# American Institute of Mathematical Sciences

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

## 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

Received  May 2013 Revised  November 2013 Published  March 2014

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].
Citation: Xuanji Jia, Yong Zhou. Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinetic & 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. doi: 10.1090/S0002-9939-02-06697-2. Google Scholar [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. doi: 10.1007/s00205-011-0439-6. Google Scholar [3] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020. Google Scholar [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. doi: 10.1007/s00220-008-0545-y. Google Scholar [5] H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Appl. Anal., 91 (2012), 947. doi: 10.1080/00036811.2011.556626. Google Scholar [6] G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. Google Scholar [7] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002. Google Scholar [8] E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics,, J. Math. Anal. Appl., 369 (2010), 317. doi: 10.1016/j.jmaa.2010.03.015. Google Scholar [9] X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives,, Kinet. Relat. Models, 5 (2012), 505. doi: 10.3934/krm.2012.5.505. Google Scholar [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. doi: 10.1016/j.jmaa.2012.06.016. Google Scholar [11] H. Lin and L. Du, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions,, Nonlinearity, 26 (2013), 219. doi: 10.1088/0951-7715/26/1/219. Google Scholar [12] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. doi: 10.1002/cpa.3160360506. Google Scholar [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. doi: 10.1016/j.nonrwa.2012.07.013. Google Scholar [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations,, J. Anal. Math., 73 (1997), 251. doi: 10.1007/BF02788146. Google Scholar [15] J. Wu, Bounds and new approaches for the 3D MHD equations,, J. Nonlinear Sci., 12 (2002), 395. doi: 10.1007/s00332-002-0486-0. Google Scholar [16] J. Wu, Regularity results for weak solutions of the 3D MHD equations,, Discrete Contin. Dyn. Syst., 10 (2004), 543. Google Scholar [17] K. Yamazaki, Remarks on the regularity criteria of generalized MHD and Navier-Stokes systems,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4773833. Google Scholar [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). doi: 10.1063/1.3589966. Google Scholar [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. doi: 10.1016/j.jmaa.2012.11.022. Google Scholar [20] Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: 10.3934/dcds.2005.12.881. Google Scholar [21] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Int. J. Non-Linear Mech., 41 (2006), 1174. doi: 10.1016/j.ijnonlinmec.2006.12.001. Google Scholar [22] Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^3$,, Proc. Amer. Math. Soc., 134 (2006), 149. doi: 10.1090/S0002-9939-05-08312-7. Google Scholar [23] 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. doi: 10.1007/s00033-005-0021-x. Google Scholar

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##### 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. doi: 10.1090/S0002-9939-02-06697-2. Google Scholar [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. doi: 10.1007/s00205-011-0439-6. Google Scholar [3] C. Cao and J. Wu, Two regularity criteria for the 3D MHD equations,, J. Differential Equations, 248 (2010), 2263. doi: 10.1016/j.jde.2009.09.020. Google Scholar [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. doi: 10.1007/s00220-008-0545-y. Google Scholar [5] H. Duan, On regularity criteria in terms of pressure for the 3D viscous MHD equations,, Appl. Anal., 91 (2012), 947. doi: 10.1080/00036811.2011.556626. Google Scholar [6] G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. Google Scholar [7] C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations,, J. Differential Equations, 213 (2005), 235. doi: 10.1016/j.jde.2004.07.002. Google Scholar [8] E. Ji and J. Lee, Some regularity criteria for the 3D incompressible magnetohydrodynamics,, J. Math. Anal. Appl., 369 (2010), 317. doi: 10.1016/j.jmaa.2010.03.015. Google Scholar [9] X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives,, Kinet. Relat. Models, 5 (2012), 505. doi: 10.3934/krm.2012.5.505. Google Scholar [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. doi: 10.1016/j.jmaa.2012.06.016. Google Scholar [11] H. Lin and L. Du, Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions,, Nonlinearity, 26 (2013), 219. doi: 10.1088/0951-7715/26/1/219. Google Scholar [12] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations,, Comm. Pure Appl. Math., 36 (1983), 635. doi: 10.1002/cpa.3160360506. Google Scholar [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. doi: 10.1016/j.nonrwa.2012.07.013. Google Scholar [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations,, J. Anal. Math., 73 (1997), 251. doi: 10.1007/BF02788146. Google Scholar [15] J. Wu, Bounds and new approaches for the 3D MHD equations,, J. Nonlinear Sci., 12 (2002), 395. doi: 10.1007/s00332-002-0486-0. Google Scholar [16] J. Wu, Regularity results for weak solutions of the 3D MHD equations,, Discrete Contin. Dyn. Syst., 10 (2004), 543. Google Scholar [17] K. Yamazaki, Remarks on the regularity criteria of generalized MHD and Navier-Stokes systems,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4773833. Google Scholar [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). doi: 10.1063/1.3589966. Google Scholar [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. doi: 10.1016/j.jmaa.2012.11.022. Google Scholar [20] Y. Zhou, Remarks on regularities for the 3D MHD equations,, Discrete Contin. Dyn. Syst., 12 (2005), 881. doi: 10.3934/dcds.2005.12.881. Google Scholar [21] Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure,, Int. J. Non-Linear Mech., 41 (2006), 1174. doi: 10.1016/j.ijnonlinmec.2006.12.001. Google Scholar [22] Y. Zhou, On regularity criteria in terms of pressure for the Navier-Stokes equations in $\mathbbR^3$,, Proc. Amer. Math. Soc., 134 (2006), 149. doi: 10.1090/S0002-9939-05-08312-7. Google Scholar [23] 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. doi: 10.1007/s00033-005-0021-x. Google Scholar
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