Regularity criteria for the 3D MHD equations via partial derivatives

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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

Abstract
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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 and 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.
[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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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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