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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

Yukang Chen 1, and  Changhua Wei 1,

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.
Keywords: partial regularity, Hausdorff measure., Navier-Stokes, fractional laplacian, suitable weak solution.
Mathematics Subject Classification: Primary: 76D03, 76D05; Secondary: 35Q3.
Citation: Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[11]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1933), 193-248. Google Scholar

[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.  Google Scholar

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J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Donud, Paris, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Scheffer, Hausdoff measure and the Navier-Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512.  Google Scholar

[17]

V. Scheffer, The Navier-Stokes equations in space dimension four, Commun. Math. Phys., 61 (1978), 41-68. doi: 10.1007/BF01609467.  Google Scholar

[18]

V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692.  Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977.  Google Scholar

[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.  Google Scholar

[25]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, New York, 2000. doi: 10.1007/BFb0103908.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr., 4 (1951), 213-231.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1933), 193-248. Google Scholar

[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.  Google Scholar

[13]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Donud, Paris, 1969. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Scheffer, Hausdoff measure and the Navier-Stokes equations, Commun. Math. Phys., 55 (1977), 97-112. doi: 10.1007/BF01626512.  Google Scholar

[17]

V. Scheffer, The Navier-Stokes equations in space dimension four, Commun. Math. Phys., 61 (1978), 41-68. doi: 10.1007/BF01609467.  Google Scholar

[18]

V. Scheffer, The Navier-Stokes equations on a bounded domain, Commun. Math. Phys., 73 (1980), 1-42. doi: 10.1007/BF01942692.  Google Scholar

[19]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland, Amsterdam, New York, 1977.  Google Scholar

[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.  Google Scholar

[25]

B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Springer, New York, 2000. doi: 10.1007/BFb0103908.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

J. Wu, Generalized MHD equations, J. Differ. Equ., 195 (2003), 284-312. doi: 10.1016/j.jde.2003.07.007.  Google Scholar

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