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Energy Cascades and Flux Locality in Physical Scales of the 3D Navier-Stokes Equations

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Abstract

Rigorous estimates for the total – (kinetic) energy plus pressure – flux in \({\mathbb{R}^3}\) are obtained from the three dimensional Navier-Stokes equations. The bounds are used to establish a condition – involving Taylor length scale and the size of the domain – sufficient for existence of the inertial range and the energy cascade in decaying turbulence (zero driving force, non-increasing global energy). Several manifestations of the locality of the flux under this condition are obtained. All the scales involved are actual physical scales in \({\mathbb{R}^3}\) and no regularity or homogeneity/scaling assumptions are made.

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References

  1. Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35(6), 771–831 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Cheskidov A., Constantin P., Friedlander S., Shvydkoy R.: Energy conservation and Onsager’s conjecture for the Euler equations. Nonlinearity 21(6), 1233–1252 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Constantin P., E W., Titi E.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equations. Communi. Math. Phys. 165(1), 207–209 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Constantin P., Foias C.: Navier-Stokes equations. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press, 1988

  5. Eyink G.L.: Locality of turbulent cascades. Phys. D 207(1–2), 91–116 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Eyink G.L., Sreenivasan K.R.: Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78(1), 87–135 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Foias C., Manley O., Rosa R., Temam R.: Navier-Stokes equations and turbulence, Volume 83 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press (2001)

  8. Foias C., Manley O.P., Rosa R.M.S., Temam R.: Estimates for the energy cascade in three-dimensional turbulent flows. C. R. Acad. Sci. Paris Sér. I Math. 333(5), 499–504 (2001)

    MathSciNet  ADS  MATH  Google Scholar 

  9. Frisch U.: Turbulence. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  10. Kolmogorov A.N.: Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (1941)

    MATH  Google Scholar 

  11. Kolmogorov A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 9–13 (1941)

    Google Scholar 

  12. Kolmogorov A.N.: On generation of isotropic turbulence in an incompressible viscous liquid. Dokl. Akad. Nauk SSSR 31, 538–540 (1941)

    Google Scholar 

  13. Kraichnan R.H.: Inertial-range transfer in two- and three-dimensional turbulence. J. Fluid Mech. 47, 525–535 (1971)

    Article  ADS  MATH  Google Scholar 

  14. Lemarié-Rieusset P.G.: Recent developments in the Navier-Stokes problem, Volume 431 of Chapman & Hall/CRC Research Notes in Mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2002

  15. L’vov V., Falkovich G.: Counterbalanced interaction locality of developed hydrodynamic turbulence. Phys. Rev. A 46(8), 4762–4772 (1992)

    Article  ADS  Google Scholar 

  16. Onsager L.: Statistical hydrodynamics. Nuovo Cimento (9) 6(Supplemento, 2(Convegno Internazionale di Meccanica Statistica)), 279–287 (1949)

    Article  MathSciNet  Google Scholar 

  17. Scheffer V.: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55(2), 97–112 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Temam R.: Navier-Stokes equations. AMS Chelsea Publishing, Providence, RI (2001)

    Google Scholar 

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  1. Department of Mathematics, University of Virginia, Charlottesville, VA, 22904, USA

    R. Dascaliuc & Z. Grujić

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  1. R. Dascaliuc
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  2. Z. Grujić
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Correspondence to Z. Grujić.

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Communicated by P. Constantin

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Dascaliuc, R., Grujić, Z. Energy Cascades and Flux Locality in Physical Scales of the 3D Navier-Stokes Equations. Commun. Math. Phys. 305, 199–220 (2011). https://doi.org/10.1007/s00220-011-1219-8

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  • DOI: https://doi.org/10.1007/s00220-011-1219-8

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