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September  2012, 5(3): 505-516. doi: 10.3934/krm.2012.5.505

Regularity criteria for the 3D MHD equations via partial derivatives

Xuanji Jia 1, and  Yong Zhou 2,

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, P. R., China
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China

Received  December 2011 Revised  February 2012 Published  August 2012

In this paper, we establish two regularity criteria for the 3D MHD equations in terms of partial derivatives of the velocity field or the pressure. It is proved that if $\partial_3 u \in L^\beta(0,T; L^\alpha(\mathbb{R}^3)),~\mbox{with}~ \frac{2}{\beta}+\frac{3}{\alpha}\leq\frac{3(\alpha+2)}{4\alpha},~\alpha>2$, or $\nabla_h P \in L^\beta(0,T; L^{\alpha}(\mathbb{R}^3)),~\mbox{with}~\frac{2}{\beta}+\frac{3}{\alpha}< 3,~\alpha>\frac{9}{7},~\beta\geq 1$, then the weak solution $(u,b)$ is regular on $[0, T]$.
Keywords: regularity criteria, partial derivatives., MHD equations.
Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D0.
Citation: Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. Kinetic & Related Models, 2012, 5 (3) : 505-516. doi: 10.3934/krm.2012.5.505
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Appl. Anal. Available from: , ().   Google Scholar

[5]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodyna-mique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^3$, Proc. Am. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08118-9.  Google Scholar

[17]

Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbbR^N$, Z. Angew. Math. Phys., 57 (2006), 384-392. doi: 10.1007/s00033-005-0021-x.  Google Scholar

[18]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Appl. Anal. Available from: , ().   Google Scholar

[5]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodyna-mique, Arch. Ration. Mech. Anal., 46 (1972), 241-279.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. Wu, Regularity results for weak solutions of the 3D MHD equations, Discrete Contin. Dyn. Syst., 10 (2004), 543-556.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^3$, Proc. Am. Math. Soc., 134 (2006), 149-156. doi: 10.1090/S0002-9939-05-08118-9.  Google Scholar

[17]

Y. Zhou, On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in $\mathbbR^N$, Z. Angew. Math. Phys., 57 (2006), 384-392. doi: 10.1007/s00033-005-0021-x.  Google Scholar

[18]

Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708. Google Scholar

[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.  Google Scholar

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