22 Mar 2017

  17. Krstulovic, G., Brachet, M.E.: Physica D 237, 2015 (2008)

    Google Scholar 

  18. Pelinovsky, E.N.: Izv. VUZov, Radiophys. 19, 373 (1976)

    Google Scholar 

  19. Chefranov, S.G.: JETP 96, 171 (1989)

    Google Scholar 

  20. Zybin, K.P., Sirota, V.A.: Uspekhi Fiz. Nauk 185, 593 (2015)

    Google Scholar 

Download references

Acknowledgements

We thank Itamar Procaccia, Pavel Lushnikov, Viktor Lvov and Gregory Falkovich for useful discussions and positive reaction on the results of this paper. The study is supported by RSF, project No. 14-17-00806P and Israel Science Foundation, Grant No. 492/18.

Author information

Authors and Affiliations

  1. A.M. Obukhov Institute of Atmospheric Physics Russian Academy of Sciences, Moscow, Russia

    S. G. Chefranov & A. S. Chefranov

  2. Physics Department, Technion-Israel Institute of Technology, Haifa, 32000, Israel

    S. G. Chefranov

Authors
  1. S. G. Chefranov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. S. Chefranov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. G. Chefranov .

Editor information

Editors and Affiliations

  1. Laboratoire de Mécanique des Fluides et d’Acoustique CNRS, Université de Lyon, École centrale de Lyon, Écully, France

    Mikhael Gorokhovski

  2. Laboratoire de Mécanique des Fluides et d’Acoustique CNRS, Université de Lyon, École centrale de Lyon, Écully, France

    Fabien S. Godeferd

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chefranov, S.G., Chefranov, A.S. (2019). The Exact Solution to the 3D Vortex Compressible Euler Equation and the Clay Millennium Problem Generalization. In: Gorokhovski, M., Godeferd, F. (eds) Turbulent Cascades II. ERCOFTAC Series, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-12547-9_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-12547-9_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-12546-2

  • Online ISBN: 978-3-030-12547-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation