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:
[1]

S. BenbernouM. 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

[2]

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

[3]

J. FanS. JiangG. 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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

show all references

References:
[1]

S. BenbernouM. 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

[2]

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

[3]

J. FanS. JiangG. 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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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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