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Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations
Partial regularity of solutions to the fractional Navier-Stokes equations
1. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China |
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. |
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