Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1983–2005 | Cite as

An Algebraic Reduction of the ‘Scaling Gap’ in the Navier–Stokes Regularity Problem

  • Zachary Bradshaw
  • Aseel Farhat
  • Zoran GrujićEmail author


It is shown—within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components, and in the context of a blow-up-type argument—that the ever-resisting ‘scaling gap’, that is, the scaling distance between a regularity criterion and a corresponding a priori bound (shortly, a measure of the super-criticality of the three dimensional Navier–Stokes regularity problem), can be reduced by an algebraic factor; since (independent) fundamental works of Ladyzhenskaya, Prodi and Serrin as well as Kato and Fujita in 1960s, all the reductions have been logarithmic in nature, regardless of the functional set up utilized. More precisely, it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy-level bound, while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria. The mathematics presented was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows, as well as by the physics of turbulent cascades and turbulent dissipation.


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The work of A.F. is supported in part by the National Science Foundation Grant DMS-1418911. The work of Z.G. is supported in part by the National Science Foundation Grant DMS-1515805 and the Lundbeck Foundation Grant R217-2016-446 (a collaborative grant with the NBI and Aarhus). Z.G. thanks the Program in Applied and Computational Mathematics at Princeton for their hospitality in Spring 2017 when the paper was finalized. The authors thank the referee for the constructive comments.


  1. 1.
    Ahlfors, L.: Conformal Invariants: Topics in Geometric Function Theory. AMS Chelsea, Providence, RI, 2010Google Scholar
  2. 2.
    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3D Euler equations. Commun. Math. Phys. 94, 61 (1984)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Bradshaw Z., Grujić Z.: A spatially localized L log L estimate on the vorticity in the 3D NSE. Indiana Univ. Math. J. 64, 433 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bradshaw Z., Grujić Z., Kukavica I.: Local analyticity radii of solutions to the 3D Navier–Stokes equations with locally analytic forcing. J. Differ. Equ. 259, 3955 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bressan A.: A lemma and a conjecture on the cost of rearrangements. Rend. Sem. Mat. Univ. Padova 110, 97 (2003)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Constantin P.: Navier–Stokes equations and area of interfaces. Commun. Math. Phys. 129, 241 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Constantin P.: Geometric statistics in turbulence. SIAM Rev. 36, 73 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Constantin P., Fefferman C.: Direction of vorticity and the problem of global regularity for the Navier–Stokes equations. Indiana Univ. Math. J. 42, 775 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    da Veiga H.B., Berselli L.C.: Differ. Integral Eqs. 15, 345 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Escauriaza L., Seregin G., Shverak V.: \({L_{3,\infty}}\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk. 58, 3 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Farhat A., Grujić Z., Leitmeyer K.: The space \({B^{-1}_{\infty, \infty}}\), volumetric sparseness, and 3D NSE. J. Math. Fluid Mech. 19, 515 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Foias C., Temam R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gallagher I., Koch G., Planchon F.: Blow-up of critical Besov norms at a potential Navier–Stokes singularity. Commun. Math. Phys. 343, 1 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Giga Y., Miyakawa T.: Navier–Stokes flow in \({\mathbb{R}^3}\) with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14, 577 (1989)CrossRefzbMATHGoogle Scholar
  15. 15.
    Giga, Y., Inui, K., Matsui, S.: On the Cauchy problem for the Navier–Stokes equations with nondecaying initial data. Hokkaido University Preprint Series in Mathematics, 1998Google Scholar
  16. 16.
    Grujić Z., Kukavica I.: Space analyticity for the Navier–Stokes and related equations with initial data in \({L^p}\). J. Funct. Anal. 152, 447 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grujić Z.: Localization and geometric depletion of vortex-stretching in the 3D NSE. Commun. Math. Phys. 290, 861 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grujić Z., Guberović R.: Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE. Commun. Math. Phys. 298, 407 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grujić Z.: A geometric measure-type regularity criterion for solutions to the 3D Navier–Stokes equations.. Nonlinearity 26, 289 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Guberović R.: Smoothness of Koch–Tataru solutions to the Navier–Stokes equations revisited. Discrete Contin. Dyn. Syst. 27, 231 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Iyer G., Kiselev A., Xu X.: Lower bound on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows. Nonlinearity 27, 973 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kida S.: Three-dimensional periodic flows with high symmetry. J. Phys. Soc. Jpn. 54, 2132 (1985)ADSCrossRefGoogle Scholar
  23. 23.
    Kukavica I.: On local uniqueness of weak solutions of the Navier–Stokes system with bounded initial data. J. Differ. Equ. 194, 39 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Leray J.: Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1 (1933)zbMATHGoogle Scholar
  25. 25.
    Leray J.: Essai sur les mouvements plans d’un liquide visqueux que limitent de parois. J. Math. Pures Appl. 13, 331 (1934)zbMATHGoogle Scholar
  26. 26.
    Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lemarie-Rieusset P.G.: Recent Developments in the Navier–Stokes Problem. Chapman and Hall/CRC, London (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Meyer, Y.: Oscillating patterns in some nonlinear evolution equations. Lecture Notes in Mathematics 1871, C. I. M. E. Foundation Subseries: Mathematical Foundation of Turbulent Viscous Flows, Springer, 2003Google Scholar
  29. 29.
    Phuc N.C.: The Navier–Stokes equations in nonendpoint borderline Lorentz spaces. J. Math. Fluid Mech. 17, 4 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ransford T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  31. 31.
    Schumacher J., Scheel J., Krasnov D., Donzis D., Yakhot V., Sreenivasan K.R.: Small-scale universality in fluid turbulence. PNAS 111, 10961 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Solynin A.Y.: Ordering of sets, hyperbolic metrics, and harmonic measure. J. Math. Sci. 95, 2256 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  34. 34.
    Taylor G.I.: Production and dissipation of vorticity in a turbulent fluid. Proc. R. Soc. Lond. A 164, 15 (1937)ADSzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA
  2. 2.Department of MathematicsFlorida State UniversityTallahasseeUSA
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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