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October  2016, 36(10): 5309-5322. doi: 10.3934/dcds.2016033

## Partial regularity of solutions to the fractional Navier-Stokes equations

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China

Received  September 2015 Revised  December 2015 Published  July 2016

We study the partial regularity of suitable weak solutions to the Navier-Stokes equations with fractional dissipation $\sqrt{-\Delta}^s$ in the critical case of $s=\frac{3}{2}$. We show that the two dimensional Hausdorff measure of space-time singular set of these solutions is zero.
Citation: Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033
##### References:
 [1] D. Barbato, F. Morandin and M. Romito, Global regularity for a logarithmically supercritical hyperdissipative dyadic equation, Dyn. Partial Differ. Equ., 11 (2014), 39-52. doi: 10.4310/DPDE.2014.v11.n1.a2. [2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [4] H. Dong and D. Du, Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time, Commun. Math. Phys., 273 (2007), 785-801. doi: 10.1007/s00220-007-0259-6. [5] H. Dong and X. Gu, Partial regularity of solutions to the four-dimensional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 53-69. doi: 10.4310/DPDE.2014.v11.n1.a3. [6] H. Dong and X. Gu, Boundary partial regularity for the high dimensional Navier-Stokes equations, J. Funct. Anal., 267 (2014), 2606-2637. doi: 10.1016/j.jfa.2014.08.001. [7] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. [8] T. Y. Hou and Z. Lei, On the partial regularity of a 3D model of the Navier-Stokes equations, Commun. Math. Phys., 287 (2009), 589-612. doi: 10.1007/s00220-008-0689-9. [9] N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z. [10] O. Ladyzhenskaya and G. A. Seregin, On the partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015. [11] J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1933), 193-248. [12] F. Lin, A new proof the Caffarelli-Kohn-Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. [13] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Donud, Paris, 1969. [14] S. A. Molchanov and E. Ostrovski, Symmetric stable processes as traces of degenerate diffusion processes, (Russian)Teor. Verojatnost. i Primenen, 14 (1969), 127-130. [15] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535. [16] V. Scheffer, Hausdoff measure and the Navier-Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512. [17] V. Scheffer, The Navier-Stokes equations in space dimension four, Commun. Math. Phys., 61 (1978), 41-68. doi: 10.1007/BF01609467. [18] V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692. [19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [20] L. Tang and Y. Yu, Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations, Commun. Math. Phys., 334 (2015), 1455-1482. doi: 10.1007/s00220-014-2149-z. [21] L. Tang and Y. Yu, Erratum to: Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations, Commun. Math. Phys., 335 (2015), 1057-1063. doi: 10.1007/s00220-015-2289-9. [22] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal.PDE., 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361. [23] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977. [24] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Commun. Anal. Geom., 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1. [25] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, New York, 2000. doi: 10.1007/BFb0103908. [26] A. F. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Differ. Equ. Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4. [27] Y. Wang and G. Wu, A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations, J. Differ. Equ., 256 (2014), 1224-1249. doi: 10.1016/j.jde.2013.10.014. [28] J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.

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##### References:
 [1] D. Barbato, F. Morandin and M. Romito, Global regularity for a logarithmically supercritical hyperdissipative dyadic equation, Dyn. Partial Differ. Equ., 11 (2014), 39-52. doi: 10.4310/DPDE.2014.v11.n1.a2. [2] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Commun. Pure Appl. Math., 35 (1982), 771-831. doi: 10.1002/cpa.3160350604. [3] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [4] H. Dong and D. Du, Partial regularity of solutions to the four-dimensional Navier-Stokes equations at the first blow-up time, Commun. Math. Phys., 273 (2007), 785-801. doi: 10.1007/s00220-007-0259-6. [5] H. Dong and X. Gu, Partial regularity of solutions to the four-dimensional Navier-Stokes equations, Dyn. Partial Differ. Equ., 11 (2014), 53-69. doi: 10.4310/DPDE.2014.v11.n1.a3. [6] H. Dong and X. Gu, Boundary partial regularity for the high dimensional Navier-Stokes equations, J. Funct. Anal., 267 (2014), 2606-2637. doi: 10.1016/j.jfa.2014.08.001. [7] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231. [8] T. Y. Hou and Z. Lei, On the partial regularity of a 3D model of the Navier-Stokes equations, Commun. Math. Phys., 287 (2009), 589-612. doi: 10.1007/s00220-008-0689-9. [9] N. H. Katz and N. Pavlović, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379. doi: 10.1007/s00039-002-8250-z. [10] O. Ladyzhenskaya and G. A. Seregin, On the partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 356-387. doi: 10.1007/s000210050015. [11] J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1933), 193-248. [12] F. Lin, A new proof the Caffarelli-Kohn-Nirenberg theorem, Commun. Pure Appl. Math., 51 (1998), 241-257. doi: 10.1002/(SICI)1097-0312(199803)51:3<241::AID-CPA2>3.0.CO;2-A. [13] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Donud, Paris, 1969. [14] S. A. Molchanov and E. Ostrovski, Symmetric stable processes as traces of degenerate diffusion processes, (Russian)Teor. Verojatnost. i Primenen, 14 (1969), 127-130. [15] V. Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. doi: 10.2140/pjm.1976.66.535. [16] V. Scheffer, Hausdoff measure and the Navier-Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512. [17] V. Scheffer, The Navier-Stokes equations in space dimension four, Commun. Math. Phys., 61 (1978), 41-68. doi: 10.1007/BF01609467. [18] V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692. [19] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. [20] L. Tang and Y. Yu, Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations, Commun. Math. Phys., 334 (2015), 1455-1482. doi: 10.1007/s00220-014-2149-z. [21] L. Tang and Y. Yu, Erratum to: Partial regularity of suitable weak solutions to the fractional Navier-Stokes equations, Commun. Math. Phys., 335 (2015), 1057-1063. doi: 10.1007/s00220-015-2289-9. [22] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal.PDE., 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361. [23] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977. [24] G. Tian and Z. Xin, Gradient estimation on Navier-Stokes equations, Commun. Anal. Geom., 7 (1999), 221-257. doi: 10.4310/CAG.1999.v7.n2.a1. [25] B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, New York, 2000. doi: 10.1007/BFb0103908. [26] A. F. Vasseur, A new proof of partial regularity of solutions to Navier-Stokes equations, NoDEA Nonlinear Differ. Equ. Appl., 14 (2007), 753-785. doi: 10.1007/s00030-007-6001-4. [27] Y. Wang and G. Wu, A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier-Stokes equations, J. Differ. Equ., 256 (2014), 1224-1249. doi: 10.1016/j.jde.2013.10.014. [28] J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.
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