American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros
  • DCDS Home
  • This Issue
  • Next Article
    Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations
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 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.

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.

[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.

[1]

Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053
[2]

Jiří Neustupa. A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1391-1400. doi: 10.3934/dcdss.2013.6.1391
[3]

Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
[4]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159
[5]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[6]

Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157
[7]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4397-4419. doi: 10.3934/dcds.2021041
[8]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[9]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[10]

Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[11]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[12]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[13]

Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic and Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545
[14]

Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141
[15]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[16]

Zijin Li, Xinghong Pan. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1333-1350. doi: 10.3934/cpaa.2019064
[17]

Xuanji Jia, Zaihong Jiang. An anisotropic regularity criterion for the 3D Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1299-1306. doi: 10.3934/cpaa.2013.12.1299
[18]

Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[19]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[20]

Hui Chen, Daoyuan Fang, Ting Zhang. Regularity of 3D axisymmetric Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1923-1939. doi: 10.3934/dcds.2017081

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (263)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy