Abstract
In this paper, we study the partial regularity of fractional Navier–Stokes equations in \({\mathbb{R}^3 \times (0, \infty)}\) with \({3/4 < s < 1}\) . We show that the suitable weak solution is regular away from a relatively closed singular set whose (5−4s)-dimentional Hausdorff measure is zero. The result is a generalization of the partial regularity for the classical Navier–Stokes system in Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
Caffarelli L., Vasseur A.: Drift diffusion equtions with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171, 1903–1930 (2010)
Constantin P., Wu. J.: Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations. Ann. I. H. Poincaré-Anal. Non Linéaire 26, 159–180 (2009)
Fabes E., Kenig C., Serapioni R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7, 77–116 (1982)
Katz N.H., Pavlovic N.: A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation. Geom. Funct. Anal. 12(2), 355–379 (2002)
Lin F.: A new proof the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51, 241–257 (1998)
Lions J.L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. Donud, Paris (1969)
Mercado J.M., Guido E.P., Sánchez-Sesma A.J., Íñiguez M., González A. et al.: Analysis of the Blasius’ formula and the Navier–Stokes fractional equation. In: Klapp, J. (ed.) Fluid Dynamics in Physics, Engineering and Environmental Applications, Environmental Science and Engineering, pp. 475–480. Springer, Berlin, Heidelberg (2013)
Palatucci, G., Savin, O., Valdinoci, E.: Local and global minimizers for a variational energy involving a fractional norm. Annali di Matematica Pura ed Applicata (2012). doi:10.1007/s10231-011-0243-9
Scheffer V.: Partial regularity of solutions to the Navier–Stokes equations. Pacif. J. Math. 66, 535–552 (1976)
Scheffer V.: Hausdoff measure and the Navier–Stokes equations. Commun. Math. Phys. 55, 97–112 (1977)
Scheffer V.: The Navier–Stokes equations on a bounded domain. Commun. Math. Phys. 73, 1–42 (1980)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Tang, L., Yu, Y.: Partial Hölder Regularity for Steady Fractional Navier–Stokes Equation (submitted)
Tao T.: Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation. Anal. PDE 2, 361–366 (2009)
Temam R.: Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam, New York (1977)
Wu J.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)
Zhang X.: Stochastic Lagrangian particle approach to fractal NavierStokes equations. Commun. Math. Phys. 311, 133–155 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Caffarelli
Rights and permissions
About this article
Cite this article
Tang, L., Yu, Y. Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations. Commun. Math. Phys. 334, 1455–1482 (2015). https://doi.org/10.1007/s00220-014-2149-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2149-z