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

Ahmad Mohammad Alghamdi; Sadek Gala; Chenyin Qian; Maria Alessandra Ragusa; Ahmad Mohammad Alghamdi; Sadek Gala; Chenyin Qian; Maria Alessandra Ragusa

doi:10.3934/era.2020012

Electronic Research Archive

2020, Volume 28, Issue 1: 183-193. doi: 10.3934/era.2020012

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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

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
$ \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
- MHD equations,
- regularity criterion,
- anisotropic Lebesgue spaces,
- a priori estimates
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

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Abstract

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

References

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