Regularity criteria for the 3D MHD equations via partial derivatives. II

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June 2014, 7(2): 291-304. doi: 10.3934/krm.2014.7.291

Regularity criteria for the 3D MHD equations via partial derivatives. II

Xuanji Jia ^1, and Yong Zhou ^2,

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

Received May 2013 Revised November 2013 Published March 2014

Abstract
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In this paper we continue studying regularity criteria for the 3D MHD equations via partial derivatives of the velocity or the pressure. We obtain some new regularity criteria which improve the related results in [1,3,9,11,17]. Precisely, we first prove that if for any $ i,\,j,\,k\in \{1,2,3\}$ there holds $ (\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k}) \in L_T^{\alpha,\gamma}$ with $\frac{2}{\alpha}+\frac{3}{\gamma}\leq 1+\frac{1}{\gamma},~2\leq \gamma\leq \infty $, then the solution $(u,b)$ is smooth on $\mathbb{R}^3\times(0,T]$. Secondly, we show that any component (resp. components) of $(\frac{\partial u_1}{\partial x_i},\,\frac{\partial u_2}{\partial x_j},\,\frac{\partial u_3}{\partial x_k})$ in the criterion above can be replaced by the corresponding velocity component (resp. components) which is (resp. are) in the space $L_T^{\alpha',\gamma'}$with $\frac{2}{\alpha'}+\frac{3}{\gamma'}\leq 1$, $3< \gamma'\leq \infty$. Fianlly, we obtain a Ladyzhenskaya-Prodi-Serrin type regularity condition involving two components of the gradient of pressure, which in fact partially answers an open question proposed in [9] and improves Theorem 3.3 in Berselli and Galdi's article [1].

Keywords: MHD equations, regularity criteria, partial derivatives..

Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76W0.

Citation: Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic and Related Models, 2014, 7 (2) : 291-304. doi: 10.3934/krm.2014.7.291

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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