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Regularity criteria for the 3D MHD equations via partial derivatives
1. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P. R., China |
2. | Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China |
References:
[1] |
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. |
[2] |
C. Cao and E. S. Titi, Global regularity criterion for the 3DNavier-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] |
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. |
[4] |
H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal. Available from: http://dx.doi.org/10.1080/00036811.2011.556626. |
[5] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodyna-mique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. |
[6] |
J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571.
doi: 10.1007/s00021-010-0039-5. |
[7] |
C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17.
doi: 10.1016/j.jde.2007.03.023. |
[8] |
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. |
[9] |
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. |
[10] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components, Nonlinear Anal. Real World Appl., 13 (2012), 410-418. |
[11] |
M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.
doi: 10.1002/mana.19971880116. |
[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] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. |
[14] |
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. |
[15] |
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. |
[16] |
Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb{R}^3$, Proc. Am. Math. Soc., 134 (2006), 149-156.
doi: 10.1090/S0002-9939-05-08118-9. |
[17] |
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. |
[18] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. |
[19] |
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
doi: 10.1088/0951-7715/23/5/004. |
show all references
References:
[1] |
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. |
[2] |
C. Cao and E. S. Titi, Global regularity criterion for the 3DNavier-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] |
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. |
[4] |
H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations, Appl. Anal. Available from: http://dx.doi.org/10.1080/00036811.2011.556626. |
[5] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodyna-mique, Arch. Ration. Mech. Anal., 46 (1972), 241-279. |
[6] |
J. Fan, S. Jiang, G. Nakamura and Y. Zhou, Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13 (2011), 557-571.
doi: 10.1007/s00021-010-0039-5. |
[7] |
C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 238 (2007), 1-17.
doi: 10.1016/j.jde.2007.03.023. |
[8] |
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. |
[9] |
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. |
[10] |
X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations involving partial components, Nonlinear Anal. Real World Appl., 13 (2012), 410-418. |
[11] |
M. A. Rojas-Medar, Magneto-micropolar fluid motion: existence and uniqueness of strong solutions, Math. Nachr., 188 (1997), 301-319.
doi: 10.1002/mana.19971880116. |
[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] |
J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556. |
[14] |
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. |
[15] |
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. |
[16] |
Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbb{R}^3$, Proc. Am. Math. Soc., 134 (2006), 149-156.
doi: 10.1090/S0002-9939-05-08118-9. |
[17] |
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. |
[18] |
Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. |
[19] |
Y. Zhou and M. Pokorný, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107.
doi: 10.1088/0951-7715/23/5/004. |
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