# American Institute of Mathematical Sciences

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. [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. [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. [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. [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. [6] G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. [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. [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. [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. [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. [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. [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. [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. [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations,, J. Anal. Math., 73 (1997), 251. doi: 10.1007/BF02788146. [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. [16] J. Wu, Regularity results for weak solutions of the 3D MHD equations,, Discrete Contin. Dyn. Syst., 10 (2004), 543. [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. [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. [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. [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. [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. [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. [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.

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. 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. 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. 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. 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. doi: 10.1080/00036811.2011.556626. [6] G. Duvaut and J. Lions, Inéquations en thermoélasticité et magnétohydrodynamique,, Arch. Ration. Mech. Anal., 46 (1972), 241. [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. [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. [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. [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. [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. [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. [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. [14] J. Wu, Viscous and inviscid magnetohydrodynamics equations,, J. Anal. Math., 73 (1997), 251. doi: 10.1007/BF02788146. [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. [16] J. Wu, Regularity results for weak solutions of the 3D MHD equations,, Discrete Contin. Dyn. Syst., 10 (2004), 543. [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. [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. [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. [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. [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. [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. [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.
 [1] 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 [2] Jishan Fan, Tohru Ozawa. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model. Kinetic & Related Models, 2009, 2 (2) : 293-305. doi: 10.3934/krm.2009.2.293 [3] Tomoyuki Suzuki. Regularity criteria in weak spaces in terms of the pressure to the MHD equations. Conference Publications, 2011, 2011 (Special) : 1335-1343. doi: 10.3934/proc.2011.2011.1335 [4] Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637 [5] Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899 [6] Jishan Fan, Tohru Ozawa. Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinetic & Related Models, 2014, 7 (1) : 45-56. doi: 10.3934/krm.2014.7.45 [7] Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973 [8] Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 [9] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [10] Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401 [11] Jishan Fan, Yasuhide Fukumoto, Yong Zhou. Logarithmically improved regularity criteria for the generalized Navier-Stokes and related equations. Kinetic & Related Models, 2013, 6 (3) : 545-556. doi: 10.3934/krm.2013.6.545 [12] Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2018117 [13] Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851 [14] Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033 [15] Quansen Jiu, Jitao Liu. Global regularity for the 3D axisymmetric MHD Equations with horizontal dissipation and vertical magnetic diffusion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 301-322. doi: 10.3934/dcds.2015.35.301 [16] Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495 [17] Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure & Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583 [18] Wendong Wang, Liqun Zhang, Zhifei Zhang. On the interior regularity criteria of the 3-D navier-stokes equations involving two velocity components. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2609-2627. doi: 10.3934/dcds.2018110 [19] Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309 [20] Hua Qiu. Regularity criteria of smooth solution to the incompressible viscoelastic flow. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2873-2888. doi: 10.3934/cpaa.2013.12.2873

2016 Impact Factor: 1.261