    March  2020, 28(1): 183-193. doi: 10.3934/era.2020012

## The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations

 1 Department of Mathematical Science, Faculty of Applied Science, Umm Alqura University, P. O. Box 14035, Makkah 21955, Saudi Arabia 2 Department of Mathematics, ENS of Mostaganem, Box 227, Mostaganem 27000, Algeria 3 Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria, 6 - 95125 Catania, Italy 4 Department of Mathematics, Zhejiang Normal University, Jinhua, China 5 RUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia

* Corresponding author: Sadek Gala, sgala793@gmail.com

Received  October 2019 Revised  January 2020 Published  March 2020

Fund Project: This research is partially supported by GNAMPA 2019. The fourth author wishes to thank the support of "RUDN University Program 5-100"

This study is devoted to investigating the regularity criterion of the 3D MHD equations in terms of pressure in the framework of anisotropic Lebesgue spaces. The result shows that if a weak solution
 $(u, b)$
satisfies
 $\begin{equation} \int_{0}^{T}{\frac{\left\Vert \left\Vert \partial _{3}\pi (\cdot , t)\right\Vert _{L_{x_{3}}^{\gamma }}\right\Vert _{L_{x_{1}x_{2}}^{\alpha }}^{q}}{1+\ln \left( e+\left\Vert \pi (\cdot , t)\right\Vert _{L^{2}}^{2}\right) }}\ dt<\infty , ~~~~~~~~~~~~~~~~~~~(1)\end{equation}$
where
 $\begin{equation*} \frac{1}{\gamma }+\frac{2}{q}+\frac{2}{\alpha } = \lambda \in \lbrack 2, 3)\text{ and }\frac{3}{\lambda }\leq \gamma \leq \alpha <\frac{1}{\lambda -2}, \end{equation*}$
then
 $(u, b)$
is regular at
 $t = T$
, which improve the previous results on the MHD equations
Citation: Ahmad Mohammad Alghamdi, Sadek Gala, Chenyin Qian, Maria Alessandra Ragusa. The anisotropic integrability logarithmic regularity criterion for the 3D MHD equations. Electronic Research Archive, 2020, 28 (1) : 183-193. doi: 10.3934/era.2020012
##### References:
  S. Benbernou, M. A. Ragusa and M. Terbeche, A logarithmically improved regularity criterion for the MHD equations in terms of one directional derivative of the pressure, Appl. Anal., 96 (2017), 2140-2148.  doi: 10.1080/00036811.2016.1207246.  Google Scholar  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  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  S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations, Math. Methods Appl. Sci., 33 (2010), 1496-1503.  doi: 10.1002/mma.1263.  Google Scholar  X. Jia and Y. Zhou, Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of $3\times 3$ mixture matrices, Nonlinearity, 28 (2015), 3289-3307.  doi: 10.1088/0951-7715/28/9/3289.  Google Scholar  X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives. Ⅱ, Kinet. Relat. Models, 7 (2014), 291-304.  doi: 10.3934/krm.2014.7.291.  Google Scholar  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.  Google Scholar  Q. Liu, The 3D Boussinesq equations with regularity in one directional derivative of the pressure, Bull. Malays. Math. Sci. Soc., 42 (2019), 3005-3019.  doi: 10.1007/s40840-018-0645-6.  Google Scholar  C. Qian, A generalized regularity criterion for the 3D Navier-Stokes equations in terms of one velocity component, J. Differential Equations, 260 (2016), 3477-3494.  doi: 10.1016/j.jde.2015.10.037.  Google Scholar  C. Qian, The anisotropic integrability regularity criterion to 3D magnetohydrodynamic equations, Math. Methods Appl. Sci., 40 (2017), 5461-5469.  doi: 10.1002/mma.4399.  Google Scholar  Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar  Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar  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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##### References:
  S. Benbernou, M. A. Ragusa and M. Terbeche, A logarithmically improved regularity criterion for the MHD equations in terms of one directional derivative of the pressure, Appl. Anal., 96 (2017), 2140-2148.  doi: 10.1080/00036811.2016.1207246.  Google Scholar  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  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  S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations, Math. Methods Appl. Sci., 33 (2010), 1496-1503.  doi: 10.1002/mma.1263.  Google Scholar  X. Jia and Y. Zhou, Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of $3\times 3$ mixture matrices, Nonlinearity, 28 (2015), 3289-3307.  doi: 10.1088/0951-7715/28/9/3289.  Google Scholar  X. Jia and Y. Zhou, Regularity criteria for the 3D MHD equations via partial derivatives. Ⅱ, Kinet. Relat. Models, 7 (2014), 291-304.  doi: 10.3934/krm.2014.7.291.  Google Scholar  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.  Google Scholar  Q. Liu, The 3D Boussinesq equations with regularity in one directional derivative of the pressure, Bull. Malays. Math. Sci. Soc., 42 (2019), 3005-3019.  doi: 10.1007/s40840-018-0645-6.  Google Scholar  C. Qian, A generalized regularity criterion for the 3D Navier-Stokes equations in terms of one velocity component, J. Differential Equations, 260 (2016), 3477-3494.  doi: 10.1016/j.jde.2015.10.037.  Google Scholar  C. Qian, The anisotropic integrability regularity criterion to 3D magnetohydrodynamic equations, Math. Methods Appl. Sci., 40 (2017), 5461-5469.  doi: 10.1002/mma.4399.  Google Scholar  Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.  doi: 10.1016/j.ijnonlinmec.2006.12.001.  Google Scholar  Y. Zhou and J. Fan, Logarithmically improved regularity criteria for the 3D viscous MHD equations, Forum Math., 24 (2012), 691-708.  doi: 10.1515/form.2011.079.  Google Scholar  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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