Abstract
In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s − 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s − 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, & Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).
Article PDF
Similar content being viewed by others
References
Beirão da Veiga H.: A new regularity class for the Navier-Stokes equations in R n. Chinese Ann. Math. Ser. B 16(4), 407–412 (1995)
Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Constantin P., Foias C.: Navier–Stokes Equations. University of Chicago Press, Chicago, IL (1988)
Escauriaza L., Seregin G., Šverák V.: L 3,∞-Solutions to the Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58(2), 211–250 (2003)
Fabes E.B., Jones B.F., Rivière M.M.: The initial value problem for the Navier-Stokes equations with data in L p. Arch. Rat. Mech. Anal. 45, 222–240 (1972)
Falconer K.: Fractal Geometry: Mathematical Foundations and Applications. (2nd edition). John Wiley & Sons, Chichester (2003)
Hopf E.: Über die Anfangswertaufgabe dür die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)
Kukavica I.: The fractal dimenson of the singular set for solutions of the Navier–Stokes system. Nonlinearity 22, 2889–2900 (2009)
Ladyzhenskaya O., Seregin G.: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech. 1, 356–387 (1999)
Leray J.: Essai sur le mouvement d’un fluide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)
Lin F.: A New Proof of the Caffarelli-Kohn-Nirenberg Theorem. Comm. Pure Appl. Math. 51, 241–257 (1998)
Robinson J.C.: Regularity and singularity in the three-dimensional Navier Stokes equations. Bole. Soc. Española de Mate. Apli. 35, 43–71 (2006)
Robinson J.C.: Dimensions, Embeddings, and Attractors. Cambridge Tracts in Mathematics, Vol. 186. Cambridge University Press, Cambridge (2011)
Robinson J.C., Sadowski W.: Decay of weak solutions and the singular set of the three-dimensional Navier-Stokes equations. Nonlinearity 20, 1185–1191 (2007)
Robinson J.C., Sadowski W.: Almost-everywhere uniqueness of Lagrangian trajectories for suitable weak solutions of the three-dimensional Navier–Stokes equations. Nonlinearity 22, 2093–2099 (2009)
Scheffer, V.: Turbulence and Hausdorff dimension. In: Turbulence and Navier-Stokes equations, Orsay 1975, Springer LNM 565, Berlin: Springer-Verlag, 1976, pp. 174–183
Serrin J.: On the interior regulariy of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 9, 187–191 (1962)
Struwe M.: On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math 41(4), 437–458 (1988)
Wolf, J.: A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations. In: Rannacher, R., Sequeira, A. (eds.) Advances in Mathematical Fluid Mechanics, Springer-Verlag, New York, USA (2010)
Acknowledgements
Both JCR and WS are supported by the EPSRC grant EP/G007470/1; WS is also supported by the Polish Government grant N N201 547 438.
Open Access
This article is distributed under the terms of theCreative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Constantin
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Robinson, J.C., Sadowski, W. On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations. Commun. Math. Phys. 309, 497–506 (2012). https://doi.org/10.1007/s00220-011-1336-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1336-4