### Electronic Research Archive

2020, Issue 1: 183-193. doi: 10.3934/era.2020012

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

• Received: 01 October 2019 Revised: 01 January 2020
• Primary: 35Q35, 35B65; Secondary: 76D05

• 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

$$$\int_{0}^{T}{\frac{\left\Vert \left\Vert \partial _{3}\pi (\cdot , t)\right\Vert _{L_{x_{3}}^{\gamma }}\right\Vert _{L_{x_{1}x_{2}}^{\alpha }}^{q}}{1+\ln \left( e+\left\Vert \pi (\cdot , t)\right\Vert _{L^{2}}^{2}\right) }}\ dt<\infty , ~~~~~~~~~~~~~~~~~~~(1)$$$

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[J]. Electronic Research Archive, 2020, 28(1): 183-193. doi: 10.3934/era.2020012

### Related Papers:

• 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

$$$\int_{0}^{T}{\frac{\left\Vert \left\Vert \partial _{3}\pi (\cdot , t)\right\Vert _{L_{x_{3}}^{\gamma }}\right\Vert _{L_{x_{1}x_{2}}^{\alpha }}^{q}}{1+\ln \left( e+\left\Vert \pi (\cdot , t)\right\Vert _{L^{2}}^{2}\right) }}\ dt<\infty , ~~~~~~~~~~~~~~~~~~~(1)$$$

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

 [1] A logarithmically improved regularity criterion for the MHD equations in terms of one directional derivative of the pressure. Appl. Anal. (2017) 96: 2140-2148. [2] Two regularity criteria for the 3D MHD equations. J. Differential Equations (2010) 248: 2263-2274. [3] Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations. J. Math. Fluid Mech. (2011) 13: 557-571. [4] Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci. (2010) 33: 1496-1503. [5] Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of $3\times 3$ mixture matrices. Nonlinearity (2015) 28: 3289-3307. [6] Regularity criteria for the 3D MHD equations via partial derivatives. Ⅱ. Kinet. Relat. Models (2014) 7: 291-304. [7] A new regularity criterion for the 3D incompressible MHD equations in terms of one component of the gradient of pressure. J. Math. Anal. Appl. (2012) 396: 345-350. [8] The 3D Boussinesq equations with regularity in one directional derivative of the pressure. Bull. Malays. Math. Sci. Soc. (2019) 42: 3005-3019. [9] A generalized regularity criterion for the 3D Navier-Stokes equations in terms of one velocity component. J. Differential Equations (2016) 260: 3477-3494. [10] The anisotropic integrability regularity criterion to 3D magnetohydrodynamic equations. Math. Methods Appl. Sci. (2017) 40: 5461-5469. [11] Regularity criteria for the 3D MHD equations in terms of the pressure. Internat. J. Non-Linear Mech. (2006) 41: 1174-1180. [12] Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. (2012) 24: 691-708. [13] On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity (2010) 23: 1097-1107.
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