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Regularity criteria for the 3D MHD equations via partial derivatives. II
1. | Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China |
2. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang |
References:
[1] |
L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595.
doi: 10.1090/S0002-9939-02-06697-2. |
[2] |
C. Cao and E. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[3] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[4] |
Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.
doi: 10.1007/s00220-008-0545-y. |
[5] |
H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal., 91 (2012), 947-952.
doi: 10.1080/00036811.2011.556626. |
[6] |
G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. |
[7] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[8] |
E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics, J. Math. Anal. Appl., 369 (2010), 317-322.
doi: 10.1016/j.jmaa.2010.03.015. |
[9] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, Kinet. Relat. Models, 5 (2012), 505-516.
doi: 10.3934/krm.2012.5.505. |
[10] |
X. Jia and Y. Zhou, A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure, J. Math. Anal. Appl., 396 (2012), 345-350.
doi: 10.1016/j.jmaa.2012.06.016. |
[11] |
H. Lin and L. Du, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity, 26 (2013), 219-239.
doi: 10.1088/0951-7715/26/1/219. |
[12] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[13] |
F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.
doi: 10.1016/j.nonrwa.2012.07.013. |
[14] |
J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265.
doi: 10.1007/BF02788146. |
[15] |
J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.
doi: 10.1007/s00332-002-0486-0. |
[16] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. |
[17] |
K. Yamazaki, Remarks on the regularity criteria of generalized MHD and Navier-Stokes systems, J. Math. Phys., 54 (2013), 011502, 16pp.
doi: 10.1063/1.4773833. |
[18] |
Z. Zhang, Z. Yao, M. Lu and L. Ni, Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Phys., 52 (2011), 053103, 7 pp.
doi: 10.1063/1.3589966. |
[19] |
Z. Zhang, P. Li and G. Yu, Regularity criteria for the 3D MHD equations via one directional derivative of the pressure, J. Math. Anal. Appl., 401 (2013), 66-71.
doi: 10.1016/j.jmaa.2012.11.022. |
[20] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
[21] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[22] |
Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb{R}^3$, Proc. Amer. Math. Soc., 134 (2006), 149-156.
doi: 10.1090/S0002-9939-05-08312-7. |
[23] |
Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb{R}^N2$, Z. Angew. Math. Phys., 57 (2006), 384-392.
doi: 10.1007/s00033-005-0021-x. |
show all references
References:
[1] |
L. Berselli and G. Galdi, Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc., 130 (2002), 3585-3595.
doi: 10.1090/S0002-9939-02-06697-2. |
[2] |
C. Cao and E. Titi, Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal., 202 (2011), 919-932.
doi: 10.1007/s00205-011-0439-6. |
[3] |
C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations, 248 (2010), 2263-2274.
doi: 10.1016/j.jde.2009.09.020. |
[4] |
Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.
doi: 10.1007/s00220-008-0545-y. |
[5] |
H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal., 91 (2012), 947-952.
doi: 10.1080/00036811.2011.556626. |
[6] |
G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. |
[7] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[8] |
E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics, J. Math. Anal. Appl., 369 (2010), 317-322.
doi: 10.1016/j.jmaa.2010.03.015. |
[9] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives, Kinet. Relat. Models, 5 (2012), 505-516.
doi: 10.3934/krm.2012.5.505. |
[10] |
X. Jia and Y. Zhou, A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure, J. Math. Anal. Appl., 396 (2012), 345-350.
doi: 10.1016/j.jmaa.2012.06.016. |
[11] |
H. Lin and L. Du, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity, 26 (2013), 219-239.
doi: 10.1088/0951-7715/26/1/219. |
[12] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[13] |
F. Wang and K. Wang, Global existence of 3D MHD equations with mixed partial dissipation and magnetic diffusion, Nonlinear Anal. Real World Appl., 14 (2013), 526-535.
doi: 10.1016/j.nonrwa.2012.07.013. |
[14] |
J. Wu, Viscous and inviscid magnetohydrodynamics equations, J. Anal. Math., 73 (1997), 251-265.
doi: 10.1007/BF02788146. |
[15] |
J. Wu, Bounds and new approaches for the 3D MHD equations, J. Nonlinear Sci., 12 (2002), 395-413.
doi: 10.1007/s00332-002-0486-0. |
[16] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. |
[17] |
K. Yamazaki, Remarks on the regularity criteria of generalized MHD and Navier-Stokes systems, J. Math. Phys., 54 (2013), 011502, 16pp.
doi: 10.1063/1.4773833. |
[18] |
Z. Zhang, Z. Yao, M. Lu and L. Ni, Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations, J. Math. Phys., 52 (2011), 053103, 7 pp.
doi: 10.1063/1.3589966. |
[19] |
Z. Zhang, P. Li and G. Yu, Regularity criteria for the 3D MHD equations via one directional derivative of the pressure, J. Math. Anal. Appl., 401 (2013), 66-71.
doi: 10.1016/j.jmaa.2012.11.022. |
[20] |
Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.
doi: 10.3934/dcds.2005.12.881. |
[21] |
Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Int. J. Non-Linear Mech., 41 (2006), 1174-1180.
doi: 10.1016/j.ijnonlinmec.2006.12.001. |
[22] |
Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb{R}^3$, Proc. Amer. Math. Soc., 134 (2006), 149-156.
doi: 10.1090/S0002-9939-05-08312-7. |
[23] |
Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbb{R}^N2$, Z. Angew. Math. Phys., 57 (2006), 384-392.
doi: 10.1007/s00033-005-0021-x. |
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